Subgroups — #27180682

Order 12928.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692285	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692275	C₂	1	1	yes	maximal, characteristic
3	#20265753	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265399	C₂	1	2	yes	maximal, complemented
5	#20265392	C₂	1	2	yes	maximal, complemented
6	#20265387	C₂	1	2	yes	maximal, complemented
7	#20265382	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×#15692178	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×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	#15692285	—	2	4	no	—
15	#15692275	—	2	4	no	—
16	C₂×#13108064	C₂²	1	1	yes	characteristic
17	C₂×#13108064	C₂²	1	2	yes	—
18	C₂×#13108062	C₂²	1	1	yes	characteristic
19	C₂×#13108062	C₂²	1	2	yes	—
20	C₂×#13108061	C₂²	1	1	yes	characteristic
21	C₂×#13108059	C₂²	1	1	yes	characteristic
22	C₂×#13108058	C₂²	1	1	yes	characteristic
23	C₂²×#11717579	C₂²	1	1	yes	characteristic
24	#15692255	C₂²	1	2	yes	—
25	#15692243	C₂²	1	2	yes	—
26	#15692239	—	2	2	no	—
27	#15692235	—	2	1	no	—
28	#15692235	—	2	1	no	—
29	C₄⋊#11717579	C₂²	1	2	yes	—
30	C₄⋊#11717579	C₂²	1	2	yes	—
31	#15692178	C₂²	1	2	yes	—
32	#15692178	C₂²	1	2	yes	—
33	#15692178	C₂²	1	2	yes	—
34	#15692175	C₂²	1	2	yes	—
35	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	—
37	C₄×D₄×C₁₀₁	—	2	2	no	—
38	C₄×D₄×C₁₀₁	C₂²	1	2	yes	—
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	#13108074	—	2	2	no	—
44	#13108074	—	2	2	no	—
45	C₂×Dic₂₀₂	—	2	2	no	—
46	C₂×Dic₂₀₂	—	2	2	no	—
47	C₂×Dic₂₀₂	—	2	1	no	—
48	C₂×Dic₂₀₂	C₂³	1	2	yes	—
49	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
50	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
52	C₂²×Dic₁₀₁	—	2	1	no	—
53	C₂×C₄×D₁₀₁	—	2	1	no	—
54	#13108064	—	2	2	no	—
55	#13108064	—	2	4	no	—
56	#13108062	—	2	2	no	—
57	#13108062	—	2	4	no	—
58	#13108061	—	2	2	no	—
59	#13108059	—	2	2	no	—
60	#13108058	—	2	2	no	—
61	C₂×#11717579	—	2	2	no	—
62	C₂×#11717579	C₂³	1	2	yes	—
63	C₂×#11717579	C₂³	1	1	yes	characteristic
64	C₂×#11717579	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₀₁	—	2	2	no	—
66	C₄⋊Dic₁₀₁	—	2	2	no	—
67	#13108051	—	2	2	no	—
68	#13108051	—	2	2	no	—
69	#41×C₁₀₁	—	2	2	no	—
70	#41×C₁₀₁	D₄	1	4	yes	—
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	1	no	—
90	Q₈×C₁₀₁	—	2	1	no	—
91	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
92	D₄×C₁₀₁	—	2	4	no	—
93	D₄×C₁₀₁	—	2	1	no	—
94	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₀₁	—	2	1	no	—
96	C₂×C₄×C₁₀₁	—	2	4	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	8	2	no	—
99	C₂×C₄×C₁₀₁	—	8	1	no	—
100	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	#41	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₀₁	—	4	2	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	4	2	no	—
110	D₄₀₄	—	2	1	no	—
111	D₄₀₄	—	2	1	no	—
112	C₄×D₁₀₁	—	4	2	no	—
113	Dic₂₀₂	—	2	2	no	—
114	Dic₂₀₂	—	2	1	no	—
115	Dic₂₀₂	—	2	2	no	—
116	Dic₂₀₂	—	2	1	no	—
117	Dic₂₀₂	—	4	2	no	—
118	#11717579	—	2	1	no	—
119	#11717579	—	2	1	no	—
120	#11717579	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	1	no	—
122	C₂²×C₁₀₁	—	2	1	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	4	1	no	—
125	C₂²×C₁₀₁	—	4	1	no	—
126	C₂²×C₁₀₁	#121	1	2	yes	—
127	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₀₁	—	4	1	no	—
129	C₂×D₁₀₁	—	4	1	no	—
130	C₂×D₁₀₁	—	4	1	no	—
131	C₄×C₁₀₁	—	2	2	no	—
132	C₄×C₁₀₁	—	2	2	no	—
133	C₄×C₁₀₁	—	2	2	no	—
134	C₄×C₁₀₁	—	4	2	no	—
135	C₄×C₁₀₁	#121	1	2	yes	—
136	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	1	no	—
140	C₂×C₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	4	2	no	—
142	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
143	C₂×C₁₀₁	#577	1	1	yes	characteristic
144	C₂×C₁₀₁	#578	1	1	yes	characteristic
145	D₁₀₁	—	8	1	no	—
146	#3046	—	101	1	no	maximal, Sylow
147	C₁₀₁	#3046	1	1	yes	characteristic, complemented, Sylow
148	C₂×#137	—	101	1	no	—
149	C₂×#135	—	101	1	no	—
150	#536	—	101	1	no	—
151	#520	D₁₀₁	1	1	yes	characteristic, complemented
152	#464	—	101	2	no	—
153	#463	—	101	2	no	—
154	#462	—	101	2	no	—
155	#461	—	101	2	no	—
156	#424	—	101	1	no	—
157	#417	—	101	1	no	—
158	C₂×#103	—	101	1	no	—
159	C₂×#41	—	101	1	no	—
160	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂²×Q₈	—	101	1	no	—
162	#137	—	202	4	no	—
163	#135	—	202	4	no	—
164	C₂×Q₁₆	—	101	1	no	—
165	C₂×Q₁₆	—	101	2	no	—
166	C₂×SD₁₆	—	101	1	no	—
167	C₂×SD₁₆	—	101	1	no	—
168	C₂×SD₁₆	—	101	2	no	—
169	C₂×D₈	—	101	1	no	—
170	C₂×M₁₆	—	101	1	no	—
171	C₂²×C₈	—	101	1	no	—
172	C₄⋊Q₈	—	101	2	no	—
173	#124	—	101	2	no	—
174	#123	—	202	2	no	—
175	C₂²⋊Q₈	—	202	1	no	—
176	C₂²⋊Q₈	—	202	1	no	—
177	C₄×Q₈	C₂×D₁₀₁	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×D₁₀₁	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	C₂×D₁₀₁	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
184	C₄⋊C₈	—	101	2	no	—
185	C₄⋊C₈	—	101	2	no	—
186	#103	—	101	2	no	—
187	#103	—	101	2	no	—
188	#103	—	101	2	no	—
189	#102	—	101	2	no	—
190	#41	—	2	2	no	—
191	#41	—	202	2	no	—
192	#41	—	202	2	no	—
193	#41	D₄₀₄	1	4	yes	—
194	C₂×Q₈	—	101	2	no	—
195	C₂×Q₈	—	101	1	no	—
196	C₂×Q₈	—	202	2	no	—
197	C₂×Q₈	—	202	2	no	—
198	C₂×Q₈	—	202	1	no	—
199	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	101	1	no	—
202	C₂×D₄	—	202	1	no	—
203	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	202	1	no	—
208	C₂²×C₄	—	202	1	no	—
209	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
210	Q₁₆	—	202	2	no	—
211	Q₁₆	—	202	4	no	—
212	SD₁₆	—	202	2	no	—
213	SD₁₆	—	202	2	no	—
214	SD₁₆	—	202	4	no	—
215	D₈	—	202	2	no	—
216	M₁₆	—	202	2	no	—
217	C₂×C₈	—	101	2	no	—
218	C₂×C₈	—	101	1	no	—
219	C₂×C₈	—	101	1	no	—
220	C₂×C₈	—	202	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	202	2	no	—
224	C₄⋊C₄	—	202	2	no	—
225	C₄⋊C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	202	2	no	—
231	C₂²⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂²×D₁₀₁	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	202	1	no	—
236	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	404	2	no	—
239	Q₈	—	202	2	no	—
240	Q₈	—	202	1	no	—
241	Q₈	—	202	2	no	—
242	Q₈	—	202	1	no	—
243	Q₈	C₂×D₄₀₄	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	404	2	no	—
247	D₄	—	202	1	no	—
248	D₄	—	202	1	no	—
249	D₄	C₂×D₄₀₄	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	404	2	no	—
256	C₂×C₄	—	404	2	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	1	no	—
259	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
263	C₂×C₄	#13108074	1	2	yes	—
264	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
265	C₈	—	202	1	no	—
266	C₈	—	202	1	no	—
267	C₈	—	202	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	404	1	no	—
274	C₂²	—	404	1	no	—
275	C₂²	—	404	1	no	—
276	C₂²	#15692229	1	2	yes	—
277	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	404	1	no	—
283	C₄	—	404	2	no	—
284	C₄	#15692229	1	2	yes	—
285	C₄	C₂²×D₄₀₄	1	1	yes	characteristic
286	C₄	C₂×#13108074	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	808	1	no	—
290	C₂	C₂×#15692229	1	1	yes	characteristic
291	C₂	#20265988	1	1	yes	characteristic
292	C₂	#20265992	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
prime radical	C₂×#15692275	C₂
hypercenter	#520	D₁₀₁
hypocenter	C₁₀₁	#3046
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×#15692285	C₂	semidirect
3. #20265753	C₂	semidirect
4. #20265399	C₂	semidirect
5. #20265392	C₂	semidirect
6. #20265387	C₂	semidirect
7. #20265382	C₂	semidirect
8. #20265204	C₂	semidirect
9. #20265176	C₂	semidirect
10. C₂×#15692178	C₂	semidirect
11. #520×C₁₀₁	C₂	semidirect
147. C₁₀₁	#3046	semidirect
151. #520	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692285	2
C₂×#15692275	2
#20265753	2
#20265399	2
#20265392	2
#20265387	2
#20265382	2
#20265204	2
#20265176	2
C₂×#15692178	2
#520×C₁₀₁	2
#3046	101