Subgroups — #26105654

Order 12032.

582 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093935	C₂	1	2	yes	maximal
2	C₂×#15688845	C₂	1	2	yes	maximal
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093935	C₄	1	8	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688845	C₄	1	8	yes	complemented
10	#15688844	C₄	1	4	yes	complemented
11	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
12	C₂×#97×C₄₇	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	#13093935	—	2	16	no	—
19	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	#97×C₄₇	—	2	8	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	2	4	no	—
36	C₂×C₈×C₄₇	C₄²	1	4	yes	—
37	C₂×C₈×C₄₇	C₄²	1	4	yes	—
38	C₂×C₈×C₄₇	M₁₆	1	8	yes	—
39	C₄²×C₄₇	—	2	8	no	—
40	C₄²×C₄₇	—	2	4	no	—
41	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
43	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄₇⋊C₈	—	2	8	no	—
45	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
46	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
47	C₂×C₄₇⋊C₈	M₁₆	1	16	yes	—
48	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	8	no	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
56	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
57	C₈×C₄₇	—	2	4	no	—
58	C₈×C₄₇	#97	1	8	yes	complemented
59	C₄₇⋊C₈	—	2	8	no	—
60	C₄₇⋊C₈	#97	1	16	yes	complemented
61	#7846	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	C₄³	1	1	yes	characteristic
63	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
64	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
65	C₄×C₄₇	—	4	4	no	—
66	C₄×C₄₇	C₂×#97	1	4	yes	—
67	C₄×C₄₇	C₂×#97	1	2	yes	—
68	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
69	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
70	C₄×#97	—	47	2	no	—
71	C₄×#97	D₄₇	1	1	yes	characteristic
72	C₂×#322	—	47	2	no	—
73	C₂×#322	—	47	1	no	—
74	C₂×C₈²	—	47	1	no	—
75	C₂×C₄₇	#1455	1	4	yes	—
76	C₂×C₄₇	C₄×#97	1	2	yes	—
77	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
78	C₂×#97	—	47	8	no	—
79	C₂×#97	Dic₄₇	1	4	yes	—
80	C₂×C₄×C₈	—	47	2	no	—
81	C₂×C₄×C₈	—	47	2	no	—
82	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
84	C₄³	C₂×D₄₇	1	1	yes	characteristic
85	#322	—	47	4	no	—
86	#322	—	47	8	no	—
87	C₈²	—	47	4	no	—
88	C₄₇	#7846	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	47	4	no	—
90	C₂²×C₈	—	47	4	no	—
91	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
92	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
93	C₂×C₄²	C₄×D₄₇	1	4	yes	—
94	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
95	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
96	#97	—	2	8	no	—
97	#97	—	94	16	no	—
98	C₄×C₈	—	47	4	no	—
99	C₄×C₈	—	47	4	no	—
100	C₄×C₈	—	47	4	no	—
101	C₄×C₈	—	47	4	no	—
102	C₄×C₈	C₄×D₄₇	1	2	yes	—
103	C₄×C₈	C₄×D₄₇	1	4	yes	—
104	C₄×C₈	C₄×D₄₇	1	2	yes	—
105	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
106	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
107	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	—	47	8	no	—
110	C₂×C₈	—	47	16	no	—
111	C₂×C₈	—	47	8	no	—
112	C₂×C₈	—	94	8	no	—
113	C₂×C₈	#10625329	1	8	yes	—
114	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
115	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
119	C₄²	C₂×C₄×D₄₇	1	2	yes	—
120	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₄²×D₄₇	1	2	yes	—
125	C₂×C₄	C₄²×D₄₇	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
129	C₂×C₄	C₂×#10625329	1	4	yes	—
130	C₈	—	2	4	no	—
131	C₈	—	47	16	no	—
132	C₈	—	94	8	no	—
133	C₈	#13093935	1	8	yes	complemented
134	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
135	C₂²	M₁₆×Dic₄₇	1	4	yes	—
136	C₂²	C₄×#10625329	1	2	yes	—
137	C₄	—	4	4	no	—
138	C₄	#97×D₄₇	1	4	yes	—
139	C₄	C₂×#13093935	1	2	yes	—
140	C₄	#15689075	1	4	yes	—
141	C₄	C₄×#10625329	1	2	yes	—
142	C₂	#20241691	1	4	yes	—
143	C₂	#97×Dic₄₇	1	2	yes	—
144	C₂	C₄×#13093935	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#7846
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
9. #15688845	C₄	semidirect
10. #15688844	C₄	semidirect
11. C₈×C₄₇⋊C₈	C₄	semidirect
58. C₈×C₄₇	#97	semidirect
60. C₄₇⋊C₈	#97	semidirect
88. C₄₇	#7846	semidirect
133. C₈	#13093935	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093935	2
C₂×#15688845	2
C₂×#15688844	2
C₂×C₈×C₄₇⋊C₈	2
C₄×#97×C₄₇	2
#7846	47