Subgroups — #26134884

Order 12032.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
2	#20243552	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal
4	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093956	—	2	1	no	—
8	#15688989	—	2	1	no	—
9	#15688899	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
11	C₄₇×#500	—	2	2	no	—
12	C₄₇×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₄₇	—	2	1	no	—
14	C₂×#107×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
20	#13093958	—	2	4	no	—
21	#13093956	—	4	2	no	—
22	C₄₇⋊#95	—	4	1	no	—
23	C₄₇⋊#95	—	4	1	no	—
24	C₂²×Q₈×C₄₇	—	4	1	no	—
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	—	2	4	no	—
28	C₄⋊Q₈×C₄₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	#107×C₄₇	—	2	4	no	—
33	#107×C₄₇	—	2	2	no	—
34	#107×C₄₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂×Q₈×C₄₇	—	4	4	no	—
38	C₂×Q₈×C₄₇	—	8	4	no	—
39	C₂²×C₄×C₄₇	—	4	2	no	—
40	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₄₇	#41	1	2	yes	—
51	C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	4	4	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₄₇	SD₁₆	1	2	yes	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄₇⋊C₁₆	—	2	4	no	—
59	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₄₇	—	8	8	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	8	4	no	—
65	C₂×C₄×C₄₇	#102	1	2	yes	—
66	C₂×C₄×C₄₇	#102	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₈×C₄₇	#103	1	4	yes	—
77	C₈×C₄₇	C₄×Q₈	1	2	yes	—
78	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
79	#13283	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
81	C₂²×C₄₇	#486	1	1	yes	characteristic
82	C₂²×C₄₇	#500	1	1	yes	characteristic
83	C₂²×C₄₇	#509	1	2	yes	—
84	C₂²×C₄₇	#510	1	2	yes	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	—	8	4	no	—
88	C₄×C₄₇	#391	1	1	yes	characteristic
89	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
90	C₄×C₄₇	#487	1	1	yes	characteristic
91	C₄×C₄₇	#498	1	1	yes	characteristic
92	Dic₄₇	—	16	2	no	—
93	C₂×#500	D₄₇	1	1	yes	characteristic
94	C₂×#346	—	47	1	no	—
95	#1846	—	47	1	no	—
96	#1380	—	47	4	no	—
97	C₂×C₄₇	#1990	1	1	yes	characteristic
98	C₂×C₄₇	#2140	1	1	yes	characteristic
99	C₂×C₄₇	#2141	1	1	yes	characteristic
100	C₂×C₄₇	#2243	1	2	yes	—
101	C₂×C₄₇	#2255	1	2	yes	—
102	C₂×C₄⋊Q₈	—	2	1	no	—
103	C₂²×C₁₆	—	47	2	no	—
104	#500	—	2	2	no	—
105	#500	Dic₄₇	1	4	yes	complemented
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	47	2	no	—
108	C₂×#107	C₂×D₄₇	1	2	yes	—
109	C₂×#106	—	94	1	no	—
110	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
111	#382	—	94	1	no	—
112	#346	—	94	2	no	—
113	C₄₇	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	94	2	no	—
122	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂×C₁₆	—	47	4	no	—
124	C₂×C₁₆	—	94	2	no	—
125	C₂×C₁₆	—	94	2	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	94	4	no	—
129	#107	C₂×Dic₄₇	1	4	yes	—
130	#106	—	188	2	no	—
131	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
133	#95	—	188	1	no	—
134	#95	—	188	1	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	188	2	no	—
139	C₂²×C₄	#10625334	1	1	yes	characteristic
140	C₂²×C₄	#10625334	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂×C₈	#10625336	1	2	yes	—
143	C₂×C₈	#10625337	1	2	yes	—
144	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
145	C₂×C₈	Q₈×D₄₇	1	2	yes	—
146	C₂×C₈	#10625354	1	2	yes	—
147	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	188	2	no	—
153	C₄⋊C₄	—	188	2	no	—
154	C₄²	#10625335	1	2	yes	—
155	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
156	C₁₆	—	94	4	no	—
157	C₂³	C₂×#10625334	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	376	2	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	376	2	no	—
166	C₂×C₄	#13093945	1	2	yes	—
167	C₂×C₄	#13093945	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
169	C₂×C₄	#13094013	1	1	yes	characteristic
170	C₂×C₄	#13094022	1	1	yes	characteristic
171	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
172	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
173	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
174	C₈	#13093948	1	4	yes	—
175	C₈	Q₈×Dic₄₇	1	2	yes	—
176	C₈	#13094000	1	2	yes	—
177	C₂²	C₂×#13093945	1	1	yes	characteristic
178	C₂²	#15689471	1	1	yes	characteristic
179	C₂²	#15689526	1	1	yes	characteristic
180	C₂²	#15689559	1	2	yes	—
181	C₂²	#15689564	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	752	2	no	—
186	C₄	#15689030	1	1	yes	characteristic
187	C₄	C₂×#13093948	1	1	yes	characteristic
188	C₄	#15689475	1	1	yes	characteristic
189	C₄	#15689519	1	1	yes	characteristic
190	C₂	#20244350	1	1	yes	characteristic
191	C₂	#20245104	1	1	yes	characteristic
192	C₂	#20245108	1	1	yes	characteristic
193	C₂	#20245627	1	2	yes	—
194	C₂	#20245671	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#13283
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₄₇×#500	C₄	semidirect
19. C₂×C₄₇⋊C₁₆	Q₈	semidirect
105. #500	Dic₄₇	semidirect
113. C₄₇	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688899	2
#20243552	2
#20241479	2
C₂×C₄₇×#500	2
#13283	47