Subgroups — C₁₀₉×#34593 #42152587

Order 27904.

2934 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2636	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2633	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#2631	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2626	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#2623	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#2617	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#2613	C₂	1	2	yes	maximal, complemented
10	C₁₀₉×#2398	C₂	1	2	yes	maximal, complemented
11	C₁₀₉×#2397	C₂	1	2	yes	maximal, complemented
12	C₁₀₉×#2392	C₂	1	4	yes	maximal, complemented
13	C₄×#120×C₁₀₉	C₂	1	4	yes	maximal, complemented
14	C₁₀₉×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₁₀₉	C₂²	1	2	yes	—
17	#529×C₁₀₉	—	2	8	no	—
18	C₂×#126×C₁₀₉	C₂²	1	2	yes	complemented
19	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×#124×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×#121×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×#120×C₁₀₉	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×D₄×C₁₀₉	—	2	8	no	—
27	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄×D₄×C₁₀₉	C₂²	1	8	yes	complemented
34	C₂×#117×C₁₀₉	C₂²	1	2	yes	complemented
35	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
37	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
38	#400×C₁₀₉	C₂²	1	4	yes	complemented
39	#399×C₁₀₉	C₂²	1	4	yes	complemented
40	#399×C₁₀₉	C₂²	1	4	yes	complemented
41	#398×C₁₀₉	C₂²	1	4	yes	complemented
42	#397×C₁₀₉	C₂²	1	4	yes	complemented
43	#397×C₁₀₉	C₂²	1	4	yes	complemented
44	#396×C₁₀₉	C₂²	1	8	yes	complemented
45	#394×C₁₀₉	C₂²	1	4	yes	—
46	#394×C₁₀₉	C₂²	1	4	yes	—
47	#394×C₁₀₉	C₂²	1	8	yes	complemented
48	#393×C₁₀₉	C₂²	1	4	yes	complemented
49	#392×C₁₀₉	C₂²	1	4	yes	—
50	#392×C₁₀₉	C₂²	1	4	yes	—
51	#392×C₁₀₉	C₂²	1	4	yes	complemented
52	#386×C₁₀₉	C₂²	1	4	yes	—
53	#386×C₁₀₉	C₂²	1	8	yes	complemented
54	#385×C₁₀₉	C₂²	1	4	yes	—
55	#385×C₁₀₉	C₂²	1	8	yes	complemented
56	#383×C₁₀₉	C₂²	1	2	yes	complemented
57	#382×C₁₀₉	C₂²	1	2	yes	complemented
58	#381×C₁₀₉	C₂²	1	2	yes	—
59	#380×C₁₀₉	C₂²	1	2	yes	—
60	#379×C₁₀₉	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
62	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
64	#376×C₁₀₉	C₂²	1	2	yes	complemented
65	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
66	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₁₀₉	—	2	8	no	—
68	C₂²×D₄×C₁₀₉	—	2	4	no	—
69	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
70	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
71	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
72	C₂²×D₄×C₁₀₉	C₂³	1	8	yes	—
73	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₁₀₉	—	2	8	no	—
75	C₂³×C₄×C₁₀₉	—	4	8	no	—
76	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
77	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
78	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
82	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	#126×C₁₀₉	—	2	8	no	—
84	#125×C₁₀₉	—	2	4	no	—
85	#124×C₁₀₉	—	2	4	no	—
86	#123×C₁₀₉	—	2	8	no	—
87	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
88	#121×C₁₀₉	—	2	4	no	—
89	#120×C₁₀₉	—	2	16	no	—
90	C₄×Q₈×C₁₀₉	—	2	4	no	—
91	C₄×D₄×C₁₀₉	—	2	8	no	—
92	C₄×D₄×C₁₀₉	—	2	8	no	—
93	C₄×D₄×C₁₀₉	—	2	8	no	—
94	C₄×D₄×C₁₀₉	—	2	8	no	—
95	C₄×D₄×C₁₀₉	—	2	8	no	—
96	C₄×D₄×C₁₀₉	—	2	4	no	—
97	C₄×D₄×C₁₀₉	—	4	16	no	—
98	#117×C₁₀₉	—	2	8	no	—
99	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
100	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
111	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
120	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
123	C₂×C₄²×C₁₀₉	D₄	1	8	yes	complemented
124	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
125	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
126	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
130	#95×C₁₀₉	C₂³	1	4	yes	—
131	#95×C₁₀₉	C₂³	1	4	yes	—
132	#95×C₁₀₉	C₂³	1	4	yes	—
133	#95×C₁₀₉	C₂³	1	8	yes	—
134	#95×C₁₀₉	C₂³	1	8	yes	—
135	#95×C₁₀₉	C₂³	1	4	yes	—
136	C₂⁴×C₁₀₉	—	2	4	no	—
137	C₂⁴×C₁₀₉	—	2	4	no	—
138	C₂⁴×C₁₀₉	—	4	4	no	—
139	C₂⁴×C₁₀₉	—	4	4	no	—
140	C₂⁴×C₁₀₉	#41	1	8	yes	—
141	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
142	C₂⁴×C₁₀₉	C₂⁴	1	4	yes	—
143	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
144	C₂×Q₈×C₁₀₉	—	2	2	no	—
145	C₂×Q₈×C₁₀₉	—	4	2	no	—
146	C₂×D₄×C₁₀₉	—	2	32	no	—
147	C₂×D₄×C₁₀₉	—	2	8	no	—
148	C₂×D₄×C₁₀₉	—	2	8	no	—
149	C₂×D₄×C₁₀₉	—	2	4	no	—
150	C₂×D₄×C₁₀₉	—	2	16	no	—
151	C₂×D₄×C₁₀₉	—	2	8	no	—
152	C₂×D₄×C₁₀₉	—	2	4	no	—
153	C₂×D₄×C₁₀₉	—	2	4	no	—
154	C₂×D₄×C₁₀₉	—	2	2	no	—
155	C₂×D₄×C₁₀₉	—	4	8	no	—
156	C₂×D₄×C₁₀₉	—	4	2	no	—
157	C₂×D₄×C₁₀₉	—	4	8	no	—
158	C₂²×C₄×C₁₀₉	—	2	4	no	—
159	C₂²×C₄×C₁₀₉	—	2	4	no	—
160	C₂²×C₄×C₁₀₉	—	2	4	no	—
161	C₂²×C₄×C₁₀₉	—	2	4	no	—
162	C₂²×C₄×C₁₀₉	—	2	8	no	—
163	C₂²×C₄×C₁₀₉	—	2	8	no	—
164	C₂²×C₄×C₁₀₉	—	2	8	no	—
165	C₂²×C₄×C₁₀₉	—	2	16	no	—
166	C₂²×C₄×C₁₀₉	—	4	8	no	—
167	C₂²×C₄×C₁₀₉	—	4	4	no	—
168	C₂²×C₄×C₁₀₉	—	4	2	no	—
169	C₂²×C₄×C₁₀₉	—	4	32	no	—
170	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
172	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₁₀₉	—	2	4	no	—
185	C₄⋊C₄×C₁₀₉	—	2	4	no	—
186	C₄⋊C₄×C₁₀₉	—	2	4	no	—
187	C₄⋊C₄×C₁₀₉	—	2	4	no	—
188	C₄⋊C₄×C₁₀₉	—	2	2	no	—
189	C₄⋊C₄×C₁₀₉	—	2	4	no	—
190	C₄⋊C₄×C₁₀₉	—	2	2	no	—
191	C₄⋊C₄×C₁₀₉	—	2	4	no	—
192	C₄⋊C₄×C₁₀₉	—	4	4	no	—
193	C₄⋊C₄×C₁₀₉	—	4	8	no	—
194	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
195	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
196	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
197	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
198	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
199	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
200	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
201	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
202	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
203	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
204	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
205	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
206	C₄²×C₁₀₉	—	2	4	no	—
207	C₄²×C₁₀₉	—	2	8	no	—
208	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
209	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
210	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
211	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
212	C₂³×C₁₀₉	—	2	2	no	—
213	C₂³×C₁₀₉	—	2	2	no	—
214	C₂³×C₁₀₉	—	2	16	no	—
215	C₂³×C₁₀₉	—	2	4	no	—
216	C₂³×C₁₀₉	—	2	4	no	—
217	C₂³×C₁₀₉	—	2	8	no	—
218	C₂³×C₁₀₉	—	2	8	no	—
219	C₂³×C₁₀₉	—	2	2	no	—
220	C₂³×C₁₀₉	—	2	4	no	—
221	C₂³×C₁₀₉	—	2	4	no	—
222	C₂³×C₁₀₉	—	2	4	no	—
223	C₂³×C₁₀₉	—	4	16	no	—
224	C₂³×C₁₀₉	—	4	8	no	—
225	C₂³×C₁₀₉	—	4	8	no	—
226	C₂³×C₁₀₉	—	4	16	no	—
227	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
228	C₂³×C₁₀₉	C₂×#41	1	8	yes	—
229	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
230	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
231	Q₈×C₁₀₉	—	4	4	no	—
232	D₄×C₁₀₉	—	2	16	no	—
233	D₄×C₁₀₉	—	2	16	no	—
234	D₄×C₁₀₉	—	4	4	no	—
235	D₄×C₁₀₉	—	4	32	no	—
236	D₄×C₁₀₉	—	4	8	no	—
237	C₂×C₄×C₁₀₉	—	2	8	no	—
238	C₂×C₄×C₁₀₉	—	2	2	no	—
239	C₂×C₄×C₁₀₉	—	2	2	no	—
240	C₂×C₄×C₁₀₉	—	2	2	no	—
241	C₂×C₄×C₁₀₉	—	2	2	no	—
242	C₂×C₄×C₁₀₉	—	2	2	no	—
243	C₂×C₄×C₁₀₉	—	2	8	no	—
244	C₂×C₄×C₁₀₉	—	2	2	no	—
245	C₂×C₄×C₁₀₉	—	2	2	no	—
246	C₂×C₄×C₁₀₉	—	2	2	no	—
247	C₂×C₄×C₁₀₉	—	2	4	no	—
248	C₂×C₄×C₁₀₉	—	2	4	no	—
249	C₂×C₄×C₁₀₉	—	2	8	no	—
250	C₂×C₄×C₁₀₉	—	2	8	no	—
251	C₂×C₄×C₁₀₉	—	2	4	no	—
252	C₂×C₄×C₁₀₉	—	2	8	no	—
253	C₂×C₄×C₁₀₉	—	2	4	no	—
254	C₂×C₄×C₁₀₉	—	2	8	no	—
255	C₂×C₄×C₁₀₉	—	4	4	no	—
256	C₂×C₄×C₁₀₉	—	4	16	no	—
257	C₂×C₄×C₁₀₉	—	4	8	no	—
258	C₂×C₄×C₁₀₉	—	4	8	no	—
259	C₂×C₄×C₁₀₉	—	4	16	no	—
260	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
267	C₂²×C₁₀₉	—	2	2	no	—
268	C₂²×C₁₀₉	—	2	2	no	—
269	C₂²×C₁₀₉	—	2	8	no	—
270	C₂²×C₁₀₉	—	2	8	no	—
271	C₂²×C₁₀₉	—	2	16	no	—
272	C₂²×C₁₀₉	—	4	32	no	—
273	C₂²×C₁₀₉	—	4	4	no	—
274	C₂²×C₁₀₉	—	4	4	no	—
275	C₂²×C₁₀₉	—	4	8	no	—
276	C₂²×C₁₀₉	—	4	2	no	—
277	C₂²×C₁₀₉	—	4	2	no	—
278	C₂²×C₁₀₉	—	4	2	no	—
279	C₂²×C₁₀₉	—	4	8	no	—
280	C₂²×C₁₀₉	#525	1	4	yes	—
281	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
282	C₂²×C₁₀₉	C₂²×#41	1	2	yes	—
283	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
284	C₂²×C₁₀₉	#585	1	2	yes	—
285	C₄×C₁₀₉	—	2	2	no	—
286	C₄×C₁₀₉	—	2	2	no	—
287	C₄×C₁₀₉	—	2	2	no	—
288	C₄×C₁₀₉	—	2	2	no	—
289	C₄×C₁₀₉	—	2	2	no	—
290	C₄×C₁₀₉	—	4	2	no	—
291	C₄×C₁₀₉	—	4	2	no	—
292	C₄×C₁₀₉	—	4	4	no	—
293	C₄×C₁₀₉	—	4	2	no	—
294	C₄×C₁₀₉	—	4	8	no	—
295	C₄×C₁₀₉	#535	1	4	yes	—
296	#34593	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂×C₁₀₉	—	2	4	no	—
298	C₂×C₁₀₉	—	4	8	no	—
299	C₂×C₁₀₉	—	8	2	no	—
300	C₂×C₁₀₉	C₂×#525	1	1	yes	characteristic
301	C₂×C₁₀₉	C₂×#535	1	1	yes	characteristic
302	C₂×C₁₀₉	#3449	1	1	yes	characteristic
303	C₂×C₁₀₉	#3459	1	4	yes	—
304	C₂×#529	C₂×C₁₀₉	1	1	yes	characteristic, complemented
305	C₂²×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
306	#2636	C₂×C₁₀₉	1	4	yes	complemented
307	#2633	C₂×C₁₀₉	1	2	yes	complemented
308	#2631	C₂×C₁₀₉	1	2	yes	complemented
309	#2626	C₂×C₁₀₉	1	2	yes	complemented
310	#2623	C₂×C₁₀₉	1	2	yes	complemented
311	#2617	C₂×C₁₀₉	1	2	yes	complemented
312	#2613	C₂×C₁₀₉	1	2	yes	complemented
313	#2398	C₂×C₁₀₉	1	2	yes	complemented
314	#2397	C₂×C₁₀₉	1	2	yes	complemented
315	#2392	C₂×C₁₀₉	1	4	yes	complemented
316	C₄×#120	C₂×C₁₀₉	1	4	yes	complemented
317	#2283	C₂×C₁₀₉	1	1	yes	characteristic, complemented
318	C₁₀₉	#34593	1	1	yes	characteristic, direct product factor, Sylow
319	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
320	C₂⁴×C₄	C₂²×C₁₀₉	1	2	yes	—
321	#529	—	2	8	no	—
322	C₂×#126	C₂²×C₁₀₉	1	2	yes	complemented
323	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
324	C₂×#124	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
325	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
326	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
327	C₂×#121	C₂²×C₁₀₉	1	1	yes	characteristic
328	C₂×#120	C₂²×C₁₀₉	1	4	yes	—
329	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
330	C₂×C₄×D₄	—	2	8	no	—
331	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
332	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
333	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
334	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
335	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
336	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
337	C₂×C₄×D₄	C₂²×C₁₀₉	1	8	yes	complemented
338	C₂×#117	C₂²×C₁₀₉	1	2	yes	complemented
339	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
340	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
341	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
342	#400	C₂²×C₁₀₉	1	4	yes	complemented
343	#399	C₂²×C₁₀₉	1	4	yes	complemented
344	#399	C₂²×C₁₀₉	1	4	yes	complemented
345	#398	C₂²×C₁₀₉	1	4	yes	complemented
346	#397	C₂²×C₁₀₉	1	4	yes	complemented
347	#397	C₂²×C₁₀₉	1	4	yes	complemented
348	#396	C₂²×C₁₀₉	1	8	yes	complemented
349	#394	C₂²×C₁₀₉	1	4	yes	—
350	#394	C₂²×C₁₀₉	1	4	yes	—
351	#394	C₂²×C₁₀₉	1	8	yes	complemented
352	#393	C₂²×C₁₀₉	1	4	yes	complemented
353	#392	C₂²×C₁₀₉	1	4	yes	—
354	#392	C₂²×C₁₀₉	1	4	yes	—
355	#392	C₂²×C₁₀₉	1	4	yes	complemented
356	#386	C₂²×C₁₀₉	1	4	yes	—
357	#386	C₂²×C₁₀₉	1	8	yes	complemented
358	#385	C₂²×C₁₀₉	1	4	yes	—
359	#385	C₂²×C₁₀₉	1	8	yes	complemented
360	#383	C₂²×C₁₀₉	1	2	yes	complemented
361	#382	C₂²×C₁₀₉	1	2	yes	complemented
362	#381	C₂²×C₁₀₉	1	2	yes	—
363	#380	C₂²×C₁₀₉	1	2	yes	—
364	#379	C₂²×C₁₀₉	1	4	yes	—
365	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
366	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
368	#376	C₂²×C₁₀₉	1	2	yes	complemented
369	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
370	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
371	C₂²×D₄	—	2	8	no	—
372	C₂²×D₄	—	2	4	no	—
373	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
374	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
375	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
376	C₂²×D₄	C₂³×C₁₀₉	1	8	yes	—
377	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
378	C₂³×C₄	—	2	8	no	—
379	C₂³×C₄	—	4	8	no	—
380	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
381	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
382	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
385	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
386	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
387	#126	—	2	8	no	—
388	#125	—	2	4	no	—
389	#124	—	2	4	no	—
390	#123	—	2	8	no	—
391	C₂²⋊Q₈	—	2	4	no	—
392	#121	—	2	4	no	—
393	#120	—	2	16	no	—
394	C₄×Q₈	—	2	4	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	2	8	no	—
399	C₄×D₄	—	2	8	no	—
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	—	4	16	no	—
402	#117	—	2	8	no	—
403	C₂×C₄⋊C₄	—	2	4	no	—
404	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
413	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
414	C₂×C₂²⋊C₄	—	2	8	no	—
415	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
427	C₂×C₄²	D₄×C₁₀₉	1	8	yes	complemented
428	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
429	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
430	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
434	#95	C₂³×C₁₀₉	1	4	yes	—
435	#95	C₂³×C₁₀₉	1	4	yes	—
436	#95	C₂³×C₁₀₉	1	4	yes	—
437	#95	C₂³×C₁₀₉	1	8	yes	—
438	#95	C₂³×C₁₀₉	1	8	yes	—
439	#95	C₂³×C₁₀₉	1	4	yes	—
440	C₂⁴	—	2	4	no	—
441	C₂⁴	—	2	4	no	—
442	C₂⁴	—	4	4	no	—
443	C₂⁴	—	4	4	no	—
444	C₂⁴	#41×C₁₀₉	1	8	yes	—
445	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
446	C₂⁴	C₂⁴×C₁₀₉	1	4	yes	—
447	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	4	2	no	—
450	C₂×D₄	—	2	32	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	16	no	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	8	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	16	no	—
470	C₂²×C₄	—	4	8	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	32	no	—
474	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
475	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
476	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
478	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	4	4	no	—
497	C₄⋊C₄	—	4	8	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	16	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	8	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	4	8	no	—
508	C₂²⋊C₄	—	4	8	no	—
509	C₂²⋊C₄	—	4	8	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	8	no	—
512	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
513	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
514	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
515	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	16	no	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	16	no	—
528	C₂³	—	4	8	no	—
529	C₂³	—	4	16	no	—
530	C₂³	—	4	8	no	—
531	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
532	C₂³	C₂×#41×C₁₀₉	1	8	yes	—
533	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
534	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
535	Q₈	—	4	4	no	—
536	D₄	—	2	16	no	—
537	D₄	—	2	16	no	—
538	D₄	—	4	4	no	—
539	D₄	—	4	32	no	—
540	D₄	—	4	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	16	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	16	no	—
563	C₂×C₄	—	4	8	no	—
564	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
570	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
571	C₂²	—	2	8	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	16	no	—
575	C₂²	—	2	8	no	—
576	C₂²	—	4	32	no	—
577	C₂²	—	4	8	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	8	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	#525×C₁₀₉	1	4	yes	—
585	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₁₀₉	1	2	yes	—
587	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	2	yes	—
588	C₂²	#585×C₁₀₉	1	2	yes	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	8	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	#535×C₁₀₉	1	4	yes	—
600	C₂	—	2	4	no	—
601	C₂	—	4	8	no	—
602	C₂	—	8	2	no	—
603	C₂	C₂×#525×C₁₀₉	1	1	yes	characteristic
604	C₂	C₂×#535×C₁₀₉	1	1	yes	characteristic
605	C₂	C₁₀₉×#3449	1	1	yes	characteristic
606	C₂	C₁₀₉×#3459	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#529×C₁₀₉	C₂	semidirect
2. C₂²×C₄×D₄×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2636	C₂	semidirect
4. C₁₀₉×#2633	C₂	semidirect
5. C₁₀₉×#2631	C₂	semidirect
6. C₁₀₉×#2626	C₂	semidirect
7. C₁₀₉×#2623	C₂	semidirect
8. C₁₀₉×#2617	C₂	semidirect
9. C₁₀₉×#2613	C₂	semidirect
10. C₁₀₉×#2398	C₂	semidirect
11. C₁₀₉×#2397	C₂	semidirect
12. C₁₀₉×#2392	C₂	semidirect
13. C₄×#120×C₁₀₉	C₂	semidirect
14. C₁₀₉×#2283	C₂	semidirect
18. C₂×#126×C₁₀₉	C₂²	semidirect
19. C₂×#125×C₁₀₉	C₂²	semidirect
20. C₂×#124×C₁₀₉	C₂²	semidirect
25. C₂×C₄×Q₈×C₁₀₉	C₂²	semidirect
32. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
33. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
34. C₂×#117×C₁₀₉	C₂²	semidirect
38. #400×C₁₀₉	C₂²	semidirect
39. #399×C₁₀₉	C₂²	semidirect
40. #399×C₁₀₉	C₂²	semidirect
41. #398×C₁₀₉	C₂²	semidirect
42. #397×C₁₀₉	C₂²	semidirect
43. #397×C₁₀₉	C₂²	semidirect
44. #396×C₁₀₉	C₂²	semidirect
47. #394×C₁₀₉	C₂²	semidirect
48. #393×C₁₀₉	C₂²	semidirect
51. #392×C₁₀₉	C₂²	semidirect
53. #386×C₁₀₉	C₂²	semidirect
55. #385×C₁₀₉	C₂²	semidirect
56. #383×C₁₀₉	C₂²	semidirect
57. #382×C₁₀₉	C₂²	semidirect
61. C₄×C₄⋊C₄×C₁₀₉	C₂²	semidirect
63. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
64. #376×C₁₀₉	C₂²	semidirect
123. C₂×C₄²×C₁₀₉	D₄	semidirect
296. #34593	C₁₀₉	direct (N × H)
304. C₂×#529	C₂×C₁₀₉	semidirect
305. C₂²×C₄×D₄	C₂×C₁₀₉	semidirect
306. #2636	C₂×C₁₀₉	semidirect
307. #2633	C₂×C₁₀₉	semidirect
308. #2631	C₂×C₁₀₉	semidirect
309. #2626	C₂×C₁₀₉	semidirect
310. #2623	C₂×C₁₀₉	semidirect
311. #2617	C₂×C₁₀₉	semidirect
312. #2613	C₂×C₁₀₉	semidirect
313. #2398	C₂×C₁₀₉	semidirect
314. #2397	C₂×C₁₀₉	semidirect
315. #2392	C₂×C₁₀₉	semidirect
316. C₄×#120	C₂×C₁₀₉	semidirect
317. #2283	C₂×C₁₀₉	semidirect
318. C₁₀₉	#34593	direct (N × H)
322. C₂×#126	C₂²×C₁₀₉	semidirect
323. C₂×#125	C₂²×C₁₀₉	semidirect
324. C₂×#124	C₂²×C₁₀₉	semidirect
329. C₂×C₄×Q₈	C₂²×C₁₀₉	semidirect
336. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
337. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
338. C₂×#117	C₂²×C₁₀₉	semidirect
342. #400	C₂²×C₁₀₉	semidirect
343. #399	C₂²×C₁₀₉	semidirect
344. #399	C₂²×C₁₀₉	semidirect
345. #398	C₂²×C₁₀₉	semidirect
346. #397	C₂²×C₁₀₉	semidirect
347. #397	C₂²×C₁₀₉	semidirect
348. #396	C₂²×C₁₀₉	semidirect
351. #394	C₂²×C₁₀₉	semidirect
352. #393	C₂²×C₁₀₉	semidirect
355. #392	C₂²×C₁₀₉	semidirect
357. #386	C₂²×C₁₀₉	semidirect
359. #385	C₂²×C₁₀₉	semidirect
360. #383	C₂²×C₁₀₉	semidirect
361. #382	C₂²×C₁₀₉	semidirect
365. C₄×C₄⋊C₄	C₂²×C₁₀₉	semidirect
367. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
368. #376	C₂²×C₁₀₉	semidirect
427. C₂×C₄²	D₄×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#529×C₁₀₉	2
C₂²×C₄×D₄×C₁₀₉	2
C₁₀₉×#2636	2
C₁₀₉×#2633	2
C₁₀₉×#2631	2
C₁₀₉×#2626	2
C₁₀₉×#2623	2
C₁₀₉×#2617	2
C₁₀₉×#2613	2
C₁₀₉×#2398	2
C₁₀₉×#2397	2
C₁₀₉×#2392	2
C₄×#120×C₁₀₉	2
C₁₀₉×#2283	2
#34593	109

Subgroups — C109×#34593 #42152587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#34593 #42152587