Subgroups — C₁₀₉×#3559 #28360813

Order 13952.

1024 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#562×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#554×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#553×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#553×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#547×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#547×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#544×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#540×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#539×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#539×C₁₀₉	C₂	1	4	yes	maximal, complemented
12	#537×C₁₀₉	C₂	1	2	yes	maximal, complemented
13	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
17	C₂×#41×C₁₀₉	C₂²	1	4	yes	—
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
21	#127×C₁₀₉	C₂²	1	1	yes	characteristic
22	#126×C₁₀₉	C₂²	1	2	yes	—
23	#126×C₁₀₉	C₂²	1	4	yes	—
24	#126×C₁₀₉	C₂²	1	4	yes	—
25	#125×C₁₀₉	C₂²	1	2	yes	—
26	#125×C₁₀₉	C₂²	1	2	yes	—
27	#125×C₁₀₉	C₂²	1	2	yes	—
28	#125×C₁₀₉	C₂²	1	2	yes	—
29	#125×C₁₀₉	C₂²	1	1	yes	characteristic
30	#125×C₁₀₉	C₂²	1	4	yes	—
31	#125×C₁₀₉	C₂²	1	2	yes	—
32	#124×C₁₀₉	C₂²	1	1	yes	characteristic
33	#124×C₁₀₉	C₂²	1	2	yes	—
34	#123×C₁₀₉	C₂²	1	2	yes	—
35	#123×C₁₀₉	C₂²	1	4	yes	—
36	#123×C₁₀₉	C₂²	1	4	yes	—
37	#123×C₁₀₉	C₂²	1	4	yes	—
38	#123×C₁₀₉	C₂²	1	4	yes	—
39	#123×C₁₀₉	C₂²	1	4	yes	—
40	#123×C₁₀₉	C₂²	1	4	yes	—
41	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
43	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
44	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
45	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
46	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
47	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
48	#121×C₁₀₉	C₂²	1	1	yes	characteristic
49	#121×C₁₀₉	C₂²	1	2	yes	—
50	#121×C₁₀₉	C₂²	1	2	yes	—
51	#121×C₁₀₉	C₂²	1	4	yes	—
52	#121×C₁₀₉	C₂²	1	4	yes	—
53	#121×C₁₀₉	C₂²	1	4	yes	—
54	#121×C₁₀₉	C₂²	1	2	yes	—
55	#121×C₁₀₉	C₂²	1	4	yes	—
56	#121×C₁₀₉	C₂²	1	2	yes	—
57	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
59	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
60	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
61	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
62	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
63	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
64	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
65	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
66	#117×C₁₀₉	C₂²	1	2	yes	—
67	#117×C₁₀₉	C₂²	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
69	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
71	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
73	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
74	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
75	#41×C₁₀₉	—	4	4	no	—
76	#41×C₁₀₉	—	4	8	no	—
77	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
79	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
80	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
82	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
83	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
84	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
86	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
87	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
88	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
91	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
96	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
97	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
98	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
99	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
100	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
102	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
104	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
105	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
107	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
108	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
109	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
110	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
111	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
112	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
113	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
114	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
115	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
116	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
118	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
121	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
122	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
123	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
126	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
127	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
130	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
131	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
132	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
133	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
134	C₄²×C₁₀₉	C₂³	1	2	yes	—
135	C₄²×C₁₀₉	C₂³	1	4	yes	—
136	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
137	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
138	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
139	Q₈×C₁₀₉	—	4	1	no	—
140	Q₈×C₁₀₉	—	4	4	no	—
141	Q₈×C₁₀₉	—	4	2	no	—
142	D₄×C₁₀₉	—	2	4	no	—
143	D₄×C₁₀₉	—	2	8	no	—
144	D₄×C₁₀₉	—	2	4	no	—
145	D₄×C₁₀₉	—	4	4	no	—
146	D₄×C₁₀₉	—	4	2	no	—
147	D₄×C₁₀₉	—	4	1	no	—
148	D₄×C₁₀₉	—	4	4	no	—
149	D₄×C₁₀₉	—	4	2	no	—
150	C₂×C₄×C₁₀₉	—	2	2	no	—
151	C₂×C₄×C₁₀₉	—	2	4	no	—
152	C₂×C₄×C₁₀₉	—	2	2	no	—
153	C₂×C₄×C₁₀₉	—	2	4	no	—
154	C₂×C₄×C₁₀₉	—	4	2	no	—
155	C₂×C₄×C₁₀₉	—	4	4	no	—
156	C₂×C₄×C₁₀₉	—	4	1	no	—
157	C₂×C₄×C₁₀₉	—	4	4	no	—
158	C₂×C₄×C₁₀₉	—	4	4	no	—
159	C₂×C₄×C₁₀₉	—	4	4	no	—
160	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
161	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
162	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
168	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
169	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
170	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
171	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
172	C₂²×C₁₀₉	—	2	1	no	—
173	C₂²×C₁₀₉	—	2	1	no	—
174	C₂²×C₁₀₉	—	2	2	no	—
175	C₂²×C₁₀₉	—	2	4	no	—
176	C₂²×C₁₀₉	—	2	2	no	—
177	C₂²×C₁₀₉	—	2	4	no	—
178	C₂²×C₁₀₉	—	2	4	no	—
179	C₂²×C₁₀₉	—	2	2	no	—
180	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
181	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
182	C₄×C₁₀₉	—	2	1	no	—
183	C₄×C₁₀₉	—	2	4	no	—
184	C₄×C₁₀₉	—	2	1	no	—
185	C₄×C₁₀₉	—	2	4	no	—
186	C₄×C₁₀₉	—	2	2	no	—
187	C₄×C₁₀₉	—	2	1	no	—
188	C₄×C₁₀₉	—	2	2	no	—
189	C₄×C₁₀₉	—	2	4	no	—
190	C₄×C₁₀₉	—	2	2	no	—
191	C₄×C₁₀₉	—	2	1	no	—
192	C₄×C₁₀₉	—	2	1	no	—
193	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
194	C₂×C₁₀₉	—	2	2	no	—
195	C₂×C₁₀₉	—	4	2	no	—
196	C₂×C₁₀₉	—	4	4	no	—
197	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₁₀₉	#585	1	1	yes	characteristic
200	#3559	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₁₀₉	#3559	1	1	yes	characteristic, direct product factor, Sylow
202	#563	C₂×C₁₀₉	1	2	yes	complemented
203	#562	C₂×C₁₀₉	1	2	yes	complemented
204	#554	C₂×C₁₀₉	1	2	yes	complemented
205	#553	C₂×C₁₀₉	1	2	yes	complemented
206	#553	C₂×C₁₀₉	1	2	yes	complemented
207	#547	C₂×C₁₀₉	1	4	yes	complemented
208	#547	C₂×C₁₀₉	1	2	yes	complemented
209	#544	C₂×C₁₀₉	1	2	yes	complemented
210	#540	C₂×C₁₀₉	1	2	yes	complemented
211	#539	C₂×C₁₀₉	1	2	yes	complemented
212	#539	C₂×C₁₀₉	1	4	yes	complemented
213	#537	C₂×C₁₀₉	1	2	yes	complemented
214	#532	C₂×C₁₀₉	1	1	yes	characteristic, complemented
215	#529	C₂×C₁₀₉	1	1	yes	characteristic, complemented
216	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic, complemented
217	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
218	C₂×#41	C₂²×C₁₀₉	1	4	yes	—
219	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
220	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
221	C₄⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
222	#127	C₂²×C₁₀₉	1	1	yes	characteristic
223	#126	C₂²×C₁₀₉	1	2	yes	—
224	#126	C₂²×C₁₀₉	1	4	yes	—
225	#126	C₂²×C₁₀₉	1	4	yes	—
226	#125	C₂²×C₁₀₉	1	2	yes	—
227	#125	C₂²×C₁₀₉	1	2	yes	—
228	#125	C₂²×C₁₀₉	1	2	yes	—
229	#125	C₂²×C₁₀₉	1	2	yes	—
230	#125	C₂²×C₁₀₉	1	1	yes	characteristic
231	#125	C₂²×C₁₀₉	1	4	yes	—
232	#125	C₂²×C₁₀₉	1	2	yes	—
233	#124	C₂²×C₁₀₉	1	1	yes	characteristic
234	#124	C₂²×C₁₀₉	1	2	yes	—
235	#123	C₂²×C₁₀₉	1	2	yes	—
236	#123	C₂²×C₁₀₉	1	4	yes	—
237	#123	C₂²×C₁₀₉	1	4	yes	—
238	#123	C₂²×C₁₀₉	1	4	yes	—
239	#123	C₂²×C₁₀₉	1	4	yes	—
240	#123	C₂²×C₁₀₉	1	4	yes	—
241	#123	C₂²×C₁₀₉	1	4	yes	—
242	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
243	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
244	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
245	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
246	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
247	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
248	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
249	#121	C₂²×C₁₀₉	1	1	yes	characteristic
250	#121	C₂²×C₁₀₉	1	2	yes	—
251	#121	C₂²×C₁₀₉	1	2	yes	—
252	#121	C₂²×C₁₀₉	1	4	yes	—
253	#121	C₂²×C₁₀₉	1	4	yes	—
254	#121	C₂²×C₁₀₉	1	4	yes	—
255	#121	C₂²×C₁₀₉	1	2	yes	—
256	#121	C₂²×C₁₀₉	1	4	yes	—
257	#121	C₂²×C₁₀₉	1	2	yes	—
258	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
259	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
260	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
261	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
262	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
263	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
264	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
265	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
266	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
267	#117	C₂²×C₁₀₉	1	2	yes	—
268	#117	C₂²×C₁₀₉	1	2	yes	—
269	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
270	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
272	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
274	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
275	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
276	#41	—	4	4	no	—
277	#41	—	4	8	no	—
278	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
279	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
280	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
281	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
282	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
283	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
284	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
285	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
286	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
287	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
288	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
289	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
290	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
291	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
292	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
293	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
294	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
295	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
296	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
297	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
298	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
299	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
300	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
301	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
302	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
303	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
304	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
305	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
306	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
307	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
308	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
309	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
310	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
311	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
312	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
313	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
314	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
315	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
316	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
317	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
318	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
319	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
320	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
321	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
322	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
323	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
324	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
325	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
326	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
327	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
328	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
329	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
330	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
331	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
332	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
333	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
334	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
335	C₄²	C₂³×C₁₀₉	1	2	yes	—
336	C₄²	C₂³×C₁₀₉	1	4	yes	—
337	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
338	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
339	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
340	Q₈	—	4	4	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	1	no	—
343	D₄	—	2	4	no	—
344	D₄	—	2	8	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	4	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	4	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
362	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
363	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
364	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
365	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
366	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
367	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
368	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
369	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
370	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
371	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
372	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	4	no	—
379	C₂²	—	2	4	no	—
380	C₂²	—	2	2	no	—
381	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
382	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	1	no	—
386	C₄	—	2	4	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	4	no	—
391	C₄	—	2	2	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	4	2	no	—
397	C₂	—	4	4	no	—
398	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
399	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
400	C₂	#585×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #563×C₁₀₉	C₂	semidirect
2. #562×C₁₀₉	C₂	semidirect
3. #554×C₁₀₉	C₂	semidirect
4. #553×C₁₀₉	C₂	semidirect
5. #553×C₁₀₉	C₂	semidirect
6. #547×C₁₀₉	C₂	semidirect
7. #547×C₁₀₉	C₂	semidirect
8. #544×C₁₀₉	C₂	semidirect
9. #540×C₁₀₉	C₂	semidirect
10. #539×C₁₀₉	C₂	semidirect
11. #539×C₁₀₉	C₂	semidirect
12. #537×C₁₀₉	C₂	semidirect
13. #532×C₁₀₉	C₂	semidirect
14. #529×C₁₀₉	C₂	semidirect
15. C₂×#125×C₁₀₉	C₂	semidirect
200. #3559	C₁₀₉	direct (N × H)
201. C₁₀₉	#3559	direct (N × H)
202. #563	C₂×C₁₀₉	semidirect
203. #562	C₂×C₁₀₉	semidirect
204. #554	C₂×C₁₀₉	semidirect
205. #553	C₂×C₁₀₉	semidirect
206. #553	C₂×C₁₀₉	semidirect
207. #547	C₂×C₁₀₉	semidirect
208. #547	C₂×C₁₀₉	semidirect
209. #544	C₂×C₁₀₉	semidirect
210. #540	C₂×C₁₀₉	semidirect
211. #539	C₂×C₁₀₉	semidirect
212. #539	C₂×C₁₀₉	semidirect
213. #537	C₂×C₁₀₉	semidirect
214. #532	C₂×C₁₀₉	semidirect
215. #529	C₂×C₁₀₉	semidirect
216. C₂×#125	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#563×C₁₀₉	2
#562×C₁₀₉	2
#554×C₁₀₉	2
#553×C₁₀₉	2
#547×C₁₀₉	2
#544×C₁₀₉	2
#540×C₁₀₉	2
#539×C₁₀₉	2
#537×C₁₀₉	2
#532×C₁₀₉	2
#529×C₁₀₉	2
C₂×#125×C₁₀₉	2
#3559	109

Subgroups — C109×#3559 #28360813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3559 #28360813