Subgroups — #27173514

Order 12928.

1088 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692213	C₂	1	2	yes	maximal, complemented
3	#20265039	C₂	1	2	yes	maximal, complemented
4	#20265037	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265034	C₂	1	2	yes	maximal, complemented
6	#20265032	C₂	1	2	yes	maximal
7	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
8	#20265024	C₂	1	2	yes	maximal, complemented
9	#380×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₀₁	—	2	8	no	—
13	C₂²×C₄×D₁₀₁	C₄	1	8	yes	complemented
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692213	—	2	8	no	—
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	4	yes	—
22	C₂×#13108051	C₂²	1	4	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	complemented
24	#15692162	C₂²	1	2	yes	complemented
25	#15692162	C₂²	1	4	yes	complemented
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
29	#95×C₁₀₁	C₂²	1	2	yes	complemented
30	C₂³×D₁₀₁	—	2	2	no	—
31	C₂³×D₁₀₁	—	2	2	no	—
32	C₂³×D₁₀₁	—	2	4	no	—
33	C₂³×D₁₀₁	—	2	4	no	—
34	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
35	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	—	2	4	no	—
38	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₀₁	—	2	4	no	—
43	C₂×C₄×D₁₀₁	—	2	4	no	—
44	C₂×C₄×D₁₀₁	—	2	4	no	—
45	C₂×C₄×D₁₀₁	—	2	32	no	—
46	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
49	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
50	C₂×C₄×D₁₀₁	D₄	1	8	yes	—
51	C₄⋊Dic₁₀₁	—	2	4	no	—
52	C₂²⋊Dic₁₀₁	—	2	4	no	—
53	#13108051	—	2	8	no	—
54	#13108051	—	2	8	no	—
55	C₄×Dic₁₀₁	—	2	8	no	—
56	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	—	2	4	no	—
58	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₁	—	2	4	no	—
63	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
64	C₂³×C₁₀₁	—	2	2	no	—
65	C₂³×C₁₀₁	—	2	2	no	—
66	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂³×C₁₀₁	#41	1	2	yes	—
68	C₂³×C₁₀₁	#41	1	2	yes	—
69	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
70	C₂²×D₁₀₁	—	2	16	no	—
71	C₂²×D₁₀₁	—	2	16	no	—
72	C₂²×D₁₀₁	—	2	8	no	—
73	C₂²×D₁₀₁	—	2	2	no	—
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	—	2	1	no	—
76	C₂²×D₁₀₁	—	2	1	no	—
77	C₂²×D₁₀₁	—	2	1	no	—
78	C₂²×D₁₀₁	—	2	1	no	—
79	C₂²×D₁₀₁	—	2	8	no	—
80	C₂²×D₁₀₁	—	2	8	no	—
81	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
84	C₂²×D₁₀₁	#41	1	4	yes	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	8	no	—
87	C₂×C₄×C₁₀₁	—	2	4	no	—
88	C₂×C₄×C₁₀₁	—	2	4	no	—
89	C₂×C₄×C₁₀₁	—	2	4	no	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	8	no	—
95	C₂×Dic₁₀₁	—	2	4	no	—
96	C₂×Dic₁₀₁	—	2	4	no	—
97	C₂×Dic₁₀₁	—	2	4	no	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
101	C₄×D₁₀₁	—	2	8	no	—
102	C₄×D₁₀₁	—	2	8	no	—
103	C₄×D₁₀₁	—	2	8	no	—
104	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	complemented
105	C₂²×C₁₀₁	—	2	2	no	—
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	—	2	4	no	—
108	C₂²×C₁₀₁	—	2	4	no	—
109	C₂²×C₁₀₁	#117	1	4	yes	complemented
110	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
112	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
113	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
115	C₂×D₁₀₁	—	2	32	no	—
116	C₂×D₁₀₁	—	2	2	no	—
117	C₂×D₁₀₁	—	2	2	no	—
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	—	2	2	no	—
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	2	4	no	—
123	C₂×D₁₀₁	—	2	4	no	—
124	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×D₁₀₁	#117	1	4	yes	complemented
126	C₄×C₁₀₁	—	2	2	no	—
127	C₄×C₁₀₁	—	4	4	no	—
128	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
129	Dic₁₀₁	—	2	2	no	—
130	Dic₁₀₁	—	4	4	no	—
131	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂×C₁₀₁	—	2	4	no	—
133	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
135	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
136	C₂×C₁₀₁	#525	1	2	yes	—
137	C₂×C₁₀₁	#525	1	2	yes	—
138	D₁₀₁	—	2	4	no	—
139	D₁₀₁	—	2	4	no	—
140	#2280	—	101	1	no	maximal, Sylow
141	C₁₀₁	#2280	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴×C₄	—	101	1	no	—
143	C₂×#117	—	101	2	no	—
144	#381	—	101	2	no	—
145	#380	—	101	1	no	—
146	#380	—	101	2	no	—
147	#380	D₁₀₁	1	1	yes	characteristic, complemented
148	#379	—	101	2	no	—
149	C₄×C₂²⋊C₄	—	101	2	no	—
150	C₄×C₂²⋊C₄	—	101	2	no	—
151	C₂⁵	—	101	1	no	—
152	C₂³×C₄	—	101	1	no	—
153	C₂³×C₄	—	101	1	no	—
154	C₂³×C₄	—	101	8	no	—
155	C₂³×C₄	—	101	1	no	—
156	C₂³×C₄	—	101	1	no	—
157	C₂³×C₄	—	101	1	no	—
158	C₂³×C₄	—	202	8	no	—
159	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
160	#117	—	202	8	no	—
161	C₂×C₄⋊C₄	—	101	2	no	—
162	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
163	C₂×C₂²⋊C₄	—	101	2	no	—
164	C₂×C₂²⋊C₄	—	101	4	no	—
165	C₂×C₂²⋊C₄	—	101	4	no	—
166	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
167	C₂×C₄²	—	101	4	no	—
168	#95	—	101	4	no	—
169	#95	—	101	2	no	—
170	#95	C₂×D₁₀₁	1	2	yes	complemented
171	C₂⁴	—	101	1	no	—
172	C₂⁴	—	101	1	no	—
173	C₂⁴	—	101	4	no	—
174	C₂⁴	—	202	4	no	—
175	C₂⁴	—	202	4	no	—
176	C₂⁴	—	202	2	no	—
177	C₂⁴	—	202	2	no	—
178	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	101	1	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	101	8	no	—
183	C₂²×C₄	—	101	4	no	—
184	C₂²×C₄	—	101	1	no	—
185	C₂²×C₄	—	101	4	no	—
186	C₂²×C₄	—	101	4	no	—
187	C₂²×C₄	—	101	4	no	—
188	C₂²×C₄	—	101	4	no	—
189	C₂²×C₄	—	202	4	no	—
190	C₂²×C₄	—	202	32	no	—
191	C₂²×C₄	—	202	4	no	—
192	C₂²×C₄	—	202	4	no	—
193	C₂²×C₄	—	202	4	no	—
194	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
195	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	202	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	202	8	no	—
202	C₂²⋊C₄	—	202	8	no	—
203	C₂²⋊C₄	—	202	4	no	—
204	C₄²	—	202	8	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	101	2	no	—
208	C₂³	—	101	2	no	—
209	C₂³	—	101	4	no	—
210	C₂³	—	101	4	no	—
211	C₂³	—	202	8	no	—
212	C₂³	—	202	8	no	—
213	C₂³	—	202	16	no	—
214	C₂³	—	202	16	no	—
215	C₂³	—	202	2	no	—
216	C₂³	—	202	2	no	—
217	C₂³	—	202	1	no	—
218	C₂³	—	202	1	no	—
219	C₂³	—	202	1	no	—
220	C₂³	—	202	1	no	—
221	C₂³	—	202	8	no	—
222	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
223	C₂³	#13108073	1	2	yes	—
224	C₂³	#13108075	1	2	yes	—
225	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	101	2	no	—
232	C₂×C₄	—	101	4	no	—
233	C₂×C₄	—	101	16	no	—
234	C₂×C₄	—	101	2	no	—
235	C₂×C₄	—	202	8	no	—
236	C₂×C₄	—	202	8	no	—
237	C₂×C₄	—	202	8	no	—
238	C₂×C₄	—	202	4	no	—
239	C₂×C₄	—	202	4	no	—
240	C₂×C₄	—	202	4	no	—
241	C₂×C₄	—	202	2	no	—
242	C₂×C₄	—	202	8	no	—
243	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
244	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
245	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	101	4	no	—
251	C₂²	—	101	4	no	—
252	C₂²	—	202	32	no	—
253	C₂²	—	202	2	no	—
254	C₂²	—	202	2	no	—
255	C₂²	—	202	2	no	—
256	C₂²	—	202	2	no	—
257	C₂²	—	202	4	no	—
258	C₂²	—	202	4	no	—
259	C₂²	—	202	4	no	—
260	C₂²	—	202	4	no	—
261	C₂²	#15692213	1	4	yes	complemented
262	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
263	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
264	C₂²	#41×D₁₀₁	1	2	yes	—
265	C₂²	C₂×#13108073	1	1	yes	characteristic
266	C₂²	C₂×#13108075	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	4	4	no	—
269	C₄	—	101	4	no	—
270	C₄	—	404	4	no	—
271	C₄	—	202	2	no	—
272	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
273	C₂	—	2	4	no	—
274	C₂	—	202	4	no	—
275	C₂	—	202	4	no	—
276	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
277	C₂	#117×D₁₀₁	1	1	yes	characteristic
278	C₂	C₂×#15692213	1	1	yes	characteristic
279	C₂	#20265687	1	2	yes	—
280	C₂	#20265690	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#380×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#380	D₁₀₁
hypocenter	C₁₀₁	#2280
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
2. C₂×#15692213	C₂	semidirect
3. #20265039	C₂	semidirect
4. #20265037	C₂	semidirect
5. #20265034	C₂	semidirect
7. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
8. #20265024	C₂	semidirect
9. #380×C₁₀₁	C₂	semidirect
13. C₂²×C₄×D₁₀₁	C₄	semidirect
19. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
23. C₂×C₄×Dic₁₀₁	C₂²	semidirect
24. #15692162	C₂²	semidirect
25. #15692162	C₂²	semidirect
27. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
29. #95×C₁₀₁	C₂²	semidirect
104. C₄×D₁₀₁	C₂²⋊C₄	semidirect
109. C₂²×C₁₀₁	#117	semidirect
125. C₂×D₁₀₁	#117	semidirect
141. C₁₀₁	#2280	semidirect
147. #380	D₁₀₁	semidirect
162. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
170. #95	C₂×D₁₀₁	semidirect
261. C₂²	#15692213	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂×#15692213	2
#20265039	2
#20265037	2
#20265034	2
#20265032	2
C₂²⋊C₄×Dic₁₀₁	2
#20265024	2
#380×C₁₀₁	2
#2280	101