Subgroups — C₂×#20245474 #26370696

Order 12032.

806 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×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689359	C₂	1	2	yes	maximal, complemented
3	#20245474	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688925	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₂²	1	2	yes	—
8	C₂×#13093998	—	2	2	no	—
9	C₂×D₇₅₂	—	2	4	no	—
10	C₂×D₇₅₂	C₂²	1	4	yes	complemented
11	#15689359	—	2	4	no	—
12	#15689359	C₂²	1	8	yes	complemented
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂×#13093946	—	2	2	no	—
15	#15688949	C₂²	1	4	yes	complemented
16	#15688925	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₇	—	2	4	no	—
19	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₇	C₂²	1	4	yes	complemented
22	C₂²×D₁₈₈	—	2	2	no	—
23	C₂²×C₂²⋊D₄₇	—	4	2	no	—
24	C₂×D₃₇₆	—	2	4	no	—
25	C₂×D₃₇₆	—	2	4	no	—
26	C₂×D₃₇₆	—	2	2	no	—
27	C₂×D₃₇₆	C₂³	1	4	yes	—
28	#13093998	—	2	8	no	—
29	#13093998	—	4	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
32	D₇₅₂	—	2	8	no	—
33	#13093958	—	2	2	no	—
34	#13093958	C₂³	1	4	yes	—
35	#13093946	—	2	8	no	—
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₄₇	—	2	2	no	—
44	C₂×C₁₆×C₄₇	—	2	8	no	—
45	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
46	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
47	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
48	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
49	C₂³×D₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	D₈	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₄₇	—	4	8	no	—
57	C₂×C₂²⋊D₄₇	—	8	8	no	—
58	C₂²×Dic₄₇	—	4	2	no	—
59	C₂×D₁₈₈	—	2	4	no	—
60	C₂×D₁₈₈	—	4	4	no	—
61	C₂×D₁₈₈	—	4	4	no	—
62	C₂×D₁₈₈	—	4	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
72	(C₂×D₄₇)⋊C₄	—	4	8	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	D₃₇₆	—	2	4	no	—
76	D₃₇₆	—	2	4	no	—
77	D₃₇₆	—	4	4	no	—
78	C₁₆×C₄₇	—	2	2	no	—
79	C₁₆×C₄₇	C₂×D₄	1	4	yes	complemented
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	D₁₆	1	4	yes	complemented
83	C₂³×C₄₇	C₂×D₈	1	2	yes	—
84	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
85	C₂²×D₄₇	—	4	4	no	—
86	C₂²×D₄₇	—	8	4	no	—
87	C₂²×D₄₇	—	8	4	no	—
88	C₂²×D₄₇	—	8	4	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	#121	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#135	1	2	yes	—
97	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₄₇	—	8	16	no	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	8	4	no	—
101	D₁₈₈	—	4	4	no	—
102	D₁₈₈	—	4	4	no	—
103	D₁₈₈	—	8	4	no	—
104	C₈×C₄₇	—	2	2	no	—
105	C₈×C₄₇	#121	1	2	yes	—
106	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×#2215	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	8	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	#466	1	2	yes	—
113	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
114	C₂²×C₄₇	C₂×D₁₆	1	4	yes	complemented
115	C₂²×C₄₇	#508	1	2	yes	—
116	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
117	C₂×D₄₇	—	8	4	no	—
118	C₂×D₄₇	—	8	4	no	—
119	C₂×D₄₇	—	8	4	no	—
120	C₂×D₄₇	—	16	8	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	#466	1	2	yes	—
123	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
124	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
125	Dic₄₇	—	8	4	no	—
126	C₂²×D₁₆	—	47	1	no	—
127	C₂³×C₁₆	D₄₇	1	1	yes	characteristic, complemented
128	C₂×#466	—	47	2	no	—
129	#2215	—	47	8	no	—
130	C₂×#366	—	47	1	no	—
131	C₂×#357	—	47	2	no	—
132	C₂×C₄₇	—	2	4	no	—
133	C₂×C₄₇	#2215	1	4	yes	complemented
134	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
135	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
136	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
137	D₄₇	—	16	4	no	—
138	C₂²×D₈	—	47	2	no	—
139	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×#121	—	94	2	no	—
141	C₂×D₁₆	—	47	4	no	—
142	C₂×D₁₆	—	94	4	no	—
143	C₂²×C₁₆	—	2	4	no	—
144	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
145	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×C₁₆	C₂×D₄₇	1	4	yes	complemented
147	#466	—	47	8	no	—
148	#466	—	94	4	no	—
149	C₂×#107	—	47	2	no	—
150	C₂×#102	—	94	2	no	—
151	#366	—	47	4	no	—
152	#357	—	47	8	no	—
153	C₄₇	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₄	—	94	2	no	—
155	C₂²×D₄	—	188	2	no	—
156	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
157	C₂×D₈	—	47	4	no	—
158	C₂×D₈	—	94	2	no	—
159	C₂×D₈	—	94	4	no	—
160	C₂×D₈	—	94	4	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
164	C₂²×C₈	D₁₈₈	1	2	yes	—
165	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
166	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
167	#121	—	94	8	no	—
168	#121	—	188	4	no	—
169	C₂×C₄⋊C₄	—	94	2	no	—
170	C₂×C₂²⋊C₄	—	188	2	no	—
171	D₁₆	—	94	8	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	8	no	—
174	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
176	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
177	#107	—	47	4	no	—
178	#107	—	94	2	no	—
179	#102	—	94	8	no	—
180	C₂⁴	—	188	2	no	—
181	C₂⁴	D₃₇₆	1	1	yes	characteristic
182	C₂×D₄	—	94	4	no	—
183	C₂×D₄	—	376	8	no	—
184	C₂×D₄	—	188	8	no	—
185	C₂×D₄	—	188	4	no	—
186	C₂×D₄	—	188	2	no	—
187	C₂×D₄	—	188	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	188	2	no	—
191	C₂²×C₄	D₃₇₆	1	2	yes	—
192	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
194	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
195	D₈	—	94	4	no	—
196	D₈	—	94	4	no	—
197	D₈	—	188	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
203	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂×C₈	#10625350	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
207	C₄⋊C₄	—	94	4	no	—
208	C₄⋊C₄	—	188	2	no	—
209	C₂²⋊C₄	—	188	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	C₂×C₂²⋊D₄₇	1	4	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	376	4	no	—
215	C₂³	—	376	4	no	—
216	C₂³	—	376	4	no	—
217	C₂³	—	188	4	no	—
218	C₂³	D₇₅₂	1	4	yes	complemented
219	C₂³	C₂×D₃₇₆	1	2	yes	—
220	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
221	D₄	—	376	16	no	—
222	D₄	—	376	4	no	—
223	D₄	—	188	4	no	—
224	D₄	—	188	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	376	4	no	—
230	C₂×C₄	—	188	4	no	—
231	C₂×C₄	#13093998	1	2	yes	—
232	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
233	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
234	C₂×C₄	#13094046	1	2	yes	—
235	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#13093998	1	2	yes	—
238	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₈	C₂×#10625350	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	376	4	no	—
244	C₂²	—	376	4	no	—
245	C₂²	—	376	4	no	—
246	C₂²	—	752	8	no	—
247	C₂²	#15689359	1	2	yes	—
248	C₂²	C₂×D₇₅₂	1	2	yes	—
249	C₂²	C₂×D₇₅₂	1	4	yes	complemented
250	C₂²	#15689555	1	2	yes	—
251	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	376	4	no	—
254	C₄	#15689359	1	2	yes	—
255	C₄	C₂×#13093998	1	1	yes	characteristic
256	C₄	C₂×#13094046	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	752	4	no	—
259	C₂	#20245474	1	4	yes	direct product factor
260	C₂	C₂×#15689359	1	1	yes	characteristic
261	C₂	C₂²×D₇₅₂	1	1	yes	characteristic
262	C₂	C₂×#15689555	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2215
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₇₅₂	C₂	semidirect
2. C₂×#15689359	C₂	semidirect
3. #20245474	C₂	direct (N × H)
4. C₂×#15688949	C₂	semidirect
5. C₂×#15688925	C₂	semidirect
6. C₂³×C₁₆×C₄₇	C₂	semidirect
10. C₂×D₇₅₂	C₂²	semidirect
12. #15689359	C₂²	semidirect
15. #15688949	C₂²	semidirect
16. #15688925	C₂²	semidirect
21. C₂²×C₁₆×C₄₇	C₂²	semidirect
45. C₂×C₁₆×C₄₇	D₄	semidirect
79. C₁₆×C₄₇	C₂×D₄	semidirect
82. C₂³×C₄₇	D₁₆	semidirect
114. C₂²×C₄₇	C₂×D₁₆	semidirect
127. C₂³×C₁₆	D₄₇	semidirect
133. C₂×C₄₇	#2215	semidirect
146. C₂²×C₁₆	C₂×D₄₇	semidirect
153. C₄₇	C₂×#2215	semidirect
174. C₂×C₁₆	C₂²⋊D₄₇	semidirect
211. C₁₆	C₂×C₂²⋊D₄₇	semidirect
218. C₂³	D₇₅₂	semidirect
249. C₂²	C₂×D₇₅₂	semidirect
259. C₂	#20245474	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×D₇₅₂	2
C₂×#15689359	2
#20245474	2
C₂×#15688949	2
C₂×#15688925	2
C₂³×C₁₆×C₄₇	2
C₂×#2215	47

Subgroups — C2×#20245474 #26370696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245474 #26370696