Subgroups — #28372867

Order 13952.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295653	C₂	1	1	yes	maximal, characteristic
2	#20295502	C₂	1	1	yes	maximal, characteristic
3	#20295336	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295331	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295319	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295318	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295065	C₂	1	1	yes	maximal, characteristic
11	#20295061	C₂	1	1	yes	maximal, characteristic
12	#20295051	C₂	1	1	yes	maximal, characteristic
13	#20295033	C₂	1	1	yes	maximal, characteristic
14	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic
15	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812697	—	2	1	no	—
21	#16812691	C₂²	1	1	yes	characteristic
22	#16812690	C₂²	1	1	yes	characteristic
23	#16812689	—	2	1	no	—
24	#16812677	—	2	1	no	—
25	#16812675	—	2	1	no	—
26	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₁₈	—	2	1	no	—
28	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
29	#16812664	—	2	1	no	—
30	#16812664	C₂²	1	1	yes	characteristic
31	#16812657	—	2	1	no	—
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812656	—	2	1	no	—
35	#16812656	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812623	C₂²	1	1	yes	characteristic
49	#16812623	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812622	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#106×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	Q₈×D₁₀₉	—	2	2	no	—
63	Q₈×D₁₀₉	Q₈	1	4	yes	complemented
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	#13182012	—	2	2	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₈×D₁₀₉	—	2	2	no	—
71	C₄⋊Dic₁₀₉	—	2	1	no	—
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	D₄	1	1	yes	characteristic
84	#13182008	D₄	1	1	yes	characteristic
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	D₄	1	1	yes	characteristic
88	#13182006	D₄	1	1	yes	characteristic
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	—	2	1	no	—
96	C₂×Q₈×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	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	8	1	no	—
109	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
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	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₀₉	—	2	4	no	—
125	C₄×D₁₀₉	—	8	1	no	—
126	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
127	C₄×D₁₀₉	#41	1	2	yes	—
128	Dic₂₁₈	—	2	1	no	—
129	Dic₂₁₈	—	2	4	no	—
130	Dic₂₁₈	—	8	1	no	—
131	Dic₂₁₈	C₂×Q₈	1	2	yes	—
132	C₈×C₁₀₉	—	2	1	no	—
133	C₈×C₁₀₉	—	2	1	no	—
134	#11724784	—	2	1	no	—
135	#11724784	—	2	1	no	—
136	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₀₉	—	2	1	no	—
138	C₂×D₁₀₉	—	2	1	no	—
139	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
140	C₄×C₁₀₉	—	2	2	no	—
141	C₄×C₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	4	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
148	Dic₁₀₉	—	2	2	no	—
149	Dic₁₀₉	—	2	1	no	—
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	4	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₁₀₉	#577	1	1	yes	characteristic
157	C₂×C₁₀₉	#578	1	1	yes	characteristic
158	D₁₀₉	—	2	2	no	—
159	#3082	—	109	1	no	maximal, Sylow
160	C₁₀₉	#3082	1	1	yes	characteristic, complemented, Sylow
161	#544	—	109	1	no	—
162	#519	—	109	1	no	—
163	#479	—	109	1	no	—
164	#479	—	109	1	no	—
165	#479	—	109	1	no	—
166	#479	—	109	1	no	—
167	#477	—	109	1	no	—
168	#477	—	109	1	no	—
169	#475	—	109	1	no	—
170	#475	D₁₀₉	1	1	yes	characteristic, complemented
171	#428	—	109	1	no	—
172	#427	—	109	1	no	—
173	#424	—	109	1	no	—
174	#419	—	109	1	no	—
175	C₂×#103	—	109	1	no	—
176	C₂²×Q₈	—	109	1	no	—
177	C₂×M₁₆	—	109	1	no	—
178	C₂²×C₈	—	109	1	no	—
179	C₄⋊Q₈	—	109	1	no	—
180	C₄⋊Q₈	—	218	1	no	—
181	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
182	#125	—	109	1	no	—
183	#125	—	109	1	no	—
184	#125	—	218	1	no	—
185	C₂²⋊Q₈	—	218	1	no	—
186	C₂²⋊Q₈	—	218	1	no	—
187	C₄×Q₈	—	109	1	no	—
188	C₄×Q₈	—	109	1	no	—
189	C₄×Q₈	—	109	1	no	—
190	C₄×Q₈	—	218	1	no	—
191	C₄×Q₈	—	218	1	no	—
192	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
193	#117	—	109	1	no	—
194	#117	—	109	1	no	—
195	#117	—	109	1	no	—
196	#117	—	218	1	no	—
197	#117	—	218	1	no	—
198	C₂×C₄⋊C₄	—	109	1	no	—
199	#107	—	109	1	no	—
200	#107	—	109	1	no	—
201	#107	—	109	1	no	—
202	#107	—	109	1	no	—
203	#106	—	109	1	no	—
204	#106	—	109	1	no	—
205	#106	C₂×D₁₀₉	1	1	yes	characteristic
206	#106	C₂×D₁₀₉	1	1	yes	characteristic
207	C₄⋊C₈	—	109	1	no	—
208	C₄⋊C₈	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
210	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
211	#103	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	C₂×D₁₀₉	1	1	yes	characteristic
218	#103	C₂×D₁₀₉	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	109	4	no	—
221	C₂×Q₈	—	109	1	no	—
222	C₂×Q₈	—	218	1	no	—
223	C₂×Q₈	—	218	1	no	—
224	C₂×Q₈	—	218	2	no	—
225	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₂²×C₄	—	109	1	no	—
227	C₂²×C₄	—	218	1	no	—
228	C₂²×C₄	—	218	1	no	—
229	M₁₆	—	218	2	no	—
230	C₂×C₈	—	109	1	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	218	2	no	—
233	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	109	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	109	1	no	—
241	C₄⋊C₄	—	109	1	no	—
242	C₄⋊C₄	—	218	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	109	1	no	—
255	C₂²⋊C₄	—	109	1	no	—
256	C₂²⋊C₄	—	218	1	no	—
257	C₂²⋊C₄	—	218	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	109	1	no	—
260	C₄²	—	218	1	no	—
261	C₄²	—	218	1	no	—
262	C₄²	—	218	1	no	—
263	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂³	—	109	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	8	1	no	—
269	Q₈	—	872	1	no	—
270	Q₈	—	109	2	no	—
271	Q₈	—	218	4	no	—
272	Q₈	—	218	1	no	—
273	Q₈	Q₈×D₁₀₉	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	872	1	no	—
279	C₂×C₄	—	109	1	no	—
280	C₂×C₄	—	109	2	no	—
281	C₂×C₄	—	109	2	no	—
282	C₂×C₄	—	109	1	no	—
283	C₂×C₄	—	109	1	no	—
284	C₂×C₄	—	218	4	no	—
285	C₂×C₄	—	218	1	no	—
286	C₂×C₄	—	218	1	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	218	1	no	—
295	C₈	—	218	1	no	—
296	C₂²	—	109	2	no	—
297	C₂²	—	218	1	no	—
298	C₂²	—	218	1	no	—
299	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	109	2	no	—
307	C₄	—	436	1	no	—
308	C₄	—	436	1	no	—
309	C₄	—	436	1	no	—
310	C₄	—	218	2	no	—
311	C₄	—	218	1	no	—
312	C₄	—	218	1	no	—
313	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
314	C₄	#41×D₁₀₉	1	1	yes	characteristic
315	C₂	—	218	2	no	—
316	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
317	C₂	#20295832	1	1	yes	characteristic
318	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#475×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#475	D₁₀₉
hypocenter	C₁₀₉	#3082
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
3. #20295336	C₂	semidirect
4. #20295334	C₂	semidirect
5. #20295333	C₂	semidirect
6. #20295331	C₂	semidirect
7. #20295319	C₂	semidirect
8. #20295318	C₂	semidirect
9. #20295310	C₂	semidirect
15. #475×C₁₀₉	C₂	semidirect
63. Q₈×D₁₀₉	Q₈	semidirect
160. C₁₀₉	#3082	semidirect
170. #475	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295653	2
#20295502	2
#20295336	2
#20295334	2
#20295333	2
#20295331	2
#20295319	2
#20295318	2
#20295310	2
#20295065	2
#20295061	2
#20295051	2
#20295033	2
#103×D₁₀₉	2
#475×C₁₀₉	2
#3082	109