Subgroups — #26134318

Order 12032.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688951	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688949	C₂	1	1	yes	maximal, characteristic
3	#20244121	C₂	1	2	yes	maximal
4	#20241480	C₂	1	2	yes	maximal
5	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	#15689021	—	2	2	no	—
9	#15688951	C₄	1	4	yes	complemented
10	#15688949	—	2	2	no	—
11	#15688886	—	2	2	no	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#346	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	2	2	no	—
18	C₂×Dic₄₇⋊C₄	—	4	2	no	—
19	#13093958	—	2	2	no	—
20	#13093958	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	C₂×C₄	1	4	yes	—
23	C₄₇⋊#95	—	4	2	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	—	2	1	no	—
28	C₂×C₁₆×C₄₇	—	2	1	no	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₄₇	Q₈	1	2	yes	—
32	C₂×C₁₆×C₄₇	Q₈	1	2	yes	—
33	C₄×C₈×C₄₇	—	2	1	no	—
34	C₄×C₈×C₄₇	—	2	1	no	—
35	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	2	no	—
39	C₂²×Dic₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	2	1	no	—
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
49	C₄²×C₄₇	—	2	1	no	—
50	C₄²×C₄₇	—	2	1	no	—
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	C₄⋊Dic₄₇	—	4	2	no	—
55	Dic₄₇⋊C₄	—	8	4	no	—
56	C₁₆×C₄₇	—	2	2	no	—
57	C₁₆×C₄₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
69	C₂×Dic₄₇	—	4	4	no	—
70	C₂×Dic₄₇	—	8	2	no	—
71	C₂×Dic₄₇	—	8	2	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₈×C₄₇	#125	1	1	yes	characteristic
80	#13167	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₄₇	#466	1	1	yes	characteristic
83	C₂²×C₄₇	#500	1	1	yes	characteristic
84	C₂²×C₄₇	#509	1	2	yes	—
85	C₂²×C₄₇	#510	1	2	yes	—
86	C₄×C₄₇	—	4	1	no	—
87	C₄×C₄₇	—	4	1	no	—
88	C₄×C₄₇	#389	1	1	yes	characteristic
89	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
90	C₄×C₄₇	#467	1	1	yes	characteristic
91	C₄×C₄₇	#499	1	1	yes	characteristic
92	Dic₄₇	—	8	4	no	—
93	Dic₄₇	—	16	2	no	—
94	C₂×#367	—	47	1	no	—
95	C₂×#366	—	47	1	no	—
96	C₂×#346	D₄₇	1	1	yes	characteristic
97	#1942	—	47	2	no	—
98	#1380	—	47	2	no	—
99	C₂×C₄₇	#1942	1	1	yes	characteristic
100	C₂×C₄₇	#2175	1	2	yes	—
101	C₂×C₄₇	#2218	1	1	yes	characteristic
102	C₂×C₄₇	#2219	1	1	yes	characteristic
103	C₂×C₄₇	#2255	1	2	yes	—
104	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
106	C₂×#107	—	47	2	no	—
107	C₂×#107	—	47	2	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	#389	—	94	2	no	—
110	#367	—	47	4	no	—
111	#366	—	94	2	no	—
112	#346	—	2	2	no	—
113	#340	—	94	2	no	—
114	C₄₇	#13167	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	94	2	no	—
118	C₂×C₄⋊C₄	—	94	2	no	—
119	C₂×C₄⋊C₄	—	188	2	no	—
120	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	Dic₉₄	1	2	yes	—
124	C₂×C₁₆	Dic₉₄	1	2	yes	—
125	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
127	#107	—	47	4	no	—
128	#107	—	94	2	no	—
129	#107	—	94	2	no	—
130	#107	—	94	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	#95	—	188	2	no	—
134	C₂²×C₄	—	188	2	no	—
135	C₂²×C₄	—	188	2	no	—
136	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
137	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	Dic₁₈₈	1	2	yes	—
142	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
145	C₂×C₈	#10625350	1	1	yes	characteristic
146	C₂×C₈	#10625350	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₄⋊C₄	—	94	4	no	—
149	C₄⋊C₄	—	376	4	no	—
150	C₄⋊C₄	—	188	2	no	—
151	C₄⋊C₄	—	188	2	no	—
152	C₄⋊C₄	—	188	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	Dic₄₇⋊C₄	1	4	yes	complemented
157	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	376	2	no	—
163	C₂×C₄	—	376	2	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	376	2	no	—
166	C₂×C₄	—	376	2	no	—
167	C₂×C₄	—	188	4	no	—
168	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
169	C₂×C₄	#13093998	1	1	yes	characteristic
170	C₂×C₄	#13094024	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
172	C₂×C₄	#13094046	1	1	yes	characteristic
173	C₂×C₄	#13094046	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
176	C₈	C₄×Dic₉₄	1	1	yes	characteristic
177	C₈	#13094003	1	1	yes	characteristic
178	C₈	#13094016	1	1	yes	characteristic
179	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
180	C₂²	#15689359	1	1	yes	characteristic
181	C₂²	#15689528	1	1	yes	characteristic
182	C₂²	#15689561	1	2	yes	—
183	C₂²	#15689566	1	2	yes	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	376	4	no	—
187	C₄	—	752	2	no	—
188	C₄	#15689021	1	1	yes	characteristic
189	C₄	C₄×Dic₁₈₈	1	1	yes	characteristic
190	C₄	#15689364	1	1	yes	characteristic
191	C₄	#15689525	1	1	yes	characteristic
192	C₂	#20244121	1	1	yes	characteristic
193	C₂	#20245254	1	2	yes	—
194	C₂	#20245491	1	1	yes	characteristic
195	C₂	#20245496	1	1	yes	characteristic
196	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	#13167
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
9. #15688951	C₄	semidirect
57. C₁₆×C₄₇	C₄⋊C₄	semidirect
114. C₄₇	#13167	semidirect
156. C₁₆	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688951	2
C₂×#15688949	2
#20244121	2
#20241480	2
C₂×C₄₇×#346	2
#13167	47