Subgroups — #26103465

Order 12032.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂×#13093958	—	2	1	no	—
6	C₂×#13093958	C₄	1	2	yes	—
7	#15688993	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#335	—	2	1	no	—
13	C₄₇×#335	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₄⋊Dic₄₇	—	4	1	no	—
16	C₂×C₄⋊Dic₄₇	—	4	1	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	4	1	no	—
19	C₂×C₄₇⋊C₁₆	—	4	1	no	—
20	#13093958	—	4	1	no	—
21	#13093958	—	4	1	no	—
22	#13093958	—	4	2	no	—
23	C₂²×C₈×C₄₇	—	4	1	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₄⋊C₈×C₄₇	—	4	1	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	—	2	1	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	8	1	no	—
41	C₂×C₈×C₄₇	—	8	1	no	—
42	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
50	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
53	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
54	C₄⋊Dic₄₇	—	4	2	no	—
55	C₄⋊Dic₄₇	—	4	2	no	—
56	C₄⋊Dic₄₇	—	8	2	no	—
57	C₄⋊Dic₄₇	—	8	1	no	—
58	C₄⋊Dic₄₇	—	8	1	no	—
59	C₄₇⋊C₁₆	—	4	2	no	—
60	C₂³×C₄₇	#102	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
73	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₄₇	—	8	2	no	—
78	C₂×Dic₄₇	—	8	2	no	—
79	C₂×Dic₄₇	—	8	2	no	—
80	C₈×C₄₇	—	2	1	no	—
81	C₈×C₄₇	—	2	1	no	—
82	C₈×C₄₇	—	8	2	no	—
83	C₈×C₄₇	#102	1	1	yes	characteristic
84	C₈×C₄₇	#103	1	1	yes	characteristic
85	C₈×C₄₇	#106	1	1	yes	characteristic
86	C₈×C₄₇	#107	1	1	yes	characteristic
87	#7274	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	#331	1	1	yes	characteristic
89	C₂²×C₄₇	#332	1	1	yes	characteristic
90	C₂²×C₄₇	#340	1	1	yes	characteristic
91	C₂²×C₄₇	#357	1	1	yes	characteristic
92	C₂²×C₄₇	#358	1	1	yes	characteristic
93	C₂²×C₄₇	#361	1	1	yes	characteristic
94	C₂²×C₄₇	#362	1	1	yes	characteristic
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	#332	1	1	yes	characteristic
98	C₄×C₄₇	#333	1	1	yes	characteristic
99	C₄×C₄₇	#340	1	1	yes	characteristic
100	C₄×C₄₇	#343	1	1	yes	characteristic
101	C₄×C₄₇	#357	1	1	yes	characteristic
102	C₄×C₄₇	#358	1	1	yes	characteristic
103	C₄×C₄₇	#366	1	1	yes	characteristic
104	C₄×C₄₇	#367	1	1	yes	characteristic
105	Dic₄₇	—	16	2	no	—
106	C₂×C₄⋊C₁₆	—	47	1	no	—
107	#1845	—	47	1	no	—
108	C₂×#335	D₄₇	1	1	yes	characteristic
109	C₂×C₄₇	#1296	1	1	yes	characteristic
110	C₂×C₄₇	#1361	1	1	yes	characteristic
111	C₂×C₄₇	#1362	1	1	yes	characteristic
112	C₂×C₄₇	#1364	1	1	yes	characteristic
113	C₂×C₄₇	#1365	1	1	yes	characteristic
114	C₂×C₄₇	#1380	1	1	yes	characteristic
115	C₂×C₄₇	#1387	1	1	yes	characteristic
116	C₂²×C₁₆	—	94	1	no	—
117	C₂×#107	—	47	2	no	—
118	C₂×#107	—	94	1	no	—
119	C₂×C₄⋊C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	#384	—	94	1	no	—
122	C₄⋊C₁₆	—	47	2	no	—
123	C₄⋊C₁₆	—	94	1	no	—
124	#335	—	2	1	no	—
125	#335	Dic₄₇	1	2	yes	complemented
126	C₄₇	#7274	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	94	2	no	—
131	C₂×C₄⋊C₄	—	188	1	no	—
132	C₂×C₄⋊C₄	—	188	1	no	—
133	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₁₆	—	94	2	no	—
135	C₂×C₁₆	—	188	1	no	—
136	C₂×C₁₆	—	188	1	no	—
137	#107	—	188	1	no	—
138	#107	—	188	1	no	—
139	#107	—	188	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄×C₈	Dic₉₄	1	1	yes	characteristic
143	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
146	C₂²×C₄	—	188	2	no	—
147	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	#10625334	1	1	yes	characteristic
149	C₂²×C₄	#10625336	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	#10625325	1	1	yes	characteristic
156	C₂×C₈	#10625325	1	1	yes	characteristic
157	C₂×C₈	D₃₇₆	1	1	yes	characteristic
158	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
159	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
160	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
162	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
163	C₄⋊C₄	—	376	2	no	—
164	C₄⋊C₄	—	376	1	no	—
165	C₄⋊C₄	—	376	1	no	—
166	C₄⋊C₄	—	188	2	no	—
167	C₄⋊C₄	—	188	2	no	—
168	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
169	C₄²	#10625335	1	1	yes	characteristic
170	C₄²	#10625337	1	1	yes	characteristic
171	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₁₆	—	188	2	no	—
173	C₂³	#13093944	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	376	2	no	—
177	C₂×C₄	—	376	2	no	—
178	C₂×C₄	—	376	2	no	—
179	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
180	C₂×C₄	#13093940	1	1	yes	characteristic
181	C₂×C₄	#13093942	1	1	yes	characteristic
182	C₂×C₄	#13093945	1	1	yes	characteristic
183	C₂×C₄	#13093946	1	1	yes	characteristic
184	C₂×C₄	#13093947	1	1	yes	characteristic
185	C₂×C₄	#13093948	1	1	yes	characteristic
186	C₂×C₄	#13093949	1	1	yes	characteristic
187	C₂×C₄	#13093955	1	1	yes	characteristic
188	C₂×C₄	#13093957	1	1	yes	characteristic
189	C₂×C₄	#13093965	1	1	yes	characteristic
190	C₂×C₄	#13093967	1	1	yes	characteristic
191	C₂×C₄	#13093968	1	1	yes	characteristic
192	C₂×C₄	#13093970	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	8	2	no	—
196	C₈	#13093946	1	1	yes	characteristic
197	C₈	#13093949	1	1	yes	characteristic
198	C₈	#13093956	1	1	yes	characteristic
199	C₈	#13093958	1	1	yes	characteristic
200	C₂²	#15688866	1	1	yes	characteristic
201	C₂²	#15688870	1	1	yes	characteristic
202	C₂²	#15688885	1	1	yes	characteristic
203	C₂²	#15688923	1	1	yes	characteristic
204	C₂²	#15688926	1	1	yes	characteristic
205	C₂²	#15688935	1	1	yes	characteristic
206	C₂²	#15688938	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	752	2	no	—
210	C₄	#15688868	1	1	yes	characteristic
211	C₄	#15688871	1	1	yes	characteristic
212	C₄	#15688886	1	1	yes	characteristic
213	C₄	#15688891	1	1	yes	characteristic
214	C₄	#15688924	1	1	yes	characteristic
215	C₄	#15688927	1	1	yes	characteristic
216	C₄	#15688948	1	1	yes	characteristic
217	C₄	#15688950	1	1	yes	characteristic
218	C₂	#20241272	1	1	yes	characteristic
219	C₂	#20241433	1	1	yes	characteristic
220	C₂	#20241436	1	1	yes	characteristic
221	C₂	#20241444	1	1	yes	characteristic
222	C₂	#20241447	1	1	yes	characteristic
223	C₂	#20241479	1	1	yes	characteristic
224	C₂	#20241498	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093944
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#7274
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₄₇⋊C₁₆)	C₄	semidirect
13. C₄₇×#335	C₄	semidirect
125. #335	Dic₄₇	semidirect
126. C₄₇	#7274	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20243548	2
C₂×C₄₇×#335	2
#7274	47