Subgroups — #26134883

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×#15689528	C₂	1	1	yes	maximal, characteristic
2	#20243552	C₂	1	1	yes	maximal, characteristic
3	#20241480	C₂	1	4	yes	maximal
4	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	1	no	—
6	#15689528	—	2	2	no	—
7	#15689528	C₄	1	4	yes	complemented
8	C₂×#13093958	—	2	1	no	—
9	C₂×#13093958	C₂²	1	2	yes	—
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093956	—	2	1	no	—
12	#15688989	—	2	1	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#346	—	2	2	no	—
16	C₂²×Dic₉₄	—	4	1	no	—
17	#13094024	—	2	4	no	—
18	#13094024	—	4	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	4	1	no	—
22	#13093958	—	2	4	no	—
23	#13093958	—	2	2	no	—
24	#13093958	—	2	4	no	—
25	#13093958	C₂×C₄	1	4	yes	—
26	#13093956	—	4	2	no	—
27	C₄₇⋊#95	—	4	1	no	—
28	C₄₇⋊#95	—	4	1	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₄₇	—	2	2	no	—
33	C₂×C₁₆×C₄₇	—	2	2	no	—
34	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
35	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₄₇	—	4	2	no	—
41	C₂²×Dic₄₇	—	4	2	no	—
42	C₂×Dic₉₄	—	4	4	no	—
43	C₂×Dic₉₄	—	8	4	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	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₄₇	SD₁₆	1	2	yes	—
53	C₄⋊Dic₄₇	—	2	4	no	—
54	C₄⋊Dic₄₇	—	4	4	no	—
55	C₄⋊Dic₄₇	—	4	2	no	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₁₆×C₄₇	—	2	4	no	—
59	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	#102	1	2	yes	—
63	C₂×C₄×C₄₇	#102	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₄₇	—	4	4	no	—
71	C₂×Dic₄₇	—	8	4	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	Dic₉₄	—	8	8	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	#391	1	1	yes	characteristic
88	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
89	C₄×C₄₇	#487	1	1	yes	characteristic
90	C₄×C₄₇	#498	1	1	yes	characteristic
91	Dic₄₇	—	8	4	no	—
92	Dic₄₇	—	16	2	no	—
93	C₂×#500	—	47	1	no	—
94	C₂×#346	D₄₇	1	1	yes	characteristic
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	—	94	1	no	—
103	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
104	#500	—	47	4	no	—
105	#500	—	94	2	no	—
106	C₂×#107	—	47	2	no	—
107	C₂×#107	—	47	2	no	—
108	C₂×#107	—	94	1	no	—
109	C₂×#106	—	94	1	no	—
110	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
111	#382	—	94	1	no	—
112	#346	—	2	2	no	—
113	C₄₇	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	188	1	no	—
115	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	94	4	no	—
118	C₄⋊Q₈	—	188	2	no	—
119	C₂×C₄⋊C₄	—	94	2	no	—
120	C₂×C₄⋊C₄	—	94	2	no	—
121	C₂×C₄⋊C₄	—	188	1	no	—
122	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
126	#107	—	47	4	no	—
127	#107	—	94	4	no	—
128	#107	—	94	2	no	—
129	#107	—	94	4	no	—
130	#106	—	188	2	no	—
131	C₄×C₈	C₄×D₄₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
133	#95	—	188	1	no	—
134	#95	—	188	1	no	—
135	C₂×Q₈	—	376	4	no	—
136	C₂×Q₈	—	188	4	no	—
137	C₂²×C₄	—	188	2	no	—
138	C₂²×C₄	—	188	2	no	—
139	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
140	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
142	C₂×C₈	#10625325	1	2	yes	—
143	C₂×C₈	Dic₁₈₈	1	2	yes	—
144	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
145	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
146	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₈	#10625350	1	2	yes	—
148	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₄⋊C₄	—	94	4	no	—
150	C₄⋊C₄	—	188	2	no	—
151	C₄⋊C₄	—	188	2	no	—
152	C₄⋊C₄	—	188	4	no	—
153	C₄⋊C₄	—	188	2	no	—
154	C₄²	#10625325	1	2	yes	—
155	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
158	Q₈	—	376	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	376	4	no	—
162	C₂×C₄	—	376	2	no	—
163	C₂×C₄	—	376	2	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	188	4	no	—
166	C₂×C₄	#13093946	1	2	yes	—
167	C₂×C₄	#13093946	1	2	yes	—
168	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#13094012	1	1	yes	characteristic
170	C₂×C₄	#13094024	1	1	yes	characteristic
171	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
172	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
174	C₈	#13093949	1	4	yes	—
175	C₈	C₄×Dic₉₄	1	2	yes	—
176	C₈	#13094003	1	2	yes	—
177	C₂²	C₂×#13093946	1	1	yes	characteristic
178	C₂²	#15689470	1	1	yes	characteristic
179	C₂²	#15689528	1	1	yes	characteristic
180	C₂²	#15689561	1	2	yes	—
181	C₂²	#15689566	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	376	4	no	—
185	C₄	—	752	2	no	—
186	C₄	#15689029	1	1	yes	characteristic
187	C₄	C₂×#13093949	1	1	yes	characteristic
188	C₄	#15689474	1	1	yes	characteristic
189	C₄	#15689521	1	1	yes	characteristic
190	C₂	#20244349	1	1	yes	characteristic
191	C₂	#20245105	1	1	yes	characteristic
192	C₂	#20245109	1	1	yes	characteristic
193	C₂	#20245625	1	2	yes	—
194	C₂	#20245674	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#13283
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #15689528	C₄	semidirect
34. C₂×C₁₆×C₄₇	Q₈	semidirect
113. C₄₇	#13283	semidirect
125. C₂×C₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
#20243552	2
#20241480	2
C₂×C₄₇×#346	2
#13283	47