Subgroups — #27181041

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265515	C₂	1	2	yes	maximal, complemented
3	#20265510	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265505	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265498	C₂	1	2	yes	maximal, complemented
6	#20265493	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265488	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265213	C₂	1	2	yes	maximal, complemented
9	#20265173	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11717579	C₂²	1	2	yes	—
13	#15692214	—	2	1	no	—
14	#15692214	—	2	1	no	—
15	#15692214	C₂²	1	2	yes	—
16	#15692214	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
20	C₂×C₄×Dic₁₀₁	—	2	1	no	—
21	#15692188	C₂²	1	2	yes	complemented
22	#15692188	C₂²	1	2	yes	complemented
23	#15692186	C₂²	1	2	yes	complemented
24	#15692186	C₂²	1	2	yes	complemented
25	#15692179	C₂²	1	2	yes	complemented
26	#15692179	C₂²	1	2	yes	complemented
27	#15692176	C₂²	1	2	yes	complemented
28	#15692176	C₂²	1	2	yes	complemented
29	C₂²⋊#11717579	C₂²	1	1	yes	characteristic
30	C₂²⋊#11717579	C₂²	1	1	yes	characteristic
31	C₂²⋊#11717579	C₂²	1	2	yes	—
32	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	#123×C₁₀₁	—	2	1	no	—
35	C₂²⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
37	#121×C₁₀₁	C₂²	1	1	yes	characteristic
38	#121×C₁₀₁	C₂²	1	1	yes	characteristic
39	#120×C₁₀₁	—	2	1	no	—
40	C₄×D₄×C₁₀₁	—	2	1	no	—
41	C₄×D₄×C₁₀₁	—	2	1	no	—
42	#117×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	—	2	1	no	—
44	C₂²×Dic₁₀₁	—	2	1	no	—
45	C₂×#11717579	—	2	4	no	—
46	C₂×#11717579	C₂³	1	2	yes	—
47	C₂×#11717579	C₂³	1	2	yes	—
48	C₄⋊Dic₁₀₁	—	2	1	no	—
49	C₄⋊Dic₁₀₁	—	2	1	no	—
50	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
51	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²⋊Dic₁₀₁	—	2	2	no	—
53	C₂²⋊Dic₁₀₁	—	2	2	no	—
54	C₄×Dic₁₀₁	—	2	2	no	—
55	C₄×Dic₁₀₁	—	2	2	no	—
56	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
57	#41×C₁₀₁	—	2	4	no	—
58	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₀₁	—	2	1	no	—
60	C₂×D₄×C₁₀₁	—	2	1	no	—
61	C₂×D₄×C₁₀₁	—	2	1	no	—
62	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	—	2	4	no	—
64	C₂²×C₄×C₁₀₁	—	2	1	no	—
65	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₁	—	2	1	no	—
72	C₄⋊C₄×C₁₀₁	—	2	1	no	—
73	C₄⋊C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
77	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
78	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
79	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
80	C₄²×C₁₀₁	—	2	1	no	—
81	C₂³×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	—	2	1	no	—
84	C₂³×C₁₀₁	—	4	1	no	—
85	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₁₀₁	—	2	2	no	—
89	D₄×C₁₀₁	—	2	2	no	—
90	D₄×C₁₀₁	—	4	2	no	—
91	D₄×C₁₀₁	—	4	2	no	—
92	D₄×C₁₀₁	—	4	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	1	no	—
101	C₂×C₄×C₁₀₁	—	4	2	no	—
102	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₁	#41	1	4	yes	—
106	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	4	2	no	—
110	C₂×Dic₁₀₁	—	4	2	no	—
111	#11717579	—	2	2	no	—
112	#11717579	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	1	no	—
114	C₂²×C₁₀₁	—	2	1	no	—
115	C₂²×C₁₀₁	—	2	1	no	—
116	C₂²×C₁₀₁	—	2	1	no	—
117	C₂²×C₁₀₁	—	2	1	no	—
118	C₂²×C₁₀₁	—	2	1	no	—
119	C₂²×C₁₀₁	—	2	1	no	—
120	C₂²×C₁₀₁	—	4	4	no	—
121	C₂²×C₁₀₁	—	4	1	no	—
122	C₂²×C₁₀₁	—	4	1	no	—
123	C₂²×C₁₀₁	—	4	1	no	—
124	C₂²×C₁₀₁	#123	1	2	yes	—
125	C₂²×C₁₀₁	#135	1	2	yes	—
126	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₀₁	—	2	1	no	—
128	C₄×C₁₀₁	—	2	1	no	—
129	C₄×C₁₀₁	—	4	1	no	—
130	C₄×C₁₀₁	—	4	1	no	—
131	C₄×C₁₀₁	—	4	1	no	—
132	C₄×C₁₀₁	#123	1	2	yes	—
133	C₄×C₁₀₁	C₂×#41	1	2	yes	—
134	Dic₁₀₁	—	4	2	no	—
135	Dic₁₀₁	—	4	2	no	—
136	C₂×C₁₀₁	—	2	2	no	—
137	C₂×C₁₀₁	—	2	2	no	—
138	C₂×C₁₀₁	—	4	1	no	—
139	C₂×C₁₀₁	—	8	1	no	—
140	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
141	C₂×C₁₀₁	C₂×#135	1	1	yes	characteristic
142	C₂×C₁₀₁	#575	1	1	yes	characteristic
143	#3091	—	101	1	no	maximal, Sylow
144	C₁₀₁	#3091	1	1	yes	characteristic, complemented, Sylow
145	#525	D₁₀₁	1	1	yes	characteristic
146	C₂×#117	—	101	1	no	—
147	#485	—	101	2	no	—
148	#484	—	101	1	no	—
149	#483	—	101	1	no	—
150	#482	—	101	2	no	—
151	#481	—	101	1	no	—
152	#480	—	101	1	no	—
153	#427	—	101	2	no	—
154	#416	—	101	2	no	—
155	C₂×C₂²⋊C₈	—	101	1	no	—
156	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
157	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂²×C₈	—	101	2	no	—
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂²⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
162	#121	C₂×D₁₀₁	1	1	yes	characteristic
163	#121	C₂×D₁₀₁	1	1	yes	characteristic
164	#120	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	#117	—	101	2	no	—
168	#117	—	101	2	no	—
169	#117	—	202	1	no	—
170	#117	—	202	1	no	—
171	#117	C₂×D₁₀₁	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	101	1	no	—
173	C₂×C₄⋊C₄	—	101	1	no	—
174	C₂×C₂²⋊C₄	—	202	1	no	—
175	C₂×C₄²	—	202	1	no	—
176	#107	—	101	2	no	—
177	#107	—	101	2	no	—
178	#106	—	101	2	no	—
179	#106	—	101	2	no	—
180	#103	—	101	2	no	—
181	#103	—	101	2	no	—
182	#102	—	101	2	no	—
183	#102	—	101	2	no	—
184	C₂²⋊C₈	—	101	1	no	—
185	C₂²⋊C₈	—	101	1	no	—
186	C₂²⋊C₈	—	101	2	no	—
187	C₂⁴	C₂²⋊D₁₀₁	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	202	1	no	—
197	C₂²×C₄	—	202	1	no	—
198	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₂×C₈	—	101	2	no	—
205	C₂×C₈	—	101	2	no	—
206	C₂×C₈	—	202	4	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	101	2	no	—
210	C₄⋊C₄	—	101	2	no	—
211	C₄⋊C₄	—	202	1	no	—
212	C₄⋊C₄	—	202	1	no	—
213	C₄⋊C₄	C₂²×D₁₀₁	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₁₀₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	202	2	no	—
221	C₂²⋊C₄	—	202	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	202	2	no	—
224	C₄²	—	202	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
231	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	404	2	no	—
247	C₂×C₄	—	404	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
253	C₂×C₄	#13108073	1	4	yes	—
254	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
255	C₈	—	202	2	no	—
256	C₈	—	202	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#15692236	1	2	yes	—
269	C₂²	#15692275	1	2	yes	—
270	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	404	2	no	—
277	C₄	—	404	2	no	—
278	C₄	#15692236	1	2	yes	—
279	C₄	C₂×#13108073	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#15692236	1	1	yes	characteristic
285	C₂	C₂×#15692275	1	1	yes	characteristic
286	C₂	#20265975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	#525×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	#525	D₁₀₁
hypocenter	C₁₀₁	#3091
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#15692214	C₂	semidirect
2. #20265515	C₂	semidirect
3. #20265510	C₂	semidirect
4. #20265505	C₂	semidirect
5. #20265498	C₂	semidirect
6. #20265493	C₂	semidirect
7. #20265488	C₂	semidirect
8. #20265213	C₂	semidirect
9. #20265173	C₂	semidirect
10. C₂×C₂²⋊#11717579	C₂	semidirect
21. #15692188	C₂²	semidirect
22. #15692188	C₂²	semidirect
23. #15692186	C₂²	semidirect
24. #15692186	C₂²	semidirect
25. #15692179	C₂²	semidirect
26. #15692179	C₂²	semidirect
27. #15692176	C₂²	semidirect
28. #15692176	C₂²	semidirect
144. C₁₀₁	#3091	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692214	2
#20265515	2
#20265510	2
#20265505	2
#20265498	2
#20265493	2
#20265488	2
#20265213	2
#20265173	2
C₂×C₂²⋊#11717579	2
#525×C₁₀₁	2
#3091	101