Subgroups — C₁₀₉×#34992 #42152986

Order 27904.

4346 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2849	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2622	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#2483	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#2412	C₂	1	8	yes	maximal, complemented
7	C₁₀₉×#2405	C₂	1	4	yes	maximal, complemented
8	C₁₀₉×#2315	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#2308	C₂	1	2	yes	maximal, complemented
10	C₂×#379×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁶×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×#123×C₁₀₉	C₂²	1	8	yes	complemented
16	C₂×#120×C₁₀₉	—	2	8	no	—
17	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#120×C₁₀₉	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
21	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
23	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
27	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
28	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
29	#397×C₁₀₉	C₂²	1	16	yes	complemented
30	#394×C₁₀₉	C₂²	1	16	yes	complemented
31	#386×C₁₀₉	C₂²	1	8	yes	complemented
32	#386×C₁₀₉	C₂²	1	8	yes	complemented
33	#386×C₁₀₉	C₂²	1	8	yes	complemented
34	#385×C₁₀₉	C₂²	1	8	yes	complemented
35	#385×C₁₀₉	C₂²	1	8	yes	complemented
36	#385×C₁₀₉	C₂²	1	8	yes	complemented
37	#381×C₁₀₉	C₂²	1	4	yes	complemented
38	#381×C₁₀₉	C₂²	1	8	yes	complemented
39	#380×C₁₀₉	C₂²	1	8	yes	complemented
40	#379×C₁₀₉	C₂²	1	4	yes	—
41	#379×C₁₀₉	C₂²	1	2	yes	complemented
42	#379×C₁₀₉	C₂²	1	2	yes	complemented
43	#379×C₁₀₉	C₂²	1	4	yes	complemented
44	C₂⁵×C₁₀₉	—	2	8	no	—
45	C₂⁵×C₁₀₉	—	4	8	no	—
46	C₂⁵×C₁₀₉	D₄	1	8	yes	—
47	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
49	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
50	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₁₀₉	—	2	8	no	—
54	C₂²×D₄×C₁₀₉	—	2	4	no	—
55	C₂²×D₄×C₁₀₉	—	2	8	no	—
56	C₂²×D₄×C₁₀₉	—	4	8	no	—
57	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
58	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
59	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
60	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
61	C₂³×C₄×C₁₀₉	—	2	8	no	—
62	C₂³×C₄×C₁₀₉	—	4	8	no	—
63	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
65	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
68	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
69	#123×C₁₀₉	—	4	16	no	—
70	#120×C₁₀₉	—	2	8	no	—
71	#120×C₁₀₉	—	2	16	no	—
72	#120×C₁₀₉	—	2	8	no	—
73	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
74	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
75	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
77	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
78	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₁₀₉	—	4	8	no	—
82	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
94	#95×C₁₀₉	C₂³	1	16	yes	complemented
95	#95×C₁₀₉	C₂³	1	16	yes	complemented
96	C₂⁴×C₁₀₉	—	2	4	no	—
97	C₂⁴×C₁₀₉	—	2	8	no	—
98	C₂⁴×C₁₀₉	—	2	32	no	—
99	C₂⁴×C₁₀₉	—	2	8	no	—
100	C₂⁴×C₁₀₉	—	2	8	no	—
101	C₂⁴×C₁₀₉	—	2	4	no	—
102	C₂⁴×C₁₀₉	—	2	8	no	—
103	C₂⁴×C₁₀₉	—	2	8	no	—
104	C₂⁴×C₁₀₉	—	2	2	no	—
105	C₂⁴×C₁₀₉	—	2	4	no	—
106	C₂⁴×C₁₀₉	—	4	32	no	—
107	C₂⁴×C₁₀₉	—	4	4	no	—
108	C₂⁴×C₁₀₉	—	4	4	no	—
109	C₂⁴×C₁₀₉	—	4	4	no	—
110	C₂⁴×C₁₀₉	—	4	4	no	—
111	C₂⁴×C₁₀₉	—	4	4	no	—
112	C₂⁴×C₁₀₉	—	4	64	no	—
113	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
114	C₂⁴×C₁₀₉	C₂×D₄	1	8	yes	—
115	C₂⁴×C₁₀₉	C₂×D₄	1	8	yes	—
116	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
117	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
118	C₂⁴×C₁₀₉	#41	1	8	yes	—
119	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
120	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
121	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
123	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
124	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
125	C₂×D₄×C₁₀₉	—	2	8	no	—
126	C₂×D₄×C₁₀₉	—	2	4	no	—
127	C₂×D₄×C₁₀₉	—	2	4	no	—
128	C₂×D₄×C₁₀₉	—	2	8	no	—
129	C₂×D₄×C₁₀₉	—	2	8	no	—
130	C₂×D₄×C₁₀₉	—	2	8	no	—
131	C₂×D₄×C₁₀₉	—	2	16	no	—
132	C₂×D₄×C₁₀₉	—	4	8	no	—
133	C₂×D₄×C₁₀₉	—	4	32	no	—
134	C₂×D₄×C₁₀₉	—	4	16	no	—
135	C₂²×C₄×C₁₀₉	—	2	8	no	—
136	C₂²×C₄×C₁₀₉	—	2	8	no	—
137	C₂²×C₄×C₁₀₉	—	2	8	no	—
138	C₂²×C₄×C₁₀₉	—	2	4	no	—
139	C₂²×C₄×C₁₀₉	—	2	2	no	—
140	C₂²×C₄×C₁₀₉	—	2	4	no	—
141	C₂²×C₄×C₁₀₉	—	2	16	no	—
142	C₂²×C₄×C₁₀₉	—	2	8	no	—
143	C₂²×C₄×C₁₀₉	—	4	8	no	—
144	C₂²×C₄×C₁₀₉	—	4	8	no	—
145	C₂²×C₄×C₁₀₉	—	4	4	no	—
146	C₂²×C₄×C₁₀₉	—	4	8	no	—
147	C₂²×C₄×C₁₀₉	—	4	32	no	—
148	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
150	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
151	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
152	C₄⋊C₄×C₁₀₉	—	4	16	no	—
153	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
154	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
155	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
156	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
157	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
158	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
159	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
160	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
161	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
162	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
163	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
164	C₂²⋊C₄×C₁₀₉	—	4	16	no	—
165	C₂³×C₁₀₉	—	2	32	no	—
166	C₂³×C₁₀₉	—	2	8	no	—
167	C₂³×C₁₀₉	—	2	16	no	—
168	C₂³×C₁₀₉	—	2	4	no	—
169	C₂³×C₁₀₉	—	2	4	no	—
170	C₂³×C₁₀₉	—	2	4	no	—
171	C₂³×C₁₀₉	—	2	4	no	—
172	C₂³×C₁₀₉	—	2	4	no	—
173	C₂³×C₁₀₉	—	2	16	no	—
174	C₂³×C₁₀₉	—	2	16	no	—
175	C₂³×C₁₀₉	—	2	8	no	—
176	C₂³×C₁₀₉	—	2	16	no	—
177	C₂³×C₁₀₉	—	2	16	no	—
178	C₂³×C₁₀₉	—	2	4	no	—
179	C₂³×C₁₀₉	—	2	4	no	—
180	C₂³×C₁₀₉	—	2	4	no	—
181	C₂³×C₁₀₉	—	2	2	no	—
182	C₂³×C₁₀₉	—	2	4	no	—
183	C₂³×C₁₀₉	—	2	8	no	—
184	C₂³×C₁₀₉	—	4	8	no	—
185	C₂³×C₁₀₉	—	4	8	no	—
186	C₂³×C₁₀₉	—	4	8	no	—
187	C₂³×C₁₀₉	—	4	8	no	—
188	C₂³×C₁₀₉	—	4	16	no	—
189	C₂³×C₁₀₉	—	4	32	no	—
190	C₂³×C₁₀₉	—	4	32	no	—
191	C₂³×C₁₀₉	—	4	128	no	—
192	C₂³×C₁₀₉	—	4	16	no	—
193	C₂³×C₁₀₉	—	4	8	no	—
194	C₂³×C₁₀₉	—	4	4	no	—
195	C₂³×C₁₀₉	—	8	8	no	—
196	C₂³×C₁₀₉	#123	1	16	yes	complemented
197	C₂³×C₁₀₉	C₂²×D₄	1	4	yes	—
198	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
199	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
200	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
201	C₂³×C₁₀₉	C₂²×D₄	1	4	yes	—
202	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
203	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
204	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
205	C₂³×C₁₀₉	H(5,𝔽₂)	1	8	yes	—
206	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
207	D₄×C₁₀₉	—	4	16	no	—
208	D₄×C₁₀₉	—	4	8	no	—
209	D₄×C₁₀₉	—	4	8	no	—
210	D₄×C₁₀₉	—	4	16	no	—
211	C₂×C₄×C₁₀₉	—	2	8	no	—
212	C₂×C₄×C₁₀₉	—	2	2	no	—
213	C₂×C₄×C₁₀₉	—	2	4	no	—
214	C₂×C₄×C₁₀₉	—	2	8	no	—
215	C₂×C₄×C₁₀₉	—	2	4	no	—
216	C₂×C₄×C₁₀₉	—	2	16	no	—
217	C₂×C₄×C₁₀₉	—	2	16	no	—
218	C₂×C₄×C₁₀₉	—	2	8	no	—
219	C₂×C₄×C₁₀₉	—	4	16	no	—
220	C₂×C₄×C₁₀₉	—	4	16	no	—
221	C₂×C₄×C₁₀₉	—	4	8	no	—
222	C₂×C₄×C₁₀₉	—	4	4	no	—
223	C₂×C₄×C₁₀₉	—	4	16	no	—
224	C₂×C₄×C₁₀₉	—	8	8	no	—
225	C₂²×C₁₀₉	—	2	32	no	—
226	C₂²×C₁₀₉	—	2	4	no	—
227	C₂²×C₁₀₉	—	2	4	no	—
228	C₂²×C₁₀₉	—	2	4	no	—
229	C₂²×C₁₀₉	—	2	4	no	—
230	C₂²×C₁₀₉	—	2	4	no	—
231	C₂²×C₁₀₉	—	2	8	no	—
232	C₂²×C₁₀₉	—	2	8	no	—
233	C₂²×C₁₀₉	—	2	16	no	—
234	C₂²×C₁₀₉	—	2	8	no	—
235	C₂²×C₁₀₉	—	2	4	no	—
236	C₂²×C₁₀₉	—	2	4	no	—
237	C₂²×C₁₀₉	—	2	4	no	—
238	C₂²×C₁₀₉	—	4	16	no	—
239	C₂²×C₁₀₉	—	4	32	no	—
240	C₂²×C₁₀₉	—	4	32	no	—
241	C₂²×C₁₀₉	—	4	16	no	—
242	C₂²×C₁₀₉	—	4	8	no	—
243	C₂²×C₁₀₉	—	4	2	no	—
244	C₂²×C₁₀₉	—	4	4	no	—
245	C₂²×C₁₀₉	—	4	8	no	—
246	C₂²×C₁₀₉	—	8	8	no	—
247	C₂²×C₁₀₉	C₂×#123	1	4	yes	—
248	C₂²×C₁₀₉	C₂×#123	1	4	yes	—
249	C₂²×C₁₀₉	C₂×#123	1	4	yes	—
250	C₂²×C₁₀₉	#534	1	8	yes	—
251	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
252	C₂²×C₁₀₉	C₂²×#41	1	2	yes	—
253	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	4	yes	—
254	C₄×C₁₀₉	—	4	2	no	—
255	C₄×C₁₀₉	—	4	8	no	—
256	C₄×C₁₀₉	—	4	4	no	—
257	C₄×C₁₀₉	—	4	8	no	—
258	#34992	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₁₀₉	—	2	4	no	—
260	C₂×C₁₀₉	—	2	8	no	—
261	C₂×C₁₀₉	—	2	4	no	—
262	C₂×C₁₀₉	—	2	4	no	—
263	C₂×C₁₀₉	—	4	4	no	—
264	C₂×C₁₀₉	—	8	2	no	—
265	C₂×C₁₀₉	C₂²×#123	1	1	yes	characteristic
266	C₂×C₁₀₉	C₂×#534	1	2	yes	—
267	C₂×C₁₀₉	#3490	1	4	yes	—
268	C₂²×#120	C₂×C₁₀₉	1	1	yes	characteristic
269	C₂³×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
270	#2849	C₂×C₁₀₉	1	4	yes	complemented
271	#2622	C₂×C₁₀₉	1	4	yes	complemented
272	#2483	C₂×C₁₀₉	1	4	yes	complemented
273	#2412	C₂×C₁₀₉	1	8	yes	complemented
274	#2405	C₂×C₁₀₉	1	4	yes	complemented
275	#2315	C₂×C₁₀₉	1	2	yes	complemented
276	#2308	C₂×C₁₀₉	1	2	yes	complemented
277	C₂×#379	C₂×C₁₀₉	1	1	yes	characteristic, complemented
278	C₁₀₉	#34992	1	1	yes	characteristic, direct product factor, Sylow
279	C₂⁶	C₂²×C₁₀₉	1	1	yes	characteristic
280	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
281	C₂³×D₄	C₂²×C₁₀₉	1	2	yes	—
282	C₂⁴×C₄	C₂²×C₁₀₉	1	2	yes	complemented
283	C₂×#123	C₂²×C₁₀₉	1	8	yes	complemented
284	C₂×#120	—	2	8	no	—
285	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
286	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
287	C₂×#120	C₂²×C₁₀₉	1	4	yes	—
288	C₂²×C₂²⋊C₄	—	2	8	no	—
289	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
290	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
291	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
292	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
293	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
294	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
295	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
296	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
297	#397	C₂²×C₁₀₉	1	16	yes	complemented
298	#394	C₂²×C₁₀₉	1	16	yes	complemented
299	#386	C₂²×C₁₀₉	1	8	yes	complemented
300	#386	C₂²×C₁₀₉	1	8	yes	complemented
301	#386	C₂²×C₁₀₉	1	8	yes	complemented
302	#385	C₂²×C₁₀₉	1	8	yes	complemented
303	#385	C₂²×C₁₀₉	1	8	yes	complemented
304	#385	C₂²×C₁₀₉	1	8	yes	complemented
305	#381	C₂²×C₁₀₉	1	4	yes	complemented
306	#381	C₂²×C₁₀₉	1	8	yes	complemented
307	#380	C₂²×C₁₀₉	1	8	yes	complemented
308	#379	C₂²×C₁₀₉	1	4	yes	—
309	#379	C₂²×C₁₀₉	1	2	yes	complemented
310	#379	C₂²×C₁₀₉	1	2	yes	complemented
311	#379	C₂²×C₁₀₉	1	4	yes	complemented
312	C₂⁵	—	2	8	no	—
313	C₂⁵	—	4	8	no	—
314	C₂⁵	D₄×C₁₀₉	1	8	yes	—
315	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
316	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
317	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
318	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
319	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
320	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
321	C₂²×D₄	—	2	8	no	—
322	C₂²×D₄	—	2	4	no	—
323	C₂²×D₄	—	2	8	no	—
324	C₂²×D₄	—	4	8	no	—
325	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
326	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
327	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
328	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
329	C₂³×C₄	—	2	8	no	—
330	C₂³×C₄	—	4	8	no	—
331	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
333	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
334	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
335	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
336	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
337	#123	—	4	16	no	—
338	#120	—	2	8	no	—
339	#120	—	2	16	no	—
340	#120	—	2	8	no	—
341	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	complemented
342	C₂×C₂²⋊C₄	—	2	16	no	—
343	C₂×C₂²⋊C₄	—	2	8	no	—
344	C₂×C₂²⋊C₄	—	2	16	no	—
345	C₂×C₂²⋊C₄	—	2	8	no	—
346	C₂×C₂²⋊C₄	—	2	8	no	—
347	C₂×C₂²⋊C₄	—	2	4	no	—
348	C₂×C₂²⋊C₄	—	2	8	no	—
349	C₂×C₂²⋊C₄	—	4	8	no	—
350	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	complemented
362	#95	C₂³×C₁₀₉	1	16	yes	complemented
363	#95	C₂³×C₁₀₉	1	16	yes	complemented
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	8	no	—
366	C₂⁴	—	2	32	no	—
367	C₂⁴	—	2	8	no	—
368	C₂⁴	—	2	8	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	2	8	no	—
372	C₂⁴	—	2	2	no	—
373	C₂⁴	—	2	4	no	—
374	C₂⁴	—	4	32	no	—
375	C₂⁴	—	4	4	no	—
376	C₂⁴	—	4	4	no	—
377	C₂⁴	—	4	4	no	—
378	C₂⁴	—	4	4	no	—
379	C₂⁴	—	4	4	no	—
380	C₂⁴	—	4	64	no	—
381	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
382	C₂⁴	C₂×D₄×C₁₀₉	1	8	yes	—
383	C₂⁴	C₂×D₄×C₁₀₉	1	8	yes	—
384	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
385	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
386	C₂⁴	#41×C₁₀₉	1	8	yes	—
387	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
388	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
389	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
390	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
391	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
392	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	8	no	—
397	C₂×D₄	—	2	8	no	—
398	C₂×D₄	—	2	8	no	—
399	C₂×D₄	—	2	16	no	—
400	C₂×D₄	—	4	8	no	—
401	C₂×D₄	—	4	32	no	—
402	C₂×D₄	—	4	16	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	—	2	8	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	16	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	—	4	8	no	—
413	C₂²×C₄	—	4	4	no	—
414	C₂²×C₄	—	4	8	no	—
415	C₂²×C₄	—	4	32	no	—
416	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
418	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
419	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
420	C₄⋊C₄	—	4	16	no	—
421	C₂²⋊C₄	—	2	16	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	16	no	—
431	C₂²⋊C₄	—	4	8	no	—
432	C₂²⋊C₄	—	4	16	no	—
433	C₂³	—	2	32	no	—
434	C₂³	—	2	8	no	—
435	C₂³	—	2	16	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	2	16	no	—
443	C₂³	—	2	8	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	16	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	8	no	—
452	C₂³	—	4	8	no	—
453	C₂³	—	4	8	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	8	no	—
456	C₂³	—	4	16	no	—
457	C₂³	—	4	32	no	—
458	C₂³	—	4	32	no	—
459	C₂³	—	4	128	no	—
460	C₂³	—	4	16	no	—
461	C₂³	—	4	8	no	—
462	C₂³	—	4	4	no	—
463	C₂³	—	8	8	no	—
464	C₂³	#123×C₁₀₉	1	16	yes	complemented
465	C₂³	C₂²×D₄×C₁₀₉	1	4	yes	—
466	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
467	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
468	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
469	C₂³	C₂²×D₄×C₁₀₉	1	4	yes	—
470	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
471	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
472	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
473	C₂³	H(5,𝔽₂)×C₁₀₉	1	8	yes	—
474	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
475	D₄	—	4	16	no	—
476	D₄	—	4	8	no	—
477	D₄	—	4	8	no	—
478	D₄	—	4	16	no	—
479	C₂×C₄	—	2	8	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	8	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	16	no	—
485	C₂×C₄	—	2	16	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	—	4	16	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	—	4	16	no	—
492	C₂×C₄	—	8	8	no	—
493	C₂²	—	2	32	no	—
494	C₂²	—	2	8	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	16	no	—
497	C₂²	—	2	8	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	4	16	no	—
507	C₂²	—	4	32	no	—
508	C₂²	—	4	32	no	—
509	C₂²	—	4	16	no	—
510	C₂²	—	4	8	no	—
511	C₂²	—	4	8	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	8	8	no	—
515	C₂²	C₂×#123×C₁₀₉	1	4	yes	—
516	C₂²	C₂×#123×C₁₀₉	1	4	yes	—
517	C₂²	C₂×#123×C₁₀₉	1	4	yes	—
518	C₂²	#534×C₁₀₉	1	8	yes	—
519	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
520	C₂²	C₂²×#41×C₁₀₉	1	2	yes	—
521	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	4	yes	—
522	C₄	—	4	8	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	8	no	—
525	C₄	—	4	4	no	—
526	C₂	—	2	4	no	—
527	C₂	—	2	8	no	—
528	C₂	—	2	4	no	—
529	C₂	—	2	4	no	—
530	C₂	—	4	4	no	—
531	C₂	—	8	2	no	—
532	C₂	C₂²×#123×C₁₀₉	1	1	yes	characteristic
533	C₂	C₂×#534×C₁₀₉	1	2	yes	—
534	C₂	C₁₀₉×#3490	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂²×#120×C₁₀₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂²⋊C₄×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2849	C₂	semidirect
4. C₁₀₉×#2622	C₂	semidirect
5. C₁₀₉×#2483	C₂	semidirect
6. C₁₀₉×#2412	C₂	semidirect
7. C₁₀₉×#2405	C₂	semidirect
8. C₁₀₉×#2315	C₂	semidirect
9. C₁₀₉×#2308	C₂	semidirect
10. C₂×#379×C₁₀₉	C₂	semidirect
14. C₂⁴×C₄×C₁₀₉	C₂²	semidirect
15. C₂×#123×C₁₀₉	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
28. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
29. #397×C₁₀₉	C₂²	semidirect
30. #394×C₁₀₉	C₂²	semidirect
31. #386×C₁₀₉	C₂²	semidirect
32. #386×C₁₀₉	C₂²	semidirect
33. #386×C₁₀₉	C₂²	semidirect
34. #385×C₁₀₉	C₂²	semidirect
35. #385×C₁₀₉	C₂²	semidirect
36. #385×C₁₀₉	C₂²	semidirect
37. #381×C₁₀₉	C₂²	semidirect
38. #381×C₁₀₉	C₂²	semidirect
39. #380×C₁₀₉	C₂²	semidirect
41. #379×C₁₀₉	C₂²	semidirect
42. #379×C₁₀₉	C₂²	semidirect
43. #379×C₁₀₉	C₂²	semidirect
73. C₂×C₄⋊C₄×C₁₀₉	C₂³	semidirect
93. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
94. #95×C₁₀₉	C₂³	semidirect
95. #95×C₁₀₉	C₂³	semidirect
196. C₂³×C₁₀₉	#123	semidirect
258. #34992	C₁₀₉	direct (N × H)
269. C₂³×C₂²⋊C₄	C₂×C₁₀₉	semidirect
270. #2849	C₂×C₁₀₉	semidirect
271. #2622	C₂×C₁₀₉	semidirect
272. #2483	C₂×C₁₀₉	semidirect
273. #2412	C₂×C₁₀₉	semidirect
274. #2405	C₂×C₁₀₉	semidirect
275. #2315	C₂×C₁₀₉	semidirect
276. #2308	C₂×C₁₀₉	semidirect
277. C₂×#379	C₂×C₁₀₉	semidirect
278. C₁₀₉	#34992	direct (N × H)
282. C₂⁴×C₄	C₂²×C₁₀₉	semidirect
283. C₂×#123	C₂²×C₁₀₉	semidirect
294. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
295. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
296. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
297. #397	C₂²×C₁₀₉	semidirect
298. #394	C₂²×C₁₀₉	semidirect
299. #386	C₂²×C₁₀₉	semidirect
300. #386	C₂²×C₁₀₉	semidirect
301. #386	C₂²×C₁₀₉	semidirect
302. #385	C₂²×C₁₀₉	semidirect
303. #385	C₂²×C₁₀₉	semidirect
304. #385	C₂²×C₁₀₉	semidirect
305. #381	C₂²×C₁₀₉	semidirect
306. #381	C₂²×C₁₀₉	semidirect
307. #380	C₂²×C₁₀₉	semidirect
309. #379	C₂²×C₁₀₉	semidirect
310. #379	C₂²×C₁₀₉	semidirect
311. #379	C₂²×C₁₀₉	semidirect
341. C₂×C₄⋊C₄	C₂³×C₁₀₉	semidirect
361. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
362. #95	C₂³×C₁₀₉	semidirect
363. #95	C₂³×C₁₀₉	semidirect
464. C₂³	#123×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#120×C₁₀₉	2
C₂³×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#2849	2
C₁₀₉×#2622	2
C₁₀₉×#2483	2
C₁₀₉×#2412	2
C₁₀₉×#2405	2
C₁₀₉×#2315	2
C₁₀₉×#2308	2
C₂×#379×C₁₀₉	2
#34992	109

Subgroups — C109×#34992 #42152986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#34992 #42152986