Subgroups — #28374458

Order 13952.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295638	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295318	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295308	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295282	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295270	C₂	1	1	yes	maximal, characteristic
8	#20295265	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295253	C₂	1	1	yes	maximal, characteristic
10	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
15	#542×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×#13182013	C₂²	1	1	yes	characteristic
21	C₂²×#11724784	C₂²	1	1	yes	characteristic
22	#16812699	C₂²	1	1	yes	characteristic
23	#16812692	—	2	1	no	—
24	#16812692	—	2	1	no	—
25	#16812691	C₂²	1	1	yes	characteristic
26	#16812689	—	2	1	no	—
27	#16812689	—	2	1	no	—
28	#16812678	C₂²	1	1	yes	characteristic
29	#16812678	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	#16812632	C₂²	1	1	yes	characteristic
33	#16812632	C₂²	1	1	yes	characteristic
34	#16812630	C₂²	1	1	yes	characteristic
35	#16812630	C₂²	1	1	yes	characteristic
36	C₄⋊#11724784	C₂²	1	1	yes	characteristic
37	C₄⋊#11724784	C₂²	1	1	yes	characteristic
38	C₄⋊#11724784	C₂²	1	1	yes	characteristic
39	C₄⋊#11724784	C₂²	1	1	yes	characteristic
40	#16812623	C₂²	1	1	yes	characteristic
41	#16812623	C₂²	1	1	yes	characteristic
42	#16812622	C₂²	1	1	yes	characteristic
43	#16812622	C₂²	1	1	yes	characteristic
44	#16812622	C₂²	1	1	yes	characteristic
45	#16812622	C₂²	1	1	yes	characteristic
46	#16812620	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
49	#126×C₁₀₉	—	2	1	no	—
50	#124×C₁₀₉	—	2	1	no	—
51	#124×C₁₀₉	C₂²	1	1	yes	characteristic
52	#123×C₁₀₉	—	2	1	no	—
53	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
54	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#121×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
58	#117×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	#13182019	—	4	1	no	—
62	#13182019	—	4	1	no	—
63	#13182017	—	4	1	no	—
64	#13182017	—	4	1	no	—
65	#13182013	—	4	1	no	—
66	C₂×#11724784	—	4	1	no	—
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₂×#11724784	C₂³	1	1	yes	characteristic
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	#13182008	—	2	1	no	—
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	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₄×Dic₁₀₉	—	2	1	no	—
84	C₄×Dic₁₀₉	—	2	1	no	—
85	C₂×Q₈×C₁₀₉	—	2	1	no	—
86	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂×D₄×C₁₀₉	—	2	1	no	—
88	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₀₉	—	2	1	no	—
90	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₀₉	—	2	1	no	—
92	C₄⋊C₄×C₁₀₉	—	2	1	no	—
93	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	8	1	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	—	8	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	8	1	no	—
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	#11724784	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
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	H(5,𝔽₂)	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	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	C₂×C₁₀₉	—	4	1	no	—
162	C₂×C₁₀₉	—	8	1	no	—
163	C₂×C₁₀₉	#537	1	1	yes	characteristic
164	C₂×C₁₀₉	#577	1	1	yes	characteristic
165	C₂×C₁₀₉	#578	1	1	yes	characteristic
166	#3258	—	109	1	no	maximal, Sylow
167	C₁₀₉	#3258	1	1	yes	characteristic, complemented, Sylow
168	#542	D₁₀₉	1	1	yes	characteristic
169	#541	—	109	1	no	—
170	#479	—	109	1	no	—
171	#478	—	109	1	no	—
172	#477	—	109	1	no	—
173	#475	—	109	1	no	—
174	#470	—	109	1	no	—
175	#468	—	109	1	no	—
176	#467	—	109	1	no	—
177	#465	—	109	1	no	—
178	#464	—	109	1	no	—
179	#464	—	109	1	no	—
180	#462	—	109	1	no	—
181	#461	—	109	1	no	—
182	#424	—	109	1	no	—
183	C₂×Q₁₆	—	109	1	no	—
184	C₂×Q₁₆	—	109	1	no	—
185	C₂×SD₁₆	—	109	1	no	—
186	C₂×SD₁₆	—	109	1	no	—
187	C₂×M₁₆	—	109	1	no	—
188	C₂²×C₈	—	109	1	no	—
189	C₄⋊Q₈	—	109	1	no	—
190	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
191	#126	—	2	1	no	—
192	#126	—	218	1	no	—
193	#126	—	218	1	no	—
194	#125	—	109	1	no	—
195	#125	—	218	1	no	—
196	#125	—	218	1	no	—
197	#124	—	2	1	no	—
198	#124	C₂×D₁₀₉	1	1	yes	characteristic
199	#123	—	2	1	no	—
200	C₂²⋊Q₈	—	2	1	no	—
201	C₂²⋊Q₈	—	109	1	no	—
202	C₂²⋊Q₈	—	109	1	no	—
203	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
204	#121	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
208	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
209	#117	C₂×D₁₀₉	1	1	yes	characteristic
210	#107	—	109	1	no	—
211	#107	—	109	1	no	—
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	C₄⋊C₈	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#103	—	109	1	no	—
223	#103	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	109	1	no	—
229	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
234	Q₁₆	—	436	1	no	—
235	Q₁₆	—	436	1	no	—
236	SD₁₆	—	436	1	no	—
237	SD₁₆	—	436	1	no	—
238	M₁₆	—	436	1	no	—
239	C₂×C₈	—	109	1	no	—
240	C₂×C₈	—	109	1	no	—
241	C₂×C₈	—	109	1	no	—
242	C₂×C₈	—	109	1	no	—
243	C₂×C₈	—	436	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	109	1	no	—
248	C₄⋊C₄	—	109	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
269	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
270	C₄²	—	2	1	no	—
271	C₄²	—	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	C₂³	—	2	1	no	—
276	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	8	1	no	—
280	Q₈	—	218	1	no	—
281	Q₈	—	218	1	no	—
282	Q₈	—	218	1	no	—
283	Q₈	—	218	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	218	1	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
303	C₈	—	218	1	no	—
304	C₈	—	218	1	no	—
305	C₈	—	218	1	no	—
306	C₈	—	218	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	436	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	#16812745	1	1	yes	characteristic
327	C₄	#16812746	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#20295604	1	1	yes	characteristic
331	C₂	#20295837	1	1	yes	characteristic
332	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#542×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#121×C₁₀₉	C₂²
hypercenter	#542	D₁₀₉
hypocenter	C₁₀₉	#3258
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #542

Complements

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

N	H ≅ G/N	type
1. #20295638	C₂	semidirect
2. #20295333	C₂	semidirect
3. #20295326	C₂	semidirect
4. #20295318	C₂	semidirect
5. #20295308	C₂	semidirect
6. #20295282	C₂	semidirect
8. #20295265	C₂	semidirect
10. #20295245	C₂	semidirect
11. #20295245	C₂	semidirect
12. #20295236	C₂	semidirect
13. #20295231	C₂	semidirect
14. #20295053	C₂	semidirect
167. C₁₀₉	#3258	semidirect

Maximal subgroups

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

subgroup	index
#20295638	2
#20295333	2
#20295326	2
#20295318	2
#20295308	2
#20295282	2
#20295270	2
#20295265	2
#20295253	2
#20295245	2
#20295236	2
#20295231	2
#20295053	2
#542×C₁₀₉	2
#3258	109