Subgroups — #27173966

Order 12928.

820 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692220	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265084	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265081	C₂	1	2	yes	maximal, complemented
6	#20265081	C₂	1	2	yes	maximal, complemented
7	#20265036	C₂	1	2	yes	maximal
8	#20265034	C₂	1	1	yes	maximal, characteristic
9	C₄×#13108051	C₂	1	1	yes	maximal, characteristic
10	#20265024	C₂	1	2	yes	maximal
11	#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂×Q₈×D₁₀₁	—	2	4	no	—
15	C₂×Q₈×D₁₀₁	—	2	4	no	—
16	C₂×Q₈×D₁₀₁	C₄	1	8	yes	complemented
17	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₀₂	—	2	4	no	—
20	#15692220	—	2	4	no	—
21	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#13108053	C₂²	1	2	yes	—
23	C₂×#13108053	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂×#13108051	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	1	yes	characteristic
32	#15692162	C₂²	1	1	yes	characteristic
33	#15692162	C₂²	1	2	yes	—
34	#15692162	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	2	yes	—
39	#95×C₁₀₁	C₂²	1	2	yes	—
40	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₂₀₂	—	2	4	no	—
42	C₂×Dic₂₀₂	—	2	4	no	—
43	C₂×Dic₂₀₂	C₂×C₄	1	8	yes	—
44	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
45	C₂×Dic₂₀₂	D₄	1	4	yes	—
46	C₂×Dic₂₀₂	D₄	1	4	yes	—
47	Q₈×D₁₀₁	—	2	32	no	—
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
53	C₂×C₄×D₁₀₁	—	2	2	no	—
54	C₂×C₄×D₁₀₁	—	2	2	no	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	—	2	4	no	—
57	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
58	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	—
59	C₄⋊Dic₁₀₁	—	2	2	no	—
60	#13108053	—	2	4	no	—
61	#13108053	—	2	4	no	—
62	#13108051	—	2	4	no	—
63	#13108051	—	2	2	no	—
64	#13108051	Q₈	1	4	yes	complemented
65	C₄×Dic₁₀₁	—	2	2	no	—
66	C₄×Dic₁₀₁	—	2	4	no	—
67	C₄×Dic₁₀₁	—	2	4	no	—
68	C₂×Q₈×C₁₀₁	—	2	2	no	—
69	C₂×Q₈×C₁₀₁	—	2	2	no	—
70	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₁	—	2	2	no	—
77	C₄²×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₀₁	—	2	1	no	—
80	C₂²×D₁₀₁	—	2	1	no	—
81	C₂²×D₁₀₁	—	2	1	no	—
82	C₂²×D₁₀₁	—	2	1	no	—
83	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
85	C₂²×D₁₀₁	#41	1	2	yes	—
86	Q₈×C₁₀₁	—	2	8	no	—
87	C₂×C₄×C₁₀₁	—	2	1	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	—	2	1	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	1	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	1	no	—
97	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
98	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	complemented
102	C₂×Dic₁₀₁	—	2	1	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	1	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	1	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
113	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
114	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
115	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
116	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	complemented
117	C₄×D₁₀₁	—	2	8	no	—
118	C₄×D₁₀₁	—	2	16	no	—
119	Dic₂₀₂	—	2	16	no	—
120	Dic₂₀₂	C₂²⋊C₄	1	16	yes	complemented
121	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
122	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×D₁₀₁	—	2	2	no	—
129	C₂×D₁₀₁	—	2	2	no	—
130	C₂×D₁₀₁	—	2	2	no	—
131	C₂×D₁₀₁	—	2	2	no	—
132	C₂×D₁₀₁	—	2	2	no	—
133	C₂×D₁₀₁	—	2	2	no	—
134	C₂×D₁₀₁	C₄×Q₈	1	4	yes	complemented
135	C₄×C₁₀₁	—	2	4	no	—
136	C₄×C₁₀₁	—	2	2	no	—
137	C₄×C₁₀₁	—	4	2	no	—
138	C₄×C₁₀₁	—	4	1	no	—
139	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
140	Dic₁₀₁	—	2	2	no	—
141	Dic₁₀₁	—	2	2	no	—
142	Dic₁₀₁	—	4	1	no	—
143	Dic₁₀₁	—	4	2	no	—
144	Dic₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
145	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₂×C₁₀₁	#519	1	1	yes	characteristic
148	C₂×C₁₀₁	#548	1	1	yes	characteristic
149	C₂×C₁₀₁	#548	1	1	yes	characteristic
150	C₂×C₁₀₁	D₄×Q₈	1	1	yes	characteristic
151	C₂×C₁₀₁	D₄×Q₈	1	1	yes	characteristic
152	D₁₀₁	—	2	4	no	—
153	Q₈×C₂²⋊C₄	—	101	1	no	maximal, Sylow
154	C₁₀₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
155	C₂³×Q₈	—	101	1	no	—
156	C₂×C₄×Q₈	—	101	1	no	—
157	C₂×C₄×Q₈	—	101	1	no	—
158	#391	—	101	1	no	—
159	#391	—	101	2	no	—
160	#391	—	101	2	no	—
161	#391	D₁₀₁	1	1	yes	characteristic, complemented
162	#380	—	101	1	no	—
163	#380	—	101	2	no	—
164	C₄×C₂²⋊C₄	—	101	1	no	—
165	C₄×C₂²⋊C₄	—	101	2	no	—
166	C₂²×Q₈	—	101	8	no	—
167	C₂²×Q₈	—	101	1	no	—
168	C₂²×Q₈	—	101	2	no	—
169	C₂²×Q₈	—	202	4	no	—
170	C₂²×Q₈	—	202	4	no	—
171	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
172	C₂³×C₄	—	101	2	no	—
173	C₂³×C₄	—	101	1	no	—
174	C₄×Q₈	—	202	4	no	—
175	C₄×Q₈	—	202	4	no	—
176	C₂×C₄⋊C₄	—	101	2	no	—
177	C₂×C₄⋊C₄	—	101	2	no	—
178	C₂×C₄⋊C₄	—	101	1	no	—
179	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	101	1	no	—
181	C₂×C₂²⋊C₄	—	101	2	no	—
182	C₂×C₂²⋊C₄	—	101	1	no	—
183	C₂×C₄²	—	101	2	no	—
184	C₂×C₄²	—	101	2	no	—
185	C₂×C₄²	—	101	1	no	—
186	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
187	#95	—	101	1	no	—
188	#95	—	101	1	no	—
189	#95	—	101	2	no	—
190	#95	—	101	2	no	—
191	#95	—	101	2	no	—
192	#95	C₂×D₁₀₁	1	2	yes	—
193	#95	C₂×D₁₀₁	1	2	yes	—
194	C₂⁴	—	101	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	101	4	no	—
198	C₂×Q₈	—	101	8	no	—
199	C₂×Q₈	—	101	4	no	—
200	C₂×Q₈	—	101	4	no	—
201	C₂×Q₈	—	202	4	no	—
202	C₂×Q₈	—	202	4	no	—
203	C₂×Q₈	—	202	32	no	—
204	C₂×Q₈	C₄×D₁₀₁	1	4	yes	—
205	C₂²×C₄	—	101	2	no	—
206	C₂²×C₄	—	101	1	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	8	no	—
209	C₂²×C₄	—	101	4	no	—
210	C₂²×C₄	—	101	1	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	202	4	no	—
213	C₂²×C₄	—	202	4	no	—
214	C₂²×C₄	—	202	2	no	—
215	C₂²×C₄	—	202	2	no	—
216	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	202	4	no	—
223	C₄⋊C₄	—	202	4	no	—
224	C₄⋊C₄	—	202	2	no	—
225	C₂²⋊C₄	—	101	4	no	—
226	C₂²⋊C₄	—	202	4	no	—
227	C₂²⋊C₄	—	202	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	202	4	no	—
230	C₄²	—	202	4	no	—
231	C₄²	—	202	2	no	—
232	C₂³	—	101	2	no	—
233	C₂³	—	101	2	no	—
234	C₂³	—	101	2	no	—
235	C₂³	—	202	1	no	—
236	C₂³	—	202	1	no	—
237	C₂³	—	202	1	no	—
238	C₂³	—	202	1	no	—
239	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	Q₈	—	101	16	no	—
242	Q₈	—	202	16	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	4	no	—
255	C₂×C₄	—	101	4	no	—
256	C₂×C₄	—	101	2	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	202	16	no	—
259	C₂×C₄	—	202	1	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	1	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	1	no	—
264	C₂×C₄	—	202	2	no	—
265	C₂×C₄	—	202	1	no	—
266	C₂×C₄	—	202	1	no	—
267	C₂×C₄	—	202	1	no	—
268	C₂×C₄	—	202	1	no	—
269	C₂×C₄	—	202	8	no	—
270	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
271	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
272	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
273	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
274	C₂×C₄	Q₈×D₁₀₁	1	2	yes	complemented
275	C₂²	—	101	4	no	—
276	C₂²	—	202	2	no	—
277	C₂²	—	202	2	no	—
278	C₂²	—	202	2	no	—
279	C₂²	—	202	2	no	—
280	C₂²	—	202	2	no	—
281	C₂²	—	202	2	no	—
282	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
283	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
284	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
285	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
286	C₂²	#41×D₁₀₁	1	1	yes	characteristic
287	C₂²	#15692302	1	1	yes	characteristic
288	C₂²	#15692303	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	101	8	no	—
294	C₄	—	404	1	no	—
295	C₄	—	404	2	no	—
296	C₄	—	202	2	no	—
297	C₄	—	202	2	no	—
298	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
299	C₂	—	202	4	no	—
300	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
301	C₂	C₄×Q₈×D₁₀₁	1	1	yes	characteristic
302	C₂	#20265653	1	1	yes	characteristic
303	C₂	#20265835	1	1	yes	characteristic
304	C₂	#20265837	1	1	yes	characteristic
305	C₂	#20265840	1	1	yes	characteristic
306	C₂	#20265842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#391×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#391	D₁₀₁
hypocenter	C₁₀₁	Q₈×C₂²⋊C₄
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₀₂	C₂	semidirect
3. C₂×#15692220	C₂	semidirect
4. #20265084	C₂	semidirect
5. #20265081	C₂	semidirect
6. #20265081	C₂	semidirect
11. #391×C₁₀₁	C₂	semidirect
16. C₂×Q₈×D₁₀₁	C₄	semidirect
64. #13108051	Q₈	semidirect
101. C₂×C₄×C₁₀₁	C₂×Q₈	semidirect
116. C₂×Dic₁₀₁	C₂×Q₈	semidirect
120. Dic₂₀₂	C₂²⋊C₄	semidirect
134. C₂×D₁₀₁	C₄×Q₈	semidirect
154. C₁₀₁	Q₈×C₂²⋊C₄	semidirect
161. #391	D₁₀₁	semidirect
274. C₂×C₄	Q₈×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₀₁	2
C₂×C₄×Dic₂₀₂	2
C₂×#15692220	2
#20265084	2
#20265081	2
#20265036	2
#20265034	2
C₄×#13108051	2
#20265024	2
#391×C₁₀₁	2
Q₈×C₂²⋊C₄	101