Subgroups — #26132664

Order 12032.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243552	C₂	1	1	yes	maximal, characteristic
2	#20243548	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×#13093958	—	2	1	no	—
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₂×#13093956	—	2	1	no	—
9	#15688993	—	2	1	no	—
10	#15688989	—	2	1	no	—
11	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#346	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₄⋊Dic₄₇	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	4	2	no	—
17	#13093958	—	2	4	no	—
18	#13093958	—	2	4	no	—
19	#13093958	—	4	2	no	—
20	#13093956	—	4	2	no	—
21	C₄₇⋊#95	—	4	2	no	—
22	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	—	2	2	no	—
26	C₂×C₁₆×C₄₇	—	2	2	no	—
27	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	D₈	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₄₇	—	4	2	no	—
33	C₂²×Dic₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₇	Q₁₆	1	2	yes	—
43	C₄⋊Dic₄₇	—	4	2	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	C₄⋊Dic₄₇	—	4	2	no	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	C₄⋊Dic₄₇	—	8	4	no	—
48	C₁₆×C₄₇	—	2	4	no	—
49	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	#107	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₄₇	—	8	2	no	—
60	C₂×Dic₄₇	—	8	2	no	—
61	C₂×Dic₄₇	—	8	4	no	—
62	C₂×Dic₄₇	—	8	2	no	—
63	C₂×Dic₄₇	—	8	2	no	—
64	C₈×C₄₇	#106	1	4	yes	—
65	C₈×C₄₇	#117	1	2	yes	—
66	C₈×C₄₇	#125	1	2	yes	—
67	#12890	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
69	C₂²×C₄₇	#486	1	1	yes	characteristic
70	C₂²×C₄₇	#486	1	1	yes	characteristic
71	C₂²×C₄₇	#509	1	2	yes	—
72	C₂²×C₄₇	#510	1	2	yes	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	#382	1	1	yes	characteristic
75	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
76	C₄×C₄₇	#487	1	1	yes	characteristic
77	C₄×C₄₇	#487	1	1	yes	characteristic
78	Dic₄₇	—	16	2	no	—
79	Dic₄₇	—	16	2	no	—
80	C₂×#346	D₄₇	1	1	yes	characteristic
81	#1846	—	47	1	no	—
82	#1845	—	47	1	no	—
83	#1380	—	47	4	no	—
84	C₂×C₄₇	#1834	1	1	yes	characteristic
85	C₂×C₄₇	#2159	1	2	yes	—
86	C₂×C₄₇	#2243	1	2	yes	—
87	C₂×C₄₇	#2244	1	1	yes	characteristic
88	C₂×C₄₇	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
90	C₂×#107	—	47	2	no	—
91	C₂×#107	—	47	2	no	—
92	C₂×#107	—	94	1	no	—
93	C₂×#106	—	94	1	no	—
94	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
95	#384	—	94	1	no	—
96	#382	—	94	1	no	—
97	#346	Dic₄₇	1	4	yes	complemented
98	C₄₇	#12890	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	94	2	no	—
102	C₂×C₄⋊C₄	—	94	2	no	—
103	C₂×C₄⋊C₄	—	188	2	no	—
104	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
108	#107	—	94	4	no	—
109	#107	—	94	4	no	—
110	#107	—	188	2	no	—
111	#106	—	188	2	no	—
112	C₄×C₈	Dic₉₄	1	2	yes	—
113	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
114	#95	—	188	2	no	—
115	C₂²×C₄	—	188	2	no	—
116	C₂²×C₄	—	188	2	no	—
117	C₂²×C₄	D₃₇₆	1	2	yes	—
118	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
119	C₂×C₈	#10625325	1	2	yes	—
120	C₂×C₈	#10625325	1	2	yes	—
121	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
122	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
123	C₂×C₈	#10625350	1	2	yes	—
124	C₂×C₈	#10625350	1	2	yes	—
125	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
126	C₄⋊C₄	—	376	4	no	—
127	C₄⋊C₄	—	188	2	no	—
128	C₄⋊C₄	—	188	2	no	—
129	C₄⋊C₄	—	188	2	no	—
130	C₄⋊C₄	—	188	2	no	—
131	C₄²	Dic₁₈₈	1	2	yes	—
132	C₄²	C₄⋊Dic₄₇	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	376	4	no	—
138	C₂×C₄	—	376	2	no	—
139	C₂×C₄	—	376	2	no	—
140	C₂×C₄	—	376	2	no	—
141	C₂×C₄	—	376	2	no	—
142	C₂×C₄	#13093958	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₄	#13094012	1	1	yes	characteristic
145	C₂×C₄	#13094012	1	1	yes	characteristic
146	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
147	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
149	C₈	#13093956	1	4	yes	—
150	C₈	#13093983	1	2	yes	—
151	C₈	#13094016	1	2	yes	—
152	C₂²	C₂×#13093958	1	1	yes	characteristic
153	C₂²	#15689470	1	1	yes	characteristic
154	C₂²	#15689470	1	1	yes	characteristic
155	C₂²	#15689561	1	2	yes	—
156	C₂²	#15689566	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	752	2	no	—
159	C₄	—	752	2	no	—
160	C₄	#15688989	1	1	yes	characteristic
161	C₄	C₂×#13093956	1	1	yes	characteristic
162	C₄	#15689474	1	1	yes	characteristic
163	C₄	#15689474	1	1	yes	characteristic
164	C₂	#20243516	1	1	yes	characteristic
165	C₂	#20245176	1	2	yes	—
166	C₂	#20245625	1	2	yes	—
167	C₂	#20245630	1	1	yes	characteristic
168	C₂	#20245634	1	1	yes	characteristic

	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	#12890
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
13. C₄₇×#346	C₄	semidirect
97. #346	Dic₄₇	semidirect
98. C₄₇	#12890	semidirect

Maximal subgroups

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

subgroup	index
#20243552	2
#20243548	2
#20241480	2
C₂×C₄₇×#346	2
#12890	47