Subgroups — C₂×#20265745 #27184442

Order 12928.

1896 conjugacy classes of subgroups.

The subgroup lattice has 288 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, complemented
2	C₂²×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265745	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692228	C₂	1	4	yes	maximal, complemented
8	C₂×#15692225	C₂	1	4	yes	maximal, complemented
9	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₀₁	C₂²	1	2	yes	—
12	C₂×#13108075	—	2	8	no	—
13	C₂×#13108075	C₂²	1	8	yes	complemented
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₄	C₂²	1	2	yes	—
16	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₄₀₄	C₂²	1	4	yes	complemented
18	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊D₁₀₁	C₂²	1	4	yes	complemented
20	C₂×D₄×D₁₀₁	—	2	8	no	—
21	C₂×D₄×D₁₀₁	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	complemented
24	#15692252	C₂²	1	8	yes	complemented
25	#15692251	C₂²	1	8	yes	complemented
26	#15692244	C₂²	1	8	yes	complemented
27	#15692228	C₂²	1	32	yes	complemented
28	#15692225	C₂²	1	32	yes	complemented
29	#15692212	C₂²	1	8	yes	complemented
30	C₂×#13108053	C₂²	1	2	yes	complemented
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×#13108051	C₂²	1	4	yes	complemented
33	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	#124×C₁₀₁	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	complemented
38	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₁₀₁	—	2	8	no	—
40	C₂³×D₁₀₁	—	4	8	no	—
41	C₂³×D₁₀₁	C₂³	1	2	yes	—
42	C₂³×D₁₀₁	C₂³	1	2	yes	—
43	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
44	C₂³×D₁₀₁	C₂³	1	8	yes	—
45	#13108075	—	2	32	no	—
46	C₂×D₄₀₄	—	2	8	no	—
47	C₂×D₄₀₄	—	2	4	no	—
48	C₂×D₄₀₄	—	2	4	no	—
49	C₂×D₄₀₄	—	4	8	no	—
50	C₂×D₄₀₄	C₂³	1	4	yes	—
51	C₂×D₄₀₄	C₂³	1	8	yes	—
52	C₂×D₄₀₄	C₂³	1	4	yes	complemented
53	C₂×D₄₀₄	C₂³	1	16	yes	complemented
54	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
55	C₂×C₂²⋊D₁₀₁	—	4	8	no	—
56	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	16	yes	complemented
58	D₄×D₁₀₁	—	2	32	no	—
59	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
60	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₀₁	—	2	8	no	—
62	C₂×C₄×D₁₀₁	D₄	1	8	yes	complemented
63	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
64	C₂×C₄×D₁₀₁	C₂³	1	8	yes	complemented
65	#13108053	C₂³	1	8	yes	complemented
66	#13108051	C₂³	1	8	yes	—
67	#13108051	C₂³	1	16	yes	complemented
68	C₄×Dic₁₀₁	C₂³	1	4	yes	complemented
69	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
70	C₂×Q₈×C₁₀₁	—	2	4	no	—
71	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	complemented
72	C₂×D₄×C₁₀₁	—	2	4	no	—
73	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₀₁	C₂³	1	16	yes	complemented
78	C₄²×C₁₀₁	C₂³	1	4	yes	complemented
79	C₂³×C₁₀₁	—	2	4	no	—
80	C₂³×C₁₀₁	—	2	8	no	—
81	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
82	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₀₁	—	2	4	no	—
84	C₂²×D₁₀₁	—	2	4	no	—
85	C₂²×D₁₀₁	—	2	4	no	—
86	C₂²×D₁₀₁	—	2	8	no	—
87	C₂²×D₁₀₁	—	2	8	no	—
88	C₂²×D₁₀₁	—	2	16	no	—
89	C₂²×D₁₀₁	—	4	16	no	—
90	C₂²×D₁₀₁	—	4	32	no	—
91	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
92	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
93	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
94	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
95	Q₈×C₁₀₁	—	2	8	no	—
96	D₄×C₁₀₁	—	2	8	no	—
97	C₂×C₄×C₁₀₁	—	2	4	no	—
98	C₂×C₄×C₁₀₁	—	2	4	no	—
99	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂²⋊D₁₀₁	—	2	16	no	—
104	C₂²⋊D₁₀₁	—	4	16	no	—
105	C₂×Dic₁₀₁	—	2	4	no	—
106	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
107	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
108	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
109	D₄₀₄	—	2	16	no	—
110	D₄₀₄	—	2	8	no	—
111	D₄₀₄	—	2	8	no	—
112	D₄₀₄	—	4	16	no	—
113	C₄×D₁₀₁	—	2	16	no	—
114	C₄×D₁₀₁	C₂×D₄	1	16	yes	complemented
115	C₂²×C₁₀₁	—	2	4	no	—
116	C₂²×C₁₀₁	—	2	8	no	—
117	C₂²×C₁₀₁	—	4	8	no	—
118	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
119	C₂²×C₁₀₁	H(5,𝔽₂)	1	4	yes	—
120	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
121	C₂×D₁₀₁	—	2	8	no	—
122	C₂×D₁₀₁	—	2	4	no	—
123	C₂×D₁₀₁	—	2	4	no	—
124	C₂×D₁₀₁	—	2	4	no	—
125	C₂×D₁₀₁	—	2	8	no	—
126	C₂×D₁₀₁	—	2	8	no	—
127	C₂×D₁₀₁	—	4	32	no	—
128	C₂×D₁₀₁	—	4	8	no	—
129	C₂×D₁₀₁	—	4	8	no	—
130	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
131	C₄×C₁₀₁	—	2	4	no	—
132	C₄×C₁₀₁	—	2	4	no	—
133	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
134	Dic₁₀₁	—	2	4	no	—
135	Dic₁₀₁	C₂²×D₄	1	4	yes	—
136	C₂×C₁₀₁	—	4	4	no	—
137	C₂×C₁₀₁	#535	1	4	yes	complemented
138	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
139	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	2	yes	—
140	D₁₀₁	—	2	4	no	—
141	D₁₀₁	—	4	4	no	—
142	D₁₀₁	—	4	4	no	—
143	C₂×#535	—	101	1	no	maximal, Sylow
144	C₁₀₁	C₂×#535	1	1	yes	characteristic, complemented, Sylow
145	C₂²×#41	—	101	1	no	—
146	C₂³×D₄	—	101	1	no	—
147	#535	—	101	16	no	—
148	C₂×#127	—	101	1	no	—
149	C₂×#127	—	101	1	no	—
150	C₂×#124	—	101	1	no	—
151	C₂×#124	D₁₀₁	1	1	yes	characteristic, complemented
152	C₂×#121	—	101	4	no	—
153	C₂×#120	—	101	4	no	—
154	C₂×#117	—	101	1	no	—
155	C₂⁵	—	101	2	no	—
156	C₂×#41	—	101	8	no	—
157	C₂×#41	—	202	8	no	—
158	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
159	C₂²×D₄	—	101	4	no	—
160	C₂²×D₄	—	101	2	no	—
161	C₂²×D₄	—	101	8	no	—
162	C₂²×D₄	—	101	4	no	—
163	C₂²×D₄	—	101	2	no	—
164	C₂²×D₄	—	101	1	no	—
165	C₂²×D₄	—	101	1	no	—
166	C₂²×D₄	—	202	8	no	—
167	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
168	C₂³×C₄	—	101	1	no	—
169	C₂³×C₄	—	101	2	no	—
170	#127	—	101	8	no	—
171	#127	—	101	8	no	—
172	#124	—	101	8	no	—
173	#124	C₂×D₁₀₁	1	8	yes	complemented
174	#121	—	101	32	no	—
175	#120	—	101	32	no	—
176	#117	—	101	8	no	—
177	C₂×C₄⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	—	101	2	no	—
179	C₂×C₂²⋊C₄	—	101	4	no	—
180	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	complemented
181	C₂×C₄²	—	101	1	no	—
182	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
183	C₂⁴	—	101	1	no	—
184	C₂⁴	—	101	8	no	—
185	C₂⁴	—	101	2	no	—
186	C₂⁴	—	101	2	no	—
187	C₂⁴	—	404	8	no	—
188	C₂⁴	—	202	8	no	—
189	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
190	#41	—	202	32	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	complemented
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	101	16	no	—
195	C₂×D₄	—	101	8	no	—
196	C₂×D₄	—	101	16	no	—
197	C₂×D₄	—	101	8	no	—
198	C₂×D₄	—	101	4	no	—
199	C₂×D₄	—	101	4	no	—
200	C₂×D₄	—	404	8	no	—
201	C₂×D₄	—	404	8	no	—
202	C₂×D₄	—	202	8	no	—
203	C₂×D₄	—	202	32	no	—
204	C₂×D₄	—	202	8	no	—
205	C₂×D₄	—	202	4	no	—
206	C₂×D₄	—	202	4	no	—
207	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	101	8	no	—
210	C₂²×C₄	—	101	4	no	—
211	C₂²×C₄	—	101	2	no	—
212	C₂²×C₄	—	101	8	no	—
213	C₂²×C₄	—	202	8	no	—
214	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
215	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
217	C₄⋊C₄	—	101	8	no	—
218	C₂²⋊C₄	—	101	8	no	—
219	C₂²⋊C₄	—	101	16	no	—
220	C₂²⋊C₄	C₂²×D₁₀₁	1	16	yes	complemented
221	C₄²	—	101	4	no	—
222	C₄²	C₂²×D₁₀₁	1	4	yes	complemented
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	101	4	no	—
226	C₂³	—	101	2	no	—
227	C₂³	—	101	4	no	—
228	C₂³	—	101	4	no	—
229	C₂³	—	404	16	no	—
230	C₂³	—	404	32	no	—
231	C₂³	—	202	16	no	—
232	C₂³	—	202	4	no	—
233	C₂³	—	202	4	no	—
234	C₂³	—	202	8	no	—
235	C₂³	—	202	8	no	—
236	C₂³	—	202	4	no	—
237	C₂³	C₂³×D₁₀₁	1	4	yes	—
238	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	D₄	—	2	8	no	—
241	D₄	—	404	16	no	—
242	D₄	—	404	16	no	—
243	D₄	—	202	16	no	—
244	D₄	—	202	16	no	—
245	D₄	—	202	8	no	—
246	D₄	—	202	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	101	4	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	16	no	—
252	C₂×C₄	—	101	4	no	—
253	C₂×C₄	—	202	16	no	—
254	C₂×C₄	—	202	4	no	—
255	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
256	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
257	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
258	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	101	4	no	—
263	C₂²	—	404	32	no	—
264	C₂²	—	404	8	no	—
265	C₂²	—	404	8	no	—
266	C₂²	—	202	4	no	—
267	C₂²	—	202	4	no	—
268	C₂²	—	202	8	no	—
269	C₂²	—	202	8	no	—
270	C₂²	—	202	8	no	—
271	C₂²	—	202	4	no	—
272	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
273	C₂²	#15692301	1	4	yes	—
274	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	101	4	no	—
278	C₄	—	202	4	no	—
279	C₄	C₂×D₄×D₁₀₁	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	404	4	no	—
282	C₂	—	404	4	no	—
283	C₂	—	202	4	no	—
284	C₂	#20265745	1	4	yes	direct product factor
285	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
286	C₂	C₂×#15692301	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	C₂×#535
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108075	C₂	semidirect
2. C₂²×D₄×D₁₀₁	C₂	semidirect
3. #20265745	C₂	direct (N × H)
4. C₂×#15692252	C₂	semidirect
5. C₂×#15692251	C₂	semidirect
6. C₂×#15692244	C₂	semidirect
7. C₂×#15692228	C₂	semidirect
8. C₂×#15692225	C₂	semidirect
9. C₂×#15692212	C₂	semidirect
10. C₂×#124×C₁₀₁	C₂	semidirect
13. C₂×#13108075	C₂²	semidirect
16. C₂²×D₄₀₄	C₂²	semidirect
17. C₂²×D₄₀₄	C₂²	semidirect
19. C₂²×C₂²⋊D₁₀₁	C₂²	semidirect
21. C₂×D₄×D₁₀₁	C₂²	semidirect
23. C₂²×C₄×D₁₀₁	C₂²	semidirect
24. #15692252	C₂²	semidirect
25. #15692251	C₂²	semidirect
26. #15692244	C₂²	semidirect
27. #15692228	C₂²	semidirect
28. #15692225	C₂²	semidirect
29. #15692212	C₂²	semidirect
30. C₂×#13108053	C₂²	semidirect
32. C₂×#13108051	C₂²	semidirect
33. C₂×C₄×Dic₁₀₁	C₂²	semidirect
34. C₂²×Q₈×C₁₀₁	C₂²	semidirect
36. #124×C₁₀₁	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₁₀₁	C₂²	semidirect
38. C₂×C₄²×C₁₀₁	C₂²	semidirect
52. C₂×D₄₀₄	C₂³	semidirect
53. C₂×D₄₀₄	C₂³	semidirect
57. C₂×C₂²⋊D₁₀₁	C₂³	semidirect
62. C₂×C₄×D₁₀₁	D₄	semidirect
64. C₂×C₄×D₁₀₁	C₂³	semidirect
65. #13108053	C₂³	semidirect
67. #13108051	C₂³	semidirect
68. C₄×Dic₁₀₁	C₂³	semidirect
71. C₂×Q₈×C₁₀₁	C₂³	semidirect
77. C₂²⋊C₄×C₁₀₁	C₂³	semidirect
78. C₄²×C₁₀₁	C₂³	semidirect
114. C₄×D₁₀₁	C₂×D₄	semidirect
137. C₂×C₁₀₁	#535	semidirect
144. C₁₀₁	C₂×#535	semidirect
151. C₂×#124	D₁₀₁	semidirect
158. C₂²×Q₈	C₂×D₁₀₁	semidirect
173. #124	C₂×D₁₀₁	semidirect
180. C₂×C₂²⋊C₄	C₂×D₁₀₁	semidirect
182. C₂×C₄²	C₂×D₁₀₁	semidirect
192. C₂×Q₈	C₂²×D₁₀₁	semidirect
220. C₂²⋊C₄	C₂²×D₁₀₁	semidirect
222. C₄²	C₂²×D₁₀₁	semidirect
284. C₂	#20265745	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×#13108075	2
C₂²×D₄×D₁₀₁	2
#20265745	2
C₂×#15692252	2
C₂×#15692251	2
C₂×#15692244	2
C₂×#15692228	2
C₂×#15692225	2
C₂×#15692212	2
C₂×#124×C₁₀₁	2
C₂×#535	101

Subgroups — C2×#20265745 #27184442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265745 #27184442