Subgroups — #26369641

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 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
2	#20252241	C₂	1	2	yes	maximal, complemented
3	#20245265	C₂	1	4	yes	maximal, complemented
4	C₄×D₇₅₂	C₂	1	4	yes	maximal, complemented
5	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245105	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₇₆	C₂²	1	2	yes	—
9	#15689597	—	2	2	no	—
10	C₂×D₇₅₂	C₄	1	8	yes	complemented
11	C₂×D₇₅₂	C₂²	1	2	yes	—
12	C₂×D₇₅₂	C₂²	1	2	yes	—
13	#15689236	—	2	4	no	—
14	C₄×D₃₇₆	C₂²	1	8	yes	complemented
15	#15689161	C₂²	1	2	yes	—
16	#15689129	—	2	2	no	—
17	#15688951	C₂²	1	2	yes	complemented
18	#15688949	C₂²	1	2	yes	complemented
19	#15688925	C₂²	1	8	yes	complemented
20	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#405	C₂²	1	1	yes	characteristic
23	C₄₇×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
25	C₂²×D₁₈₈	—	2	2	no	—
26	C₂×D₃₇₆	—	2	2	no	—
27	C₂×D₃₇₆	—	2	4	no	—
28	C₂×D₃₇₆	C₂×C₄	1	8	yes	complemented
29	C₂×D₃₇₆	C₂³	1	4	yes	—
30	C₄×D₁₈₈	—	2	8	no	—
31	#13093986	—	4	4	no	—
32	#13093983	—	2	2	no	—
33	C₂²⋊C₄×D₄₇	—	4	4	no	—
34	D₇₅₂	C₂×C₄	1	8	yes	complemented
35	D₇₅₂	C₂×C₄	1	8	yes	complemented
36	#13093958	C₂³	1	4	yes	—
37	#13093956	—	2	2	no	—
38	#13093946	—	2	8	no	—
39	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#117×C₄₇	D₄	1	1	yes	characteristic
42	M₃₂×C₄₇	—	2	2	no	—
43	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
45	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
47	#97×C₄₇	D₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂³×D₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×D₁₈₈	—	2	8	no	—
52	C₂×D₁₈₈	—	2	4	no	—
53	C₂×D₁₈₈	—	4	2	no	—
54	C₂×D₁₈₈	—	4	4	no	—
55	C₂×C₄×D₄₇	—	4	8	no	—
56	M₁₆×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	#41	1	2	yes	—
61	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
63	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
64	C₄²×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₄₇	—	4	2	no	—
66	(C₂×D₄₇)⋊C₄	—	4	8	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄×Dic₄₇	—	4	2	no	—
69	D₃₇₆	—	2	8	no	—
70	D₃₇₆	—	2	4	no	—
71	D₃₇₆	C₂²×C₄	1	8	yes	—
72	C₁₆×C₄₇	—	2	2	no	—
73	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₄₇	—	4	4	no	—
77	C₂²×D₄₇	—	4	8	no	—
78	C₂²×D₄₇	—	16	4	no	—
79	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#135	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₇	—	4	4	no	—
85	D₁₈₈	—	2	8	no	—
86	D₁₈₈	—	4	4	no	—
87	D₁₈₈	—	4	8	no	—
88	C₄×D₄₇	—	16	8	no	—
89	C₈×C₄₇	—	2	2	no	—
90	C₈×C₄₇	C₄×D₄	1	2	yes	—
91	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
93	#33456	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	C₄×D₈	1	2	yes	—
96	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₄₇	—	4	8	no	—
98	C₂×D₄₇	—	8	8	no	—
99	C₂×D₄₇	—	16	8	no	—
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	C₄×D₈	1	2	yes	—
102	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
103	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
104	Dic₄₇	—	8	4	no	—
105	C₂²×D₁₆	—	47	1	no	—
106	#2948	—	47	2	no	—
107	#2177	—	47	4	no	—
108	C₄×D₁₆	—	47	4	no	—
109	#2158	—	47	1	no	—
110	#2140	—	47	2	no	—
111	#2108	D₄₇	1	1	yes	characteristic, complemented
112	C₂×C₄₇	—	2	2	no	—
113	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
114	C₂×C₄₇	#3415	1	2	yes	—
115	D₄₇	—	8	8	no	—
116	C₂²×D₈	—	47	2	no	—
117	#518	—	94	2	no	—
118	C₂×D₁₆	—	47	8	no	—
119	C₂×D₁₆	—	47	2	no	—
120	C₂×D₁₆	—	47	2	no	—
121	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
123	#442	—	94	4	no	—
124	C₄×D₈	—	47	8	no	—
125	#427	—	47	2	no	—
126	#418	—	94	2	no	—
127	#405	C₂×D₄₇	1	1	yes	characteristic
128	#367	—	47	2	no	—
129	#366	—	47	2	no	—
130	#357	—	47	8	no	—
131	#346	C₂×D₄₇	1	2	yes	complemented
132	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
133	C₄₇	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	94	2	no	—
135	C₂×D₈	—	47	8	no	—
136	C₂×D₈	—	47	4	no	—
137	C₂×D₈	—	94	2	no	—
138	C₂×D₈	—	94	4	no	—
139	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
141	C₄×D₄	—	94	8	no	—
142	C₄×D₄	—	188	4	no	—
143	#117	—	94	2	no	—
144	#117	D₁₈₈	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	188	4	no	—
146	D₁₆	—	47	8	no	—
147	D₁₆	—	47	8	no	—
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₄×D₄₇	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
151	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
153	#107	—	47	4	no	—
154	#106	—	94	2	no	—
155	#102	—	94	8	no	—
156	#97	D₁₈₈	1	2	yes	—
157	C₄×C₈	C₂²×D₄₇	1	2	yes	—
158	C₂⁴	—	188	2	no	—
159	C₂×D₄	—	94	8	no	—
160	C₂×D₄	—	94	4	no	—
161	C₂×D₄	—	188	2	no	—
162	C₂×D₄	—	188	4	no	—
163	C₂²×C₄	—	188	8	no	—
164	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
165	D₈	—	47	8	no	—
166	D₈	—	94	8	no	—
167	D₈	—	94	4	no	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
170	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
171	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
172	C₂×C₈	#10625350	1	2	yes	—
173	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
174	C₄⋊C₄	—	94	4	no	—
175	C₄⋊C₄	D₃₇₆	1	2	yes	—
176	C₂²⋊C₄	—	188	2	no	—
177	C₂²⋊C₄	—	188	8	no	—
178	C₂²⋊C₄	D₃₇₆	1	2	yes	—
179	C₄²	—	188	2	no	—
180	C₄²	C₂×D₁₈₈	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
183	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
184	C₂³	—	188	4	no	—
185	C₂³	—	188	8	no	—
186	C₂³	—	752	4	no	—
187	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
188	D₄	—	94	8	no	—
189	D₄	—	188	4	no	—
190	D₄	—	188	8	no	—
191	C₂×C₄	—	188	4	no	—
192	C₂×C₄	—	752	8	no	—
193	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
194	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
195	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
196	C₂×C₄	#13094046	1	2	yes	—
197	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₁₈₈	1	2	yes	—
200	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
201	C₈	C₂×#10625350	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	376	8	no	—
204	C₂²	—	188	8	no	—
205	C₂²	—	752	8	no	—
206	C₂²	C₄×D₃₇₆	1	2	yes	—
207	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	376	4	no	—
210	C₄	C₄×D₃₇₆	1	2	yes	—
211	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
212	C₄	C₂×#13094046	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	376	8	no	—
215	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
216	C₂	#20256529	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₇₅₂	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33456
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20252241	C₂	semidirect
3. #20245265	C₂	semidirect
4. C₄×D₇₅₂	C₂	semidirect
5. #20245172	C₂	semidirect
6. #20245105	C₂	semidirect
7. C₄₇×#2108	C₂	semidirect
10. C₂×D₇₅₂	C₄	semidirect
14. C₄×D₃₇₆	C₂²	semidirect
17. #15688951	C₂²	semidirect
18. #15688949	C₂²	semidirect
19. #15688925	C₂²	semidirect
23. C₄₇×#346	C₂²	semidirect
24. C₄×C₁₆×C₄₇	C₂²	semidirect
28. C₂×D₃₇₆	C₂×C₄	semidirect
34. D₇₅₂	C₂×C₄	semidirect
35. D₇₅₂	C₂×C₄	semidirect
43. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
111. #2108	D₄₇	semidirect
131. #346	C₂×D₄₇	semidirect
132. C₄×C₁₆	C₂×D₄₇	semidirect
133. C₄₇	#33456	semidirect
149. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₇₅₂	2
#20252241	2
#20245265	2
C₄×D₇₅₂	2
#20245172	2
#20245105	2
C₄₇×#2108	2
#33456	47