Subgroups — #28375800

Order 13952.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 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
2	#20295392	C₂	1	1	yes	maximal, characteristic
3	#20295388	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295383	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295313	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295248	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295248	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295226	C₂	1	1	yes	maximal, characteristic
11	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295131	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
15	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182016	C₂²	1	1	yes	characteristic
18	C₂×#13182016	C₂²	1	1	yes	characteristic
19	#16812696	C₂²	1	1	yes	characteristic
20	#16812688	—	2	1	no	—
21	#16812687	C₂²	1	1	yes	characteristic
22	#16812687	C₂²	1	1	yes	characteristic
23	#16812684	—	2	1	no	—
24	#16812684	—	2	1	no	—
25	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
26	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
27	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
28	#16812661	—	2	1	no	—
29	#16812632	C₂²	1	1	yes	characteristic
30	#16812630	C₂²	1	1	yes	characteristic
31	#16812630	C₂²	1	1	yes	characteristic
32	C₄⋊#11724784	C₂²	1	1	yes	characteristic
33	C₄⋊#11724784	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic
35	#16812623	C₂²	1	1	yes	characteristic
36	#16812622	C₂²	1	1	yes	characteristic
37	#16812622	C₂²	1	1	yes	characteristic
38	#16812622	C₂²	1	1	yes	characteristic
39	#16812619	C₂²	1	1	yes	characteristic
40	#16812619	C₂²	1	1	yes	characteristic
41	#16812619	C₂²	1	1	yes	characteristic
42	#16812619	C₂²	1	1	yes	characteristic
43	#16812619	C₂²	1	1	yes	characteristic
44	#16812619	C₂²	1	1	yes	characteristic
45	#16812610	C₂²	1	1	yes	characteristic
46	#16812610	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
49	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
50	#125×C₁₀₉	—	2	1	no	—
51	#125×C₁₀₉	—	2	1	no	—
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	#125×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	—	2	1	no	—
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×D₄₃₆	—	2	1	no	—
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182019	—	2	2	no	—
67	#13182016	—	2	2	no	—
68	#13182016	—	2	2	no	—
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	#13182008	—	2	1	no	—
75	#13182006	—	2	1	no	—
76	#13182006	—	2	1	no	—
77	#13182006	—	2	1	no	—
78	#13182006	—	2	1	no	—
79	#13182006	—	2	1	no	—
80	#13182006	—	2	1	no	—
81	C₄×Dic₁₀₉	—	2	1	no	—
82	C₂×Q₈×C₁₀₉	—	2	1	no	—
83	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₀₉	—	2	1	no	—
86	C₄⋊C₄×C₁₀₉	—	2	1	no	—
87	C₄⋊C₄×C₁₀₉	—	2	1	no	—
88	C₄⋊C₄×C₁₀₉	—	2	1	no	—
89	C₄⋊C₄×C₁₀₉	—	2	1	no	—
90	C₄⋊C₄×C₁₀₉	—	2	1	no	—
91	C₄⋊C₄×C₁₀₉	—	2	1	no	—
92	C₄⋊C₄×C₁₀₉	—	2	1	no	—
93	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄²×C₁₀₉	—	2	1	no	—
100	C₄²×C₁₀₉	—	2	1	no	—
101	C₄²×C₁₀₉	—	2	1	no	—
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	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	4	1	no	—
109	Q₈×C₁₀₉	—	4	2	no	—
110	Q₈×C₁₀₉	#41	1	2	yes	—
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	—	2	1	no	—
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	D₄₃₆	—	2	1	no	—
125	D₄₃₆	—	2	1	no	—
126	D₄₃₆	—	2	1	no	—
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	4	2	no	—
129	D₄₃₆	—	8	1	no	—
130	C₄×D₁₀₉	—	8	1	no	—
131	C₄×D₁₀₉	—	8	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	2	1	no	—
134	#11724784	—	2	1	no	—
135	#11724784	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	4	1	no	—
140	C₂×D₁₀₉	—	4	1	no	—
141	C₂×D₁₀₉	—	4	1	no	—
142	C₂×D₁₀₉	—	4	1	no	—
143	C₂×D₁₀₉	—	4	1	no	—
144	C₂×D₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	2	2	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	2	1	no	—
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	2	1	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	C₂×C₁₀₉	#548	1	1	yes	characteristic
160	C₂×C₁₀₉	#575	1	1	yes	characteristic
161	C₂×C₁₀₉	#577	1	1	yes	characteristic
162	D₁₀₉	—	8	1	no	—
163	D₁₀₉	—	8	1	no	—
164	#3371	—	109	1	no	maximal, Sylow
165	C₁₀₉	#3371	1	1	yes	characteristic, complemented, Sylow
166	#559	—	109	1	no	—
167	#557	D₁₀₉	1	1	yes	characteristic, complemented
168	#490	—	109	1	no	—
169	#489	—	109	1	no	—
170	#488	—	109	1	no	—
171	#479	—	109	1	no	—
172	#478	—	109	1	no	—
173	#476	—	109	1	no	—
174	#464	—	109	1	no	—
175	#464	—	109	1	no	—
176	#460	—	109	1	no	—
177	#446	—	109	1	no	—
178	#441	—	109	1	no	—
179	#440	—	109	1	no	—
180	C₄×SD₁₆	—	109	1	no	—
181	C₂×Q₁₆	—	109	1	no	—
182	C₂×SD₁₆	—	109	1	no	—
183	C₂×SD₁₆	—	109	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
186	#127	—	109	1	no	—
187	#125	—	2	1	no	—
188	#125	—	2	1	no	—
189	#125	C₂×D₁₀₉	1	1	yes	characteristic
190	#125	C₂×D₁₀₉	1	1	yes	characteristic
191	#124	—	109	1	no	—
192	#124	—	109	1	no	—
193	#124	—	218	1	no	—
194	#123	—	218	1	no	—
195	#123	—	218	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	109	1	no	—
198	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₄×D₄	—	109	1	no	—
203	C₄×D₄	—	109	1	no	—
204	C₄×D₄	—	218	1	no	—
205	#107	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	—	109	1	no	—
208	C₄⋊C₈	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
210	C₄⋊C₈	—	109	1	no	—
211	#103	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#102	—	109	1	no	—
216	#102	—	109	1	no	—
217	#102	—	109	1	no	—
218	#102	—	109	1	no	—
219	#102	—	109	1	no	—
220	#102	—	109	1	no	—
221	#97	—	109	1	no	—
222	#97	—	109	1	no	—
223	C₄×C₈	—	109	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	109	1	no	—
226	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	109	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	—	218	1	no	—
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	—	218	1	no	—
234	Q₁₆	—	218	2	no	—
235	SD₁₆	—	218	2	no	—
236	SD₁₆	—	218	2	no	—
237	C₂×C₈	—	109	1	no	—
238	C₂×C₈	—	109	1	no	—
239	C₂×C₈	—	109	1	no	—
240	C₂×C₈	—	109	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	109	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	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	218	1	no	—
258	C₂²⋊C₄	—	218	1	no	—
259	C₂²⋊C₄	—	218	1	no	—
260	C₂²⋊C₄	—	218	1	no	—
261	C₂²⋊C₄	—	218	1	no	—
262	C₂²⋊C₄	—	218	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
270	C₂³	—	218	1	no	—
271	C₂³	—	218	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	218	1	no	—
276	Q₈	—	218	1	no	—
277	Q₈	#13182030	1	2	yes	—
278	D₄	—	872	1	no	—
279	D₄	—	436	2	no	—
280	D₄	—	218	1	no	—
281	D₄	—	218	1	no	—
282	D₄	—	218	1	no	—
283	D₄	—	218	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	872	1	no	—
289	C₂×C₄	—	872	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	—	218	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
299	C₈	—	218	1	no	—
300	C₈	—	218	1	no	—
301	C₈	—	218	1	no	—
302	C₈	—	218	1	no	—
303	C₂²	—	436	1	no	—
304	C₂²	—	436	1	no	—
305	C₂²	—	436	1	no	—
306	C₂²	—	436	1	no	—
307	C₂²	—	436	1	no	—
308	C₂²	—	436	1	no	—
309	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	436	1	no	—
321	C₄	—	436	1	no	—
322	C₄	C₂×#13182030	1	1	yes	characteristic
323	C₄	#16812747	1	1	yes	characteristic
324	C₂	—	872	1	no	—
325	C₂	—	872	1	no	—
326	C₂	#20295687	1	1	yes	characteristic
327	C₂	#20295824	1	1	yes	characteristic
328	C₂	#20295834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#557×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812696	C₂²
hypercenter	#557	D₁₀₉
hypocenter	C₁₀₉	#3371
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
3. #20295388	C₂	semidirect
4. #20295383	C₂	semidirect
5. #20295330	C₂	semidirect
6. #20295323	C₂	semidirect
7. #20295313	C₂	semidirect
8. #20295248	C₂	semidirect
9. #20295248	C₂	semidirect
11. #20295157	C₂	semidirect
12. #20295131	C₂	semidirect
13. #20295124	C₂	semidirect
14. C₄×#13182016	C₂	semidirect
15. #557×C₁₀₉	C₂	semidirect
165. C₁₀₉	#3371	semidirect
167. #557	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
#20295392	2
#20295388	2
#20295383	2
#20295330	2
#20295323	2
#20295313	2
#20295248	2
#20295226	2
#20295157	2
#20295131	2
#20295124	2
C₄×#13182016	2
#557×C₁₀₉	2
#3371	109