Subgroups — C₂×#20265688 #27184374

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	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265688	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692238	C₂	1	2	yes	maximal, complemented
4	C₂×#15692234	C₂	1	2	yes	maximal, complemented
5	C₂×#15692229	C₂	1	2	yes	maximal, complemented
6	C₂×#15692224	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	C₂⁴×C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108074	—	2	8	no	—
11	C₂×#13108074	C₂²	1	8	yes	complemented
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	complemented
17	#15692238	C₂²	1	16	yes	complemented
18	#15692234	C₂²	1	16	yes	complemented
19	#15692229	C₂²	1	16	yes	complemented
20	#15692224	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	—
27	C₂×#13108051	C₂²	1	4	yes	complemented
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	complemented
29	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₁	—	2	8	no	—
31	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₀₁	C₂²	1	8	yes	complemented
36	C₂³×D₁₀₁	C₂³	1	2	yes	—
37	#13108074	—	2	32	no	—
38	C₂×Dic₂₀₂	—	2	4	no	—
39	C₂×Dic₂₀₂	C₂³	1	4	yes	complemented
40	C₂×D₄₀₄	—	2	4	no	—
41	C₂×D₄₀₄	C₂³	1	4	yes	complemented
42	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
43	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
44	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
45	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
48	C₂×C₄×D₁₀₁	—	2	8	no	—
49	C₂×C₄×D₁₀₁	C₂³	1	8	yes	complemented
50	C₄⋊Dic₁₀₁	C₂³	1	8	yes	complemented
51	#13108053	C₂³	1	16	yes	complemented
52	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
53	C₂²⋊Dic₁₀₁	C₂³	1	16	yes	—
54	#13108051	C₂³	1	16	yes	complemented
55	C₄×Dic₁₀₁	C₂³	1	8	yes	complemented
56	C₂⁴×C₁₀₁	—	2	4	no	—
57	C₂⁴×C₁₀₁	—	2	8	no	—
58	C₂⁴×C₁₀₁	C₂³	1	4	yes	—
59	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	8	no	—
63	C₂²×C₄×C₁₀₁	—	2	8	no	—
64	C₂²×C₄×C₁₀₁	—	2	32	no	—
65	C₂²×C₄×C₁₀₁	D₄	1	8	yes	complemented
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂³×C₁₀₁	—	2	32	no	—
74	C₂³×C₁₀₁	—	2	8	no	—
75	C₂³×C₁₀₁	—	2	4	no	—
76	C₂³×C₁₀₁	—	2	4	no	—
77	C₂³×C₁₀₁	—	2	4	no	—
78	C₂³×C₁₀₁	—	2	16	no	—
79	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
80	C₂³×C₁₀₁	#41	1	8	yes	complemented
81	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
83	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	—	2	8	no	—
86	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
87	C₂×C₄×C₁₀₁	—	2	8	no	—
88	C₂×C₄×C₁₀₁	—	2	4	no	—
89	C₂×C₄×C₁₀₁	—	2	16	no	—
90	C₂×C₄×C₁₀₁	—	2	16	no	—
91	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
94	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
97	C₂²⋊D₁₀₁	—	2	16	no	—
98	C₂²⋊D₁₀₁	—	2	32	no	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	2	8	no	—
101	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
102	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
103	D₄₀₄	—	2	8	no	—
104	C₄×D₁₀₁	—	2	16	no	—
105	Dic₂₀₂	—	2	8	no	—
106	C₂²×C₁₀₁	—	2	32	no	—
107	C₂²×C₁₀₁	—	2	8	no	—
108	C₂²×C₁₀₁	—	2	16	no	—
109	C₂²×C₁₀₁	—	2	4	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
113	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
115	C₂²×C₁₀₁	C₂×#41	1	8	yes	complemented
116	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₀₁	—	2	4	no	—
118	C₂×D₁₀₁	—	2	8	no	—
119	C₂×D₁₀₁	—	4	8	no	—
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	—	2	8	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	—	4	4	no	—
131	C₂×#525	—	101	1	no	maximal, Sylow
132	C₁₀₁	C₂×#525	1	1	yes	characteristic, complemented, Sylow
133	C₂²×#41	—	101	1	no	—
134	C₂⁴×C₄	D₁₀₁	1	1	yes	characteristic, complemented
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	C₂×D₁₀₁	1	1	yes	characteristic
143	C₂×#41	—	101	8	no	—
144	C₂×#41	—	202	8	no	—
145	C₂²×Q₈	—	101	1	no	—
146	C₂²×D₄	—	101	4	no	—
147	C₂²×D₄	—	101	2	no	—
148	C₂²×D₄	—	101	1	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	101	2	no	—
151	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
152	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
154	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₀₁	1	8	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	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	—	101	2	no	—
171	C₂⁴	C₂²×D₁₀₁	1	4	yes	—
172	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
173	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
174	#41	—	202	32	no	—
175	C₂×Q₈	—	101	4	no	—
176	C₂×Q₈	—	202	4	no	—
177	C₂×D₄	—	101	16	no	—
178	C₂×D₄	—	101	8	no	—
179	C₂×D₄	—	101	4	no	—
180	C₂×D₄	—	202	8	no	—
181	C₂×D₄	—	202	16	no	—
182	C₂×D₄	—	202	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	101	4	no	—
188	C₂²×C₄	—	101	2	no	—
189	C₂²×C₄	—	101	8	no	—
190	C₂²×C₄	—	202	8	no	—
191	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	complemented
192	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
194	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
195	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
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	32	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	101	4	no	—
212	C₂³	—	202	4	no	—
213	C₂³	—	202	8	no	—
214	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
215	C₂³	#13108074	1	8	yes	complemented
216	C₂³	C₂³×D₁₀₁	1	2	yes	—
217	C₂³	C₂³×D₁₀₁	1	4	yes	—
218	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
219	Q₈	—	202	8	no	—
220	D₄	—	202	16	no	—
221	D₄	—	202	32	no	—
222	D₄	—	202	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	101	8	no	—
228	C₂×C₄	—	101	4	no	—
229	C₂×C₄	—	202	4	no	—
230	C₂×C₄	—	202	8	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	32	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	16	no	—
244	C₂²	—	404	8	no	—
245	C₂²	—	202	4	no	—
246	C₂²	—	202	8	no	—
247	C₂²	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
248	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
249	C₂²	C₂×#13108074	1	4	yes	—
250	C₂²	C₂×#13108074	1	8	yes	complemented
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	—	2	8	no	—
259	C₂	—	404	4	no	—
260	C₂	#20265688	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
262	C₂	C₂²×#13108074	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108074	C₂	semidirect
2. #20265688	C₂	direct (N × H)
3. C₂×#15692238	C₂	semidirect
4. C₂×#15692234	C₂	semidirect
5. C₂×#15692229	C₂	semidirect
6. C₂×#15692224	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
8. C₂×#15692214	C₂	semidirect
9. C₂⁴×C₄×C₁₀₁	C₂	semidirect
11. C₂×#13108074	C₂²	semidirect
12. C₂²×Dic₂₀₂	C₂²	semidirect
13. C₂²×D₄₀₄	C₂²	semidirect
16. C₂²×C₄×D₁₀₁	C₂²	semidirect
17. #15692238	C₂²	semidirect
18. #15692234	C₂²	semidirect
19. #15692229	C₂²	semidirect
20. #15692224	C₂²	semidirect
21. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
22. #15692214	C₂²	semidirect
23. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
24. C₂×#13108053	C₂²	semidirect
27. C₂×#13108051	C₂²	semidirect
28. C₂×C₄×Dic₁₀₁	C₂²	semidirect
35. C₂³×C₄×C₁₀₁	C₂²	semidirect
39. C₂×Dic₂₀₂	C₂³	semidirect
41. C₂×D₄₀₄	C₂³	semidirect
49. C₂×C₄×D₁₀₁	C₂³	semidirect
50. C₄⋊Dic₁₀₁	C₂³	semidirect
51. #13108053	C₂³	semidirect
54. #13108051	C₂³	semidirect
55. C₄×Dic₁₀₁	C₂³	semidirect
65. C₂²×C₄×C₁₀₁	D₄	semidirect
80. C₂³×C₁₀₁	#41	semidirect
93. C₂×C₄×C₁₀₁	C₂×D₄	semidirect
115. C₂²×C₁₀₁	C₂×#41	semidirect
127. C₂×C₁₀₁	#525	semidirect
132. C₁₀₁	C₂×#525	semidirect
134. C₂⁴×C₄	D₁₀₁	semidirect
155. C₂³×C₄	C₂×D₁₀₁	semidirect
191. C₂²×C₄	C₂²⋊D₁₀₁	semidirect
215. C₂³	#13108074	semidirect
234. C₂×C₄	C₂×C₂²⋊D₁₀₁	semidirect
250. C₂²	C₂×#13108074	semidirect
260. C₂	#20265688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13108074	2
#20265688	2
C₂×#15692238	2
C₂×#15692234	2
C₂×#15692229	2
C₂×#15692224	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692214	2
C₂⁴×C₄×C₁₀₁	2
C₂×#525	101

Subgroups — C2×#20265688 #27184374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265688 #27184374