Subgroups — #27180671

Order 12928.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692280	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692275	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265746	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265399	C₂	1	2	yes	maximal, complemented
5	#20265392	C₂	1	2	yes	maximal, complemented
6	#20265377	C₂	1	2	yes	maximal, complemented
7	#20265372	C₂	1	2	yes	maximal, complemented
8	#20265204	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692175	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	#15692280	—	2	4	no	—
15	#15692275	—	2	4	no	—
16	C₂×#13108064	C₂²	1	1	yes	characteristic, complemented
17	C₂×#13108062	C₂²	1	1	yes	characteristic, complemented
18	C₂×#13108061	C₂²	1	2	yes	—
19	C₂×#13108061	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13108059	C₂²	1	2	yes	—
21	C₂×#13108059	C₂²	1	1	yes	characteristic, complemented
22	C₂×#13108058	C₂²	1	1	yes	characteristic
23	C₂²×#11717579	C₂²	1	1	yes	characteristic
24	#15692252	C₂²	1	2	yes	—
25	#15692243	C₂²	1	2	yes	—
26	#15692230	—	2	1	no	—
27	#15692230	—	2	1	no	—
28	#15692225	—	2	2	no	—
29	C₄⋊#11717579	C₂²	1	2	yes	complemented
30	C₄⋊#11717579	C₂²	1	2	yes	complemented
31	#15692178	C₂²	1	2	yes	complemented
32	#15692175	C₂²	1	2	yes	—
33	#15692175	C₂²	1	2	yes	—
34	#15692175	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	complemented
37	C₄×D₄×C₁₀₁	—	2	2	no	—
38	C₄×D₄×C₁₀₁	C₂²	1	2	yes	complemented
39	#117×C₁₀₁	—	2	1	no	—
40	#117×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂³×D₁₀₁	—	2	1	no	—
44	#13108074	—	2	2	no	—
45	#13108074	—	2	2	no	—
46	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
47	C₂×D₄₀₄	—	2	2	no	—
48	C₂×D₄₀₄	—	2	2	no	—
49	C₂×D₄₀₄	—	2	1	no	—
50	C₂×D₄₀₄	—	2	2	no	—
51	C₂×D₄₀₄	—	2	2	no	—
52	C₂×D₄₀₄	C₂³	1	2	yes	—
53	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
55	C₂×C₄×D₁₀₁	—	2	1	no	—
56	#13108064	—	2	2	no	—
57	#13108062	—	2	2	no	—
58	#13108061	—	2	4	no	—
59	#13108061	—	2	2	no	—
60	#13108059	—	2	4	no	—
61	#13108059	—	2	2	no	—
62	#13108058	—	2	2	no	—
63	C₂×#11717579	—	2	2	no	—
64	C₂×#11717579	C₂³	1	1	yes	characteristic
65	C₂×#11717579	C₂³	1	1	yes	characteristic
66	C₂×#11717579	C₂³	1	2	yes	—
67	#13108051	—	2	2	no	—
68	#13108051	—	2	2	no	—
69	#41×C₁₀₁	—	2	2	no	—
70	#41×C₁₀₁	D₄	1	4	yes	complemented
71	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₀₁	—	2	1	no	—
73	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	—	2	1	no	—
75	C₂²×C₄×C₁₀₁	—	2	1	no	—
76	C₂²×C₄×C₁₀₁	—	2	2	no	—
77	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	—	2	1	no	—
79	C₄⋊C₄×C₁₀₁	—	2	2	no	—
80	C₄⋊C₄×C₁₀₁	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₀₁	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
84	C₂²⋊C₄×C₁₀₁	D₄	1	2	yes	—
85	C₄²×C₁₀₁	—	2	2	no	—
86	C₄²×C₁₀₁	C₂³	1	2	yes	—
87	C₂³×C₁₀₁	—	2	1	no	—
88	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₀₁	—	2	2	no	—
90	C₂²×D₁₀₁	—	2	1	no	—
91	C₂²×D₁₀₁	—	4	2	no	—
92	C₂²×D₁₀₁	—	8	1	no	—
93	C₂²×D₁₀₁	—	8	1	no	—
94	Q₈×C₁₀₁	—	2	1	no	—
95	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
96	D₄×C₁₀₁	—	2	4	no	—
97	D₄×C₁₀₁	—	2	1	no	—
98	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	8	2	no	—
103	C₂×C₄×C₁₀₁	—	8	1	no	—
104	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₁	#41	1	2	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₀₁	—	4	2	no	—
111	C₂×Dic₁₀₁	—	2	1	no	—
112	D₄₀₄	—	2	2	no	—
113	D₄₀₄	—	2	1	no	—
114	D₄₀₄	—	2	2	no	—
115	D₄₀₄	—	2	1	no	—
116	D₄₀₄	—	4	2	no	—
117	D₄₀₄	—	8	2	no	—
118	D₄₀₄	—	8	2	no	—
119	C₄×D₁₀₁	—	4	2	no	—
120	Dic₂₀₂	—	2	1	no	—
121	Dic₂₀₂	—	2	1	no	—
122	#11717579	—	2	1	no	—
123	#11717579	—	2	1	no	—
124	#11717579	—	2	2	no	—
125	C₂²×C₁₀₁	—	2	1	no	—
126	C₂²×C₁₀₁	—	2	1	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	4	1	no	—
129	C₂²×C₁₀₁	—	4	1	no	—
130	C₂²×C₁₀₁	#121	1	2	yes	—
131	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₀₁	—	4	1	no	—
133	C₂×D₁₀₁	—	4	2	no	—
134	C₂×D₁₀₁	—	4	1	no	—
135	C₂×D₁₀₁	—	4	2	no	—
136	C₂×D₁₀₁	—	4	1	no	—
137	C₂×D₁₀₁	—	4	2	no	—
138	C₂×D₁₀₁	—	8	4	no	—
139	C₄×C₁₀₁	—	2	2	no	—
140	C₄×C₁₀₁	—	2	2	no	—
141	C₄×C₁₀₁	—	2	2	no	—
142	C₄×C₁₀₁	—	4	2	no	—
143	C₄×C₁₀₁	#121	1	2	yes	—
144	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
146	Dic₁₀₁	—	4	1	no	—
147	C₂×C₁₀₁	—	2	2	no	—
148	C₂×C₁₀₁	—	4	2	no	—
149	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
150	C₂×C₁₀₁	#576	1	1	yes	characteristic
151	C₂×C₁₀₁	#577	1	1	yes	characteristic
152	D₁₀₁	—	8	1	no	—
153	D₁₀₁	—	8	2	no	—
154	#3045	—	101	1	no	maximal, Sylow
155	C₁₀₁	#3045	1	1	yes	characteristic, complemented, Sylow
156	C₂×AGL(1,ℤ/8)	—	101	1	no	—
157	C₂×#135	—	101	1	no	—
158	#535	—	101	1	no	—
159	#520	D₁₀₁	1	1	yes	characteristic, complemented
160	#464	—	101	2	no	—
161	#463	—	101	2	no	—
162	#460	—	101	2	no	—
163	#459	—	101	2	no	—
164	#424	—	101	1	no	—
165	#417	—	101	1	no	—
166	C₂×#102	—	101	1	no	—
167	C₂×#41	—	101	1	no	—
168	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
169	C₂²×D₄	—	101	1	no	—
170	AGL(1,ℤ/8)	—	202	4	no	—
171	#135	—	202	4	no	—
172	C₂×Q₁₆	—	101	1	no	—
173	C₂×SD₁₆	—	101	1	no	—
174	C₂×SD₁₆	—	101	2	no	—
175	C₂×SD₁₆	—	101	1	no	—
176	C₂×D₈	—	101	1	no	—
177	C₂×D₈	—	101	2	no	—
178	C₂×M₁₆	—	101	1	no	—
179	C₂²×C₈	—	101	1	no	—
180	#127	—	101	2	no	—
181	#124	—	101	2	no	—
182	#121	—	202	1	no	—
183	#121	—	202	1	no	—
184	#120	—	202	2	no	—
185	C₄×Q₈	C₂×D₁₀₁	1	2	yes	complemented
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	C₂×D₁₀₁	1	2	yes	complemented
188	#117	—	2	1	no	—
189	#117	C₂×D₁₀₁	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
192	C₄⋊C₈	—	101	2	no	—
193	C₄⋊C₈	—	101	2	no	—
194	#103	—	101	2	no	—
195	#102	—	101	2	no	—
196	#102	—	101	2	no	—
197	#102	—	101	2	no	—
198	C₂⁴	—	202	1	no	—
199	#41	—	2	2	no	—
200	#41	—	202	2	no	—
201	#41	—	202	2	no	—
202	#41	D₄₀₄	1	4	yes	complemented
203	C₂×Q₈	—	101	1	no	—
204	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	101	2	no	—
207	C₂×D₄	—	101	1	no	—
208	C₂×D₄	—	202	1	no	—
209	C₂×D₄	—	202	2	no	—
210	C₂×D₄	—	202	2	no	—
211	C₂×D₄	—	202	1	no	—
212	C₂×D₄	—	202	2	no	—
213	C₂×D₄	—	202	2	no	—
214	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	202	1	no	—
219	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
220	Q₁₆	—	202	2	no	—
221	SD₁₆	—	202	2	no	—
222	SD₁₆	—	202	4	no	—
223	SD₁₆	—	202	2	no	—
224	D₈	—	202	2	no	—
225	D₈	—	202	4	no	—
226	M₁₆	—	202	2	no	—
227	C₂×C₈	—	101	1	no	—
228	C₂×C₈	—	101	1	no	—
229	C₂×C₈	—	101	2	no	—
230	C₂×C₈	—	202	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
234	C₄⋊C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
235	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	202	2	no	—
238	C₂²⋊C₄	—	202	2	no	—
239	C₂²⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂²×D₁₀₁	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	808	1	no	—
244	C₂³	—	808	1	no	—
245	C₂³	—	404	2	no	—
246	C₂³	—	202	2	no	—
247	C₂³	—	202	1	no	—
248	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	202	1	no	—
251	Q₈	—	202	1	no	—
252	Q₈	C₂×D₄₀₄	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	808	2	no	—
256	D₄	—	808	2	no	—
257	D₄	—	404	2	no	—
258	D₄	—	404	2	no	—
259	D₄	—	202	2	no	—
260	D₄	—	202	1	no	—
261	D₄	—	202	2	no	—
262	D₄	—	202	1	no	—
263	D₄	C₂×D₄₀₄	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	404	2	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
275	C₂×C₄	#13108074	1	2	yes	—
276	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
277	C₈	—	202	1	no	—
278	C₈	—	202	1	no	—
279	C₈	—	202	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	808	4	no	—
286	C₂²	—	404	1	no	—
287	C₂²	—	404	2	no	—
288	C₂²	—	404	1	no	—
289	C₂²	—	404	2	no	—
290	C₂²	—	404	1	no	—
291	C₂²	—	404	2	no	—
292	C₂²	#15692229	1	2	yes	—
293	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	404	1	no	—
299	C₄	#15692229	1	2	yes	—
300	C₄	C₂²×D₄₀₄	1	1	yes	characteristic
301	C₄	C₂×#13108074	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	808	2	no	—
305	C₂	—	808	1	no	—
306	C₂	C₂×#15692229	1	1	yes	characteristic
307	C₂	#20265980	1	1	yes	characteristic
308	C₂	#20265985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#520×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	#520	D₁₀₁
hypocenter	C₁₀₁	#3045
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#15692280	C₂	semidirect
2. C₂×#15692275	C₂	semidirect
3. #20265746	C₂	semidirect
4. #20265399	C₂	semidirect
5. #20265392	C₂	semidirect
6. #20265377	C₂	semidirect
7. #20265372	C₂	semidirect
8. #20265204	C₂	semidirect
9. #20265176	C₂	semidirect
10. C₂×#15692175	C₂	semidirect
11. #520×C₁₀₁	C₂	semidirect
16. C₂×#13108064	C₂²	semidirect
17. C₂×#13108062	C₂²	semidirect
19. C₂×#13108061	C₂²	semidirect
21. C₂×#13108059	C₂²	semidirect
29. C₄⋊#11717579	C₂²	semidirect
30. C₄⋊#11717579	C₂²	semidirect
31. #15692178	C₂²	semidirect
34. #15692175	C₂²	semidirect
36. C₄×Q₈×C₁₀₁	C₂²	semidirect
38. C₄×D₄×C₁₀₁	C₂²	semidirect
70. #41×C₁₀₁	D₄	semidirect
155. C₁₀₁	#3045	semidirect
159. #520	D₁₀₁	semidirect
185. C₄×Q₈	C₂×D₁₀₁	semidirect
187. C₄×D₄	C₂×D₁₀₁	semidirect
202. #41	D₄₀₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692280	2
C₂×#15692275	2
#20265746	2
#20265399	2
#20265392	2
#20265377	2
#20265372	2
#20265204	2
#20265176	2
C₂×#15692175	2
#520×C₁₀₁	2
#3045	101