Subgroups — #27179650

Order 12928.

772 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108075	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265186	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265183	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265181	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265177	C₂	1	4	yes	maximal, complemented
8	#20265172	C₂	1	4	yes	maximal, complemented
9	#419×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108075	—	2	4	no	—
11	C₂×#13108075	—	2	1	no	—
12	C₂×#13108075	—	2	1	no	—
13	C₂×#13108075	C₄	1	8	yes	complemented
14	C₂×#13108075	C₂²	1	2	yes	—
15	C₂×#13108075	C₂²	1	2	yes	—
16	C₂²×D₄₀₄	—	2	1	no	—
17	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₀₁	—	2	1	no	—
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108058	C₂²	1	1	yes	characteristic
21	C₂×#13108057	C₂²	1	2	yes	complemented
22	M₁₆×D₁₀₁	—	2	4	no	—
23	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
24	#15692214	C₂²	1	1	yes	characteristic
25	#15692213	C₂²	1	2	yes	complemented
26	#15692212	—	2	1	no	—
27	#15692211	—	2	1	no	—
28	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×D₁₀₁	—	2	1	no	—
30	C₄²×D₁₀₁	—	2	1	no	—
31	#15692179	C₂²	1	4	yes	complemented
32	#15692177	C₂²	1	4	yes	complemented
33	#15692175	C₂²	1	4	yes	complemented
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
36	#117×C₁₀₁	C₂²	1	1	yes	characteristic
37	#103×C₁₀₁	C₂²	1	4	yes	complemented
38	C₂³×D₁₀₁	—	2	1	no	—
39	C₂³×D₁₀₁	D₄	1	1	yes	characteristic
40	#13108075	—	2	8	no	—
41	#13108075	—	2	8	no	—
42	#13108075	—	2	4	no	—
43	#13108075	—	2	4	no	—
44	#13108075	C₂×C₄	1	8	yes	—
45	#13108075	C₂×C₄	1	8	yes	complemented
46	C₂×D₄₀₄	—	2	8	no	—
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	C₂×C₄	1	4	yes	complemented
52	C₂×D₄₀₄	C₂³	1	2	yes	—
53	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₁	—	2	8	no	—
55	C₂×C₄×D₁₀₁	—	2	2	no	—
56	C₂×C₄×D₁₀₁	—	2	2	no	—
57	C₂×C₄×D₁₀₁	—	2	2	no	—
58	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
59	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
60	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
64	#13108058	—	2	2	no	—
65	#13108057	—	2	4	no	—
66	C₂×#11717579	C₂³	1	2	yes	—
67	C₈×D₁₀₁	—	2	4	no	—
68	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
69	#13108053	—	2	2	no	—
70	C₂²⋊Dic₁₀₁	—	2	1	no	—
71	#13108051	—	2	2	no	—
72	C₄×Dic₁₀₁	—	2	1	no	—
73	C₄×Dic₁₀₁	—	2	2	no	—
74	C₂×Q₈×C₁₀₁	—	2	2	no	—
75	C₂×Q₈×C₁₀₁	—	2	1	no	—
76	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
77	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	—	2	1	no	—
79	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₀₁	—	2	2	no	—
81	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
84	C₄²×C₁₀₁	—	2	1	no	—
85	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	4	2	no	—
89	C₂²×D₁₀₁	—	4	4	no	—
90	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
91	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
93	Q₈×C₁₀₁	—	2	4	no	—
94	Q₈×C₁₀₁	—	2	2	no	—
95	Q₈×C₁₀₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₁₀₁	—	2	4	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
105	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
107	D₄₀₄	—	2	8	no	—
108	D₄₀₄	—	2	8	no	—
109	D₄₀₄	—	2	4	no	—
110	D₄₀₄	—	2	2	no	—
111	D₄₀₄	C₂²×C₄	1	4	yes	—
112	C₄×D₁₀₁	—	2	8	no	—
113	C₄×D₁₀₁	—	2	8	no	—
114	C₄×D₁₀₁	—	4	4	no	—
115	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
116	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
117	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
118	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
119	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
120	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
121	C₈×C₁₀₁	—	2	2	no	—
122	#11717579	—	2	2	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	2	no	—
127	C₂×D₁₀₁	—	2	2	no	—
128	C₂×D₁₀₁	—	2	4	no	—
129	C₂×D₁₀₁	—	4	4	no	—
130	C₂×D₁₀₁	—	4	8	no	—
131	C₂×D₁₀₁	—	4	4	no	—
132	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₄×C₁₀₁	—	2	4	no	—
135	C₄×C₁₀₁	—	4	2	no	—
136	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
137	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
138	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
139	Dic₁₀₁	—	4	2	no	—
140	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
141	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₁₀₁	—	2	2	no	—
143	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₁₀₁	#575	1	2	yes	—
145	D₁₀₁	—	2	2	no	—
146	D₁₀₁	—	2	2	no	—
147	D₁₀₁	—	4	4	no	—
148	#2896	—	101	1	no	maximal, Sylow
149	C₁₀₁	#2896	1	1	yes	characteristic, complemented, Sylow
150	C₂²×#41	—	101	1	no	—
151	C₂²×M₁₆	—	101	1	no	—
152	C₂×#117	—	101	1	no	—
153	#419	—	101	1	no	—
154	#419	D₁₀₁	1	1	yes	characteristic, complemented
155	#418	—	101	1	no	—
156	#418	—	101	1	no	—
157	#417	—	101	4	no	—
158	#416	—	101	4	no	—
159	C₂×#41	—	101	8	no	—
160	C₂×#41	—	101	2	no	—
161	C₂×#41	—	101	2	no	—
162	C₂×#41	—	202	1	no	—
163	C₂×#41	—	202	1	no	—
164	C₂×#41	—	202	4	no	—
165	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂²×D₄	—	101	1	no	—
167	C₂²×D₄	—	202	1	no	—
168	C₂³×C₄	—	101	1	no	—
169	C₂³×C₄	—	202	1	no	—
170	C₂×M₁₆	—	101	1	no	—
171	C₂×M₁₆	—	101	2	no	—
172	C₂×M₁₆	—	202	4	no	—
173	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
174	C₂²×C₈	—	101	2	no	—
175	#117	—	101	1	no	—
176	#117	—	101	2	no	—
177	#117	—	202	1	no	—
178	#117	—	202	1	no	—
179	#117	C₂×D₁₀₁	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	101	2	no	—
181	C₂×C₂²⋊C₄	—	202	1	no	—
182	C₂×C₄²	—	202	1	no	—
183	#103	—	101	4	no	—
184	#103	C₂×D₁₀₁	1	4	yes	complemented
185	#102	—	101	4	no	—
186	#102	—	101	4	no	—
187	C₂⁴	—	101	1	no	—
188	C₂⁴	—	202	1	no	—
189	#41	—	101	8	no	—
190	#41	—	101	8	no	—
191	#41	—	202	4	no	—
192	#41	—	202	4	no	—
193	#41	—	202	8	no	—
194	#41	—	202	8	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
198	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
199	C₂×D₄	—	101	4	no	—
200	C₂×D₄	—	101	2	no	—
201	C₂×D₄	—	202	2	no	—
202	C₂×D₄	—	202	2	no	—
203	C₂×D₄	—	202	8	no	—
204	C₂×D₄	—	202	2	no	—
205	C₂×D₄	—	202	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	4	no	—
209	C₂²×C₄	—	101	1	no	—
210	C₂²×C₄	—	101	1	no	—
211	C₂²×C₄	—	101	4	no	—
212	C₂²×C₄	—	101	1	no	—
213	C₂²×C₄	—	101	1	no	—
214	C₂²×C₄	—	202	2	no	—
215	C₂²×C₄	—	202	2	no	—
216	C₂²×C₄	—	202	8	no	—
217	C₂²×C₄	—	202	2	no	—
218	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
219	M₁₆	—	2	2	no	—
220	M₁₆	—	202	4	no	—
221	M₁₆	—	202	2	no	—
222	C₂×C₈	—	101	2	no	—
223	C₂×C₈	—	202	4	no	—
224	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
225	C₄⋊C₄	—	101	2	no	—
226	C₄⋊C₄	—	202	2	no	—
227	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	202	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	202	2	no	—
233	C₄²	—	202	1	no	—
234	C₂³	—	101	4	no	—
235	C₂³	—	101	1	no	—
236	C₂³	—	101	1	no	—
237	C₂³	—	404	4	no	—
238	C₂³	—	404	2	no	—
239	C₂³	—	202	2	no	—
240	C₂³	—	202	4	no	—
241	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	2	4	no	—
244	Q₈	C₂×C₄×D₁₀₁	1	4	yes	—
245	D₄	—	101	4	no	—
246	D₄	—	202	8	no	—
247	D₄	—	202	8	no	—
248	D₄	—	202	2	no	—
249	D₄	—	202	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	101	1	no	—
254	C₂×C₄	—	101	1	no	—
255	C₂×C₄	—	101	4	no	—
256	C₂×C₄	—	101	2	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	101	2	no	—
259	C₂×C₄	—	101	2	no	—
260	C₂×C₄	—	101	4	no	—
261	C₂×C₄	—	101	4	no	—
262	C₂×C₄	—	404	4	no	—
263	C₂×C₄	—	202	8	no	—
264	C₂×C₄	—	202	8	no	—
265	C₂×C₄	—	202	2	no	—
266	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
267	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
268	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
269	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	202	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	101	2	no	—
274	C₂²	—	101	2	no	—
275	C₂²	—	404	8	no	—
276	C₂²	—	404	4	no	—
277	C₂²	—	404	4	no	—
278	C₂²	—	202	4	no	—
279	C₂²	—	202	2	no	—
280	C₂²	—	202	2	no	—
281	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
282	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	4	2	no	—
285	C₄	—	101	2	no	—
286	C₄	—	101	2	no	—
287	C₄	—	404	2	no	—
288	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
289	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
290	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	404	4	no	—
293	C₂	—	202	2	no	—
294	C₂	—	202	2	no	—
295	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
296	C₂	#20265976	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#419×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂²×#13108075	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#419	D₁₀₁
hypocenter	C₁₀₁	#2896
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×D₁₀₁	C₂	semidirect
3. #117×D₁₀₁	C₂	semidirect
4. #20265186	C₂	semidirect
5. #20265183	C₂	semidirect
6. #20265181	C₂	semidirect
7. #20265177	C₂	semidirect
8. #20265172	C₂	semidirect
9. #419×C₁₀₁	C₂	semidirect
13. C₂×#13108075	C₄	semidirect
21. C₂×#13108057	C₂²	semidirect
23. C₂×C₈×D₁₀₁	C₂²	semidirect
25. #15692213	C₂²	semidirect
28. C₄⋊C₄×D₁₀₁	C₂²	semidirect
31. #15692179	C₂²	semidirect
32. #15692177	C₂²	semidirect
33. #15692175	C₂²	semidirect
37. #103×C₁₀₁	C₂²	semidirect
45. #13108075	C₂×C₄	semidirect
51. C₂×D₄₀₄	C₂×C₄	semidirect
76. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
149. C₁₀₁	#2896	semidirect
154. #419	D₁₀₁	semidirect
184. #103	C₂×D₁₀₁	semidirect
197. C₂×Q₈	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108075	2
C₂×M₁₆×D₁₀₁	2
#117×D₁₀₁	2
#20265186	2
#20265183	2
#20265181	2
#20265177	2
#20265172	2
#419×C₁₀₁	2
#2896	101