Subgroups — C₂×#20265684 #27184370

Order 12928.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 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, complemented
2	#20265684	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692236	C₂	1	2	yes	maximal, complemented
4	C₂×#15692231	C₂	1	2	yes	maximal, complemented
5	C₂×#15692226	C₂	1	2	yes	maximal, complemented
6	C₂×#15692223	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	4	yes	maximal, complemented
8	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	—	2	8	no	—
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	8	yes	complemented
17	#15692236	C₂²	1	16	yes	complemented
18	#15692231	C₂²	1	16	yes	complemented
19	#15692226	C₂²	1	16	yes	complemented
20	#15692223	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₁₀₁	C₂²	1	32	yes	complemented
22	#15692214	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
24	C₂×#13108053	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
27	C₂×#13108051	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₁₀₁	—	2	8	no	—
30	C₂×#41×C₁₀₁	C₂²	1	8	yes	complemented
31	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
32	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
34	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	complemented
36	C₂³×D₁₀₁	—	2	4	no	—
37	C₂³×D₁₀₁	—	2	8	no	—
38	C₂³×D₁₀₁	C₂³	1	4	yes	—
39	C₂³×D₁₀₁	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
41	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
42	C₂²×Dic₁₀₁	—	2	4	no	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
45	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
46	C₂×C₄×D₁₀₁	—	2	8	no	—
47	C₂×C₄×D₁₀₁	—	2	8	no	—
48	C₂×C₄×D₁₀₁	—	2	32	no	—
49	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
50	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
51	C₄⋊Dic₁₀₁	C₂³	1	8	yes	complemented
52	#13108053	C₂³	1	16	yes	complemented
53	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
54	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	complemented
55	#13108051	C₂³	1	16	yes	—
56	C₄×Dic₁₀₁	C₂³	1	8	yes	complemented
57	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
58	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
59	#41×C₁₀₁	—	2	32	no	—
60	C₂×Q₈×C₁₀₁	—	2	4	no	—
61	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	complemented
62	C₂×D₄×C₁₀₁	—	2	8	no	—
63	C₂×D₄×C₁₀₁	—	2	4	no	—
64	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
65	C₂×D₄×C₁₀₁	C₂³	1	4	yes	complemented
66	C₂²×C₄×C₁₀₁	—	2	8	no	—
67	C₂²×C₄×C₁₀₁	D₄	1	8	yes	complemented
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	complemented
73	C₂³×C₁₀₁	—	2	4	no	—
74	C₂³×C₁₀₁	—	2	4	no	—
75	C₂³×C₁₀₁	—	2	8	no	—
76	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
77	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
78	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
79	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
80	C₂²×D₁₀₁	—	2	32	no	—
81	C₂²×D₁₀₁	—	2	8	no	—
82	C₂²×D₁₀₁	—	2	4	no	—
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	2	16	no	—
85	C₂²×D₁₀₁	#41	1	8	yes	complemented
86	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
87	Q₈×C₁₀₁	—	2	8	no	—
88	D₄×C₁₀₁	—	2	16	no	—
89	D₄×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	4	no	—
91	C₂×C₄×C₁₀₁	—	2	16	no	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
95	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂²⋊D₁₀₁	—	2	32	no	—
99	C₂×Dic₁₀₁	—	2	8	no	—
100	C₂×Dic₁₀₁	—	2	4	no	—
101	C₂×Dic₁₀₁	—	2	8	no	—
102	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
103	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
104	C₄×D₁₀₁	—	2	16	no	—
105	C₄×D₁₀₁	—	2	16	no	—
106	C₂²×C₁₀₁	—	2	8	no	—
107	C₂²×C₁₀₁	—	2	4	no	—
108	C₂²×C₁₀₁	—	2	4	no	—
109	C₂²×C₁₀₁	—	2	8	no	—
110	C₂²×C₁₀₁	—	4	8	no	—
111	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
112	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
113	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
114	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
115	C₂×D₁₀₁	—	2	32	no	—
116	C₂×D₁₀₁	—	2	16	no	—
117	C₂×D₁₀₁	—	2	4	no	—
118	C₂×D₁₀₁	—	2	4	no	—
119	C₂×D₁₀₁	C₂×#41	1	8	yes	complemented
120	C₄×C₁₀₁	—	2	4	no	—
121	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
122	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
123	Dic₁₀₁	—	2	4	no	—
124	Dic₁₀₁	—	2	8	no	—
125	C₂×C₁₀₁	—	2	4	no	—
126	C₂×C₁₀₁	—	4	4	no	—
127	C₂×C₁₀₁	#525	1	4	yes	complemented
128	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
129	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
130	D₁₀₁	—	2	8	no	—
131	C₂×#525	—	101	1	no	maximal, Sylow
132	C₁₀₁	C₂×#525	1	1	yes	characteristic, complemented, Sylow
133	C₂²×#41	D₁₀₁	1	1	yes	characteristic, complemented
134	C₂⁴×C₄	—	101	1	no	—
135	#525	—	101	16	no	—
136	C₂×#123	—	101	2	no	—
137	C₂×C₂²⋊Q₈	—	101	2	no	—
138	C₂×#121	—	101	2	no	—
139	C₂×#120	—	101	2	no	—
140	C₂×C₄×D₄	—	101	4	no	—
141	C₂×#117	—	101	1	no	—
142	C₂⁵	—	101	1	no	—
143	C₂×#41	—	2	8	no	—
144	C₂×#41	C₂×D₁₀₁	1	8	yes	complemented
145	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
146	C₂²×D₄	—	101	4	no	—
147	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
148	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
149	C₂³×C₄	—	101	1	no	—
150	C₂³×C₄	—	101	2	no	—
151	C₂³×C₄	—	101	8	no	—
152	C₂³×C₄	—	101	2	no	—
153	C₂³×C₄	—	202	8	no	—
154	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₀₁	1	2	yes	complemented
156	#123	—	101	16	no	—
157	C₂²⋊Q₈	—	101	16	no	—
158	#121	—	101	16	no	—
159	#120	—	101	16	no	—
160	C₄×D₄	—	101	32	no	—
161	#117	—	101	8	no	—
162	C₂×C₄⋊C₄	—	101	2	no	—
163	C₂×C₄⋊C₄	—	101	4	no	—
164	C₂×C₂²⋊C₄	—	101	2	no	—
165	C₂×C₂²⋊C₄	—	101	4	no	—
166	C₂×C₂²⋊C₄	—	101	4	no	—
167	C₂×C₄²	—	101	2	no	—
168	C₂⁴	—	101	2	no	—
169	C₂⁴	—	101	4	no	—
170	C₂⁴	—	202	8	no	—
171	C₂⁴	—	202	4	no	—
172	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
173	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
174	#41	—	2	32	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	complemented
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	101	16	no	—
180	C₂×D₄	—	202	16	no	—
181	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
182	C₂×D₄	C₂²×D₁₀₁	1	4	yes	complemented
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	101	4	no	—
185	C₂²×C₄	—	101	2	no	—
186	C₂²×C₄	—	101	4	no	—
187	C₂²×C₄	—	101	8	no	—
188	C₂²×C₄	—	101	8	no	—
189	C₂²×C₄	—	202	4	no	—
190	C₂²×C₄	—	202	8	no	—
191	C₂²×C₄	—	202	8	no	—
192	C₂²×C₄	—	202	32	no	—
193	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	complemented
194	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
195	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
197	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	complemented
199	C₄⋊C₄	—	101	16	no	—
200	C₄⋊C₄	—	101	8	no	—
201	C₂²⋊C₄	—	101	8	no	—
202	C₂²⋊C₄	—	101	16	no	—
203	C₂²⋊C₄	—	101	16	no	—
204	C₄²	—	101	8	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	101	8	no	—
209	C₂³	—	101	4	no	—
210	C₂³	—	202	16	no	—
211	C₂³	—	202	32	no	—
212	C₂³	—	202	8	no	—
213	C₂³	—	202	4	no	—
214	C₂³	—	202	4	no	—
215	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
216	C₂³	C₂³×D₁₀₁	1	4	yes	—
217	C₂³	C₂³×D₁₀₁	1	2	yes	—
218	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	8	no	—
222	D₄	—	202	32	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	101	8	no	—
226	C₂×C₄	—	101	4	no	—
227	C₂×C₄	—	202	8	no	—
228	C₂×C₄	—	202	4	no	—
229	C₂×C₄	—	202	8	no	—
230	C₂×C₄	—	202	16	no	—
231	C₂×C₄	—	202	16	no	—
232	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	16	yes	complemented
235	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
236	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
237	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	8	no	—
243	C₂²	—	101	8	no	—
244	C₂²	—	202	32	no	—
245	C₂²	—	202	4	no	—
246	C₂²	—	202	4	no	—
247	C₂²	—	202	16	no	—
248	C₂²	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
249	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
250	C₂²	#41×D₁₀₁	1	4	yes	—
251	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	202	4	no	—
254	C₄	—	202	8	no	—
255	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	4	4	no	—
259	C₂	—	202	8	no	—
260	C₂	#20265684	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
262	C₂	C₂×#41×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×#41×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	C₂×#525
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×D₁₀₁	C₂	semidirect
2. #20265684	C₂	direct (N × H)
3. C₂×#15692236	C₂	semidirect
4. C₂×#15692231	C₂	semidirect
5. C₂×#15692226	C₂	semidirect
6. C₂×#15692223	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
8. C₂×#15692214	C₂	semidirect
9. C₂²×#41×C₁₀₁	C₂	semidirect
16. C₂²×C₄×D₁₀₁	C₂²	semidirect
17. #15692236	C₂²	semidirect
18. #15692231	C₂²	semidirect
19. #15692226	C₂²	semidirect
20. #15692223	C₂²	semidirect
21. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
22. #15692214	C₂²	semidirect
23. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
24. C₂×#13108053	C₂²	semidirect
26. C₂×C₂²⋊Dic₁₀₁	C₂²	semidirect
28. C₂×C₄×Dic₁₀₁	C₂²	semidirect
30. C₂×#41×C₁₀₁	C₂²	semidirect
31. C₂²×Q₈×C₁₀₁	C₂²	semidirect
33. C₂²×D₄×C₁₀₁	C₂²	semidirect
35. C₂³×C₄×C₁₀₁	C₂²	semidirect
51. C₄⋊Dic₁₀₁	C₂³	semidirect
52. #13108053	C₂³	semidirect
54. C₂²⋊Dic₁₀₁	C₂³	semidirect
56. C₄×Dic₁₀₁	C₂³	semidirect
61. C₂×Q₈×C₁₀₁	C₂³	semidirect
65. C₂×D₄×C₁₀₁	C₂³	semidirect
67. C₂²×C₄×C₁₀₁	D₄	semidirect
72. C₂²×C₄×C₁₀₁	C₂³	semidirect
85. C₂²×D₁₀₁	#41	semidirect
94. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
119. C₂×D₁₀₁	C₂×#41	semidirect
127. C₂×C₁₀₁	#525	semidirect
132. C₁₀₁	C₂×#525	semidirect
133. C₂²×#41	D₁₀₁	semidirect
144. C₂×#41	C₂×D₁₀₁	semidirect
145. C₂²×Q₈	C₂×D₁₀₁	semidirect
148. C₂²×D₄	C₂×D₁₀₁	semidirect
155. C₂³×C₄	C₂×D₁₀₁	semidirect
176. C₂×Q₈	C₂²×D₁₀₁	semidirect
182. C₂×D₄	C₂²×D₁₀₁	semidirect
193. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
198. C₂²×C₄	C₂²×D₁₀₁	semidirect
234. C₂×C₄	C₂×C₂²⋊D₁₀₁	semidirect
260. C₂	#20265684	direct (N × H)

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
#20265684	2
C₂×#15692236	2
C₂×#15692231	2
C₂×#15692226	2
C₂×#15692223	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692214	2
C₂²×#41×C₁₀₁	2
C₂×#525	101

Subgroups — C2×#20265684 #27184370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265684 #27184370