Subgroups — #28375787

Order 13952.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295761	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295736	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295391	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295386	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295381	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295307	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295242	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295222	C₂	1	1	yes	maximal, characteristic
12	#20295139	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₈₇₂	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182018	C₂²	1	1	yes	characteristic
17	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
18	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
19	#16812699	C₂²	1	1	yes	characteristic
20	#16812696	C₂²	1	1	yes	characteristic
21	#16812691	C₂²	1	1	yes	characteristic
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812688	—	2	1	no	—
24	#16812687	C₂²	1	1	yes	characteristic
25	#16812687	C₂²	1	1	yes	characteristic
26	#16812684	—	2	1	no	—
27	#16812684	—	2	1	no	—
28	#16812677	—	2	1	no	—
29	#16812677	—	2	1	no	—
30	#16812675	—	2	1	no	—
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
35	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	#16812661	—	2	1	no	—
38	C₄⋊C₄×D₁₀₉	—	2	1	no	—
39	#16812633	C₂²	1	1	yes	characteristic
40	#16812633	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812621	C₂²	1	1	yes	characteristic
49	#16812621	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#97×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	—	2	1	no	—
68	C₂×C₄×D₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	#13182018	—	2	2	no	—
72	D₈₇₂	—	2	2	no	—
73	D₈₇₂	—	2	2	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
102	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₀₉	—	2	1	no	—
106	C₂²×D₁₀₉	—	2	1	no	—
107	C₂²×D₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	4	1	no	—
109	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	D₄₃₆	—	2	1	no	—
122	D₄₃₆	—	2	1	no	—
123	D₄₃₆	—	2	1	no	—
124	D₄₃₆	—	2	1	no	—
125	D₄₃₆	—	2	1	no	—
126	D₄₃₆	—	4	2	no	—
127	D₄₃₆	—	8	1	no	—
128	D₄₃₆	#41	1	2	yes	—
129	C₄×D₁₀₉	—	4	2	no	—
130	C₄×D₁₀₉	—	8	1	no	—
131	C₄×D₁₀₉	—	8	1	no	—
132	C₄×D₁₀₉	—	8	1	no	—
133	C₄×D₁₀₉	—	8	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	8	1	no	—
137	C₈×C₁₀₉	—	2	1	no	—
138	C₈×C₁₀₉	—	2	1	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₀₉	—	2	2	no	—
143	C₂×D₁₀₉	—	4	1	no	—
144	C₂×D₁₀₉	—	4	1	no	—
145	C₂×D₁₀₉	—	4	1	no	—
146	C₂×D₁₀₉	—	4	1	no	—
147	C₂×D₁₀₉	—	4	1	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	2	1	no	—
157	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	C₂×C₁₀₉	#548	1	1	yes	characteristic
165	C₂×C₁₀₉	#575	1	1	yes	characteristic
166	C₂×C₁₀₉	#576	1	1	yes	characteristic
167	D₁₀₉	—	4	2	no	—
168	D₁₀₉	—	8	1	no	—
169	D₁₀₉	—	8	1	no	—
170	#3370	—	109	1	no	maximal, Sylow
171	C₁₀₉	#3370	1	1	yes	characteristic, complemented, Sylow
172	#559	—	109	1	no	—
173	#554	—	109	1	no	—
174	#490	—	109	1	no	—
175	#489	—	109	1	no	—
176	#488	—	109	1	no	—
177	#479	—	109	1	no	—
178	#478	—	109	1	no	—
179	#474	—	109	1	no	—
180	#463	—	109	1	no	—
181	#463	—	109	1	no	—
182	#459	—	109	1	no	—
183	#446	D₁₀₉	1	1	yes	characteristic, complemented
184	#442	—	109	1	no	—
185	#440	—	109	1	no	—
186	C₄×D₈	—	109	1	no	—
187	C₂×SD₁₆	—	109	1	no	—
188	C₂×D₈	—	109	1	no	—
189	C₂×D₈	—	109	1	no	—
190	C₄⋊Q₈	—	109	1	no	—
191	#127	—	109	1	no	—
192	#125	—	109	1	no	—
193	#125	—	109	1	no	—
194	#124	—	109	1	no	—
195	#124	—	109	1	no	—
196	#124	—	218	1	no	—
197	#123	—	218	1	no	—
198	#123	—	218	1	no	—
199	C₂²⋊Q₈	—	218	1	no	—
200	C₂²⋊Q₈	—	218	1	no	—
201	C₂²⋊Q₈	—	218	1	no	—
202	C₄×Q₈	—	109	1	no	—
203	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
204	C₄×D₄	—	109	1	no	—
205	C₄×D₄	—	109	1	no	—
206	C₄×D₄	—	109	1	no	—
207	C₄×D₄	—	109	1	no	—
208	C₄×D₄	—	109	1	no	—
209	C₄×D₄	—	218	1	no	—
210	C₂×C₄⋊C₄	—	218	1	no	—
211	#107	—	109	1	no	—
212	#107	—	109	1	no	—
213	#106	—	109	1	no	—
214	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
216	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#102	—	109	1	no	—
221	#102	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#102	—	109	1	no	—
227	#97	C₂×D₁₀₉	1	1	yes	characteristic
228	#97	C₂×D₁₀₉	1	1	yes	characteristic
229	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
230	C₂×Q₈	—	109	1	no	—
231	C₂×Q₈	—	218	1	no	—
232	C₂×Q₈	D₄₃₆	1	1	yes	characteristic
233	C₂×D₄	—	109	1	no	—
234	C₂×D₄	—	109	1	no	—
235	C₂×D₄	—	109	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	—	218	1	no	—
238	C₂²×C₄	—	218	1	no	—
239	C₂²×C₄	—	218	1	no	—
240	C₂²×C₄	—	218	1	no	—
241	C₂²×C₄	—	218	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	SD₁₆	—	218	2	no	—
244	D₈	—	218	2	no	—
245	D₈	—	218	2	no	—
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	109	1	no	—
253	C₄⋊C₄	—	109	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
261	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
262	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₂²⋊C₄	—	218	1	no	—
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₄²	—	218	1	no	—
273	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
275	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₂³	—	218	1	no	—
277	C₂³	—	218	1	no	—
278	C₂³	—	218	1	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	872	1	no	—
281	Q₈	—	218	1	no	—
282	Q₈	—	218	1	no	—
283	D₄	—	872	1	no	—
284	D₄	—	109	2	no	—
285	D₄	—	436	2	no	—
286	D₄	—	218	1	no	—
287	D₄	—	218	1	no	—
288	D₄	—	218	1	no	—
289	D₄	—	218	1	no	—
290	D₄	—	218	1	no	—
291	C₂×C₄	—	872	1	no	—
292	C₂×C₄	—	872	1	no	—
293	C₂×C₄	—	872	1	no	—
294	C₂×C₄	—	872	1	no	—
295	C₂×C₄	—	436	2	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₂²	—	436	1	no	—
313	C₂²	—	436	1	no	—
314	C₂²	—	436	1	no	—
315	C₂²	—	436	1	no	—
316	C₂²	—	436	1	no	—
317	C₂²	—	436	1	no	—
318	C₂²	—	436	1	no	—
319	C₂²	—	436	1	no	—
320	C₂²	—	218	2	no	—
321	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	436	1	no	—
329	C₄	—	436	1	no	—
330	C₄	—	436	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	436	1	no	—
333	C₄	#41×D₁₀₉	1	1	yes	characteristic
334	C₄	#16812747	1	1	yes	characteristic
335	C₂	—	872	1	no	—
336	C₂	—	872	1	no	—
337	C₂	—	436	2	no	—
338	C₂	#20295688	1	1	yes	characteristic
339	C₂	#20295826	1	1	yes	characteristic
340	C₂	#20295831	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#446×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295222	C₂
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	#3370
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #20295761	C₂	semidirect
2. #20295736	C₂	semidirect
3. #20295391	C₂	semidirect
4. #20295386	C₂	semidirect
5. #20295381	C₂	semidirect
6. #20295334	C₂	semidirect
7. #20295327	C₂	semidirect
8. #20295307	C₂	semidirect
9. #20295242	C₂	semidirect
10. #20295242	C₂	semidirect
12. #20295139	C₂	semidirect
13. #20295128	C₂	semidirect
14. C₄×D₈₇₂	C₂	semidirect
15. #446×C₁₀₉	C₂	semidirect
171. C₁₀₉	#3370	semidirect
183. #446	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295761	2
#20295736	2
#20295391	2
#20295386	2
#20295381	2
#20295334	2
#20295327	2
#20295307	2
#20295242	2
#20295222	2
#20295139	2
#20295128	2
C₄×D₈₇₂	2
#446×C₁₀₉	2
#3370	109