Subgroups — #27179868

Order 12928.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₀₁	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20265255	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265251	C₂	1	2	yes	maximal, complemented
5	#20265246	C₂	1	2	yes	maximal, complemented
6	C₈×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265242	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265199	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265139	C₂	1	2	yes	maximal, complemented
10	C₄×#13108057	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₀₁	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	C₂×#13108058	C₂²	1	2	yes	complemented
15	C₂×#13108057	C₂²	1	1	yes	characteristic
16	C₂×#13108057	C₂²	1	2	yes	—
17	C₂×#13108057	C₂²	1	1	yes	characteristic
18	M₁₆×D₁₀₁	—	2	8	no	—
19	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
20	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
21	C₄×D₄₀₄	C₂²	1	1	yes	characteristic
22	D₄×Dic₁₀₁	C₄	1	1	yes	characteristic
23	#15692217	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
25	#15692215	C₄	1	2	yes	—
26	C₄⋊C₄×D₁₀₁	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₁₀₁	C₄	1	2	yes	—
28	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
29	C₄⋊#11717579	C₂²	1	1	yes	characteristic, complemented
30	#15692184	C₂²	1	2	yes	complemented
31	C₂²⋊#11717579	C₂²	1	2	yes	complemented
32	#15692167	C₂²	1	2	yes	complemented
33	#15692167	C₂²	1	2	yes	complemented
34	#15692165	C₂²	1	1	yes	characteristic, complemented
35	#15692165	C₂²	1	1	yes	characteristic, complemented
36	C₄×#11717579	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	complemented
38	C₄×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₁₀₁	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
43	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
45	D₄×D₁₀₁	—	2	4	no	—
46	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
54	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
55	#13108058	—	2	4	no	—
56	#13108057	—	2	4	no	—
57	#13108057	—	2	2	no	—
58	#13108057	D₄	1	4	yes	complemented
59	C₂×#11717579	C₂³	1	1	yes	characteristic
60	C₂×#11717579	C₂³	1	2	yes	—
61	C₂×#11717579	C₂³	1	1	yes	characteristic
62	C₈×D₁₀₁	—	2	4	no	—
63	C₈×D₁₀₁	—	2	4	no	—
64	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
65	#13108053	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
67	#13108051	C₂×C₄	1	2	yes	—
68	#13108051	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	M₁₆×C₁₀₁	—	2	4	no	—
75	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
77	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
80	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
81	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₀₁	—	2	4	no	—
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	4	2	no	—
85	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
100	D₄₀₄	M₁₆	1	4	yes	complemented
101	C₄×D₁₀₁	—	2	4	no	—
102	C₄×D₁₀₁	—	2	2	no	—
103	C₄×D₁₀₁	—	2	2	no	—
104	C₄×D₁₀₁	—	2	2	no	—
105	C₄×D₁₀₁	—	2	4	no	—
106	C₄×D₁₀₁	—	4	1	no	—
107	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
108	C₄×D₁₀₁	#41	1	2	yes	—
109	C₈×C₁₀₁	—	2	2	no	—
110	C₈×C₁₀₁	—	2	1	no	—
111	C₈×C₁₀₁	C₂×D₄	1	2	yes	complemented
112	#11717579	—	2	2	no	—
113	#11717579	—	2	1	no	—
114	#11717579	C₂×D₄	1	2	yes	complemented
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	—	2	2	no	—
118	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₀₁	—	2	2	no	—
120	C₂×D₁₀₁	—	2	1	no	—
121	C₂×D₁₀₁	—	2	1	no	—
122	C₂×D₁₀₁	—	2	2	no	—
123	C₂×D₁₀₁	—	4	8	no	—
124	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
125	C₂×D₁₀₁	C₂×M₁₆	1	4	yes	complemented
126	C₄×C₁₀₁	—	2	2	no	—
127	C₄×C₁₀₁	—	2	1	no	—
128	C₄×C₁₀₁	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	2	1	no	—
133	Dic₁₀₁	—	2	1	no	—
134	Dic₁₀₁	C₄×D₄	1	2	yes	—
135	C₂×C₁₀₁	—	4	2	no	—
136	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×C₁₀₁	#568	1	1	yes	characteristic
139	D₁₀₁	—	2	2	no	—
140	D₁₀₁	—	2	4	no	—
141	D₄×M₁₆	—	101	1	no	maximal, Sylow
142	C₁₀₁	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	101	2	no	—
144	C₂×C₄×D₄	—	101	1	no	—
145	#436	—	101	1	no	—
146	#436	D₁₀₁	1	1	yes	characteristic, complemented
147	#435	—	101	2	no	—
148	#435	—	101	2	no	—
149	C₈×D₄	—	101	1	no	—
150	C₈×D₄	—	101	1	no	—
151	#423	—	101	1	no	—
152	#407	—	101	2	no	—
153	C₄×M₁₆	—	101	1	no	—
154	C₂²×D₄	—	101	1	no	—
155	C₂³×C₄	—	101	2	no	—
156	C₂×M₁₆	—	101	2	no	—
157	C₂×M₁₆	—	101	1	no	—
158	C₂×M₁₆	—	101	2	no	—
159	C₂×M₁₆	—	101	1	no	—
160	C₂×M₁₆	—	202	8	no	—
161	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	complemented
162	C₂²×C₈	—	101	2	no	—
163	C₂²×C₈	—	101	2	no	—
164	C₄×D₄	—	101	1	no	—
165	C₄×D₄	—	101	2	no	—
166	C₄×D₄	—	101	1	no	—
167	C₄×D₄	—	101	2	no	—
168	C₄×D₄	—	101	1	no	—
169	C₄×D₄	C₂×D₁₀₁	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	101	1	no	—
171	C₂×C₂²⋊C₄	—	101	2	no	—
172	C₂×C₄²	—	101	1	no	—
173	C₄⋊C₈	—	101	2	no	—
174	C₄⋊C₈	—	101	1	no	—
175	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	101	2	no	—
177	C₂²⋊C₈	—	101	2	no	—
178	C₂²⋊C₈	—	101	2	no	—
179	C₂²⋊C₈	C₂×D₁₀₁	1	2	yes	complemented
180	#97	—	101	1	no	—
181	#97	—	101	1	no	—
182	C₄×C₈	—	101	1	no	—
183	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
184	C₂⁴	—	101	2	no	—
185	C₂×D₄	—	101	2	no	—
186	C₂×D₄	—	101	1	no	—
187	C₂×D₄	—	202	4	no	—
188	C₂×D₄	C₄×D₁₀₁	1	1	yes	characteristic
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	2	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	2	no	—
196	C₂²×C₄	—	202	4	no	—
197	C₂²×C₄	—	202	4	no	—
198	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	101	4	no	—
201	M₁₆	—	202	2	no	—
202	M₁₆	—	202	4	no	—
203	M₁₆	—	202	4	no	—
204	C₂×C₈	—	101	1	no	—
205	C₂×C₈	—	101	1	no	—
206	C₂×C₈	—	101	2	no	—
207	C₂×C₈	—	202	4	no	—
208	C₂×C₈	—	202	4	no	—
209	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
211	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
212	C₄⋊C₄	—	101	2	no	—
213	C₄⋊C₄	—	101	1	no	—
214	C₄⋊C₄	C₄×D₁₀₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	101	2	no	—
216	C₂²⋊C₄	—	101	2	no	—
217	C₂²⋊C₄	—	101	2	no	—
218	C₂²⋊C₄	C₄×D₁₀₁	1	2	yes	—
219	C₄²	—	101	1	no	—
220	C₄²	—	101	1	no	—
221	C₄²	—	101	1	no	—
222	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
223	C₂³	—	101	2	no	—
224	C₂³	—	101	1	no	—
225	C₂³	—	404	2	no	—
226	C₂³	—	202	4	no	—
227	C₂³	—	202	4	no	—
228	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	101	4	no	—
231	D₄	—	202	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	101	1	no	—
235	C₂×C₄	—	101	2	no	—
236	C₂×C₄	—	101	2	no	—
237	C₂×C₄	—	101	1	no	—
238	C₂×C₄	—	101	1	no	—
239	C₂×C₄	—	101	2	no	—
240	C₂×C₄	—	404	1	no	—
241	C₂×C₄	—	202	4	no	—
242	C₂×C₄	—	202	4	no	—
243	C₂×C₄	—	202	4	no	—
244	C₂×C₄	—	202	2	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	101	2	no	—
254	C₈	—	202	2	no	—
255	C₈	—	202	1	no	—
256	C₈	D₄×D₁₀₁	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	101	2	no	—
261	C₂²	—	101	4	no	—
262	C₂²	—	404	8	no	—
263	C₂²	—	202	2	no	—
264	C₂²	—	202	1	no	—
265	C₂²	—	202	1	no	—
266	C₂²	—	202	2	no	—
267	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	101	2	no	—
271	C₄	—	202	2	no	—
272	C₄	—	202	1	no	—
273	C₄	—	202	1	no	—
274	C₄	M₁₆×D₁₀₁	1	2	yes	complemented
275	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
276	C₄	#41×D₁₀₁	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	202	2	no	—
279	C₂	—	202	4	no	—
280	C₂	C₄×D₄×D₁₀₁	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₁₀₁	1	1	yes	characteristic
282	C₂	#20265942	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#436×C₁₀₁	C₂
prime radical	C₂×D₄×D₁₀₁	C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#436	D₁₀₁
hypocenter	C₁₀₁	D₄×M₁₆
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₀₁	C₂	semidirect
3. #20265255	C₂	semidirect
4. #20265251	C₂	semidirect
5. #20265246	C₂	semidirect
6. C₈×D₄₀₄	C₂	semidirect
7. #20265242	C₂	semidirect
8. #20265199	C₂	semidirect
9. #20265139	C₂	semidirect
10. C₄×#13108057	C₂	semidirect
11. #436×C₁₀₁	C₂	semidirect
14. C₂×#13108058	C₂²	semidirect
19. C₂×C₈×D₁₀₁	C₂²	semidirect
20. C₂×C₈×D₁₀₁	C₂²	semidirect
29. C₄⋊#11717579	C₂²	semidirect
30. #15692184	C₂²	semidirect
31. C₂²⋊#11717579	C₂²	semidirect
32. #15692167	C₂²	semidirect
33. #15692167	C₂²	semidirect
34. #15692165	C₂²	semidirect
35. #15692165	C₂²	semidirect
36. C₄×#11717579	C₂²	semidirect
37. C₂×M₁₆×C₁₀₁	C₂²	semidirect
39. C₄⋊C₈×C₁₀₁	C₂²	semidirect
40. C₂²⋊C₈×C₁₀₁	C₂²	semidirect
41. C₄×C₈×C₁₀₁	C₂²	semidirect
58. #13108057	D₄	semidirect
100. D₄₀₄	M₁₆	semidirect
111. C₈×C₁₀₁	C₂×D₄	semidirect
114. #11717579	C₂×D₄	semidirect
125. C₂×D₁₀₁	C₂×M₁₆	semidirect
128. C₄×C₁₀₁	C₂×M₁₆	semidirect
142. C₁₀₁	D₄×M₁₆	semidirect
146. #436	D₁₀₁	semidirect
161. C₂×M₁₆	C₂×D₁₀₁	semidirect
175. C₄⋊C₈	C₂×D₁₀₁	semidirect
179. C₂²⋊C₈	C₂×D₁₀₁	semidirect
183. C₄×C₈	C₂×D₁₀₁	semidirect
256. C₈	D₄×D₁₀₁	semidirect
274. C₄	M₁₆×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₀₁	2
C₄×D₄×D₁₀₁	2
#20265255	2
#20265251	2
#20265246	2
C₈×D₄₀₄	2
#20265242	2
#20265199	2
#20265139	2
C₄×#13108057	2
#436×C₁₀₁	2
D₄×M₁₆	101