Subgroups — #26216480

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	2	yes	maximal
2	#20243522	C₂	1	8	yes	maximal, complemented
3	#20243105	C₂	1	4	yes	maximal
4	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	#15688993	C₂²	1	4	yes	—
7	#15688989	C₂²	1	4	yes	—
8	#15688983	—	2	4	no	—
9	C₄×C₄⋊Dic₄₇	—	2	4	no	—
10	#15688886	C₂²	1	16	yes	—
11	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#97×C₄₇	C₄	1	8	yes	complemented
14	C₂×#97×C₄₇	C₂²	1	4	yes	—
15	#13093983	—	2	16	no	—
16	C₂×C₄⋊Dic₄₇	—	2	8	no	—
17	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
18	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
19	C₂×C₄×Dic₄₇	—	2	4	no	—
20	C₄₇⋊#95	—	2	8	no	—
21	C₂³×C₄×C₄₇	D₄	1	2	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	—	2	8	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
30	#97×C₄₇	C₂×C₄	1	16	yes	—
31	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	—	4	4	no	—
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	#41	1	8	yes	—
39	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₄₇	—	2	8	no	—
41	C₂×C₈×C₄₇	—	2	16	no	—
42	C₂×C₈×C₄₇	—	2	8	no	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
44	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
45	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
46	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
47	C₂²⋊Dic₄₇	—	4	8	no	—
48	C₄⋊Dic₄₇	—	2	16	no	—
49	C₄⋊Dic₄₇	—	4	16	no	—
50	C₄×Dic₄₇	—	4	8	no	—
51	C₂³×C₄₇	—	2	4	no	—
52	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂³×C₄₇	#124	1	4	yes	—
54	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	4	8	no	—
57	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₄₇	#117	1	8	yes	—
60	C₂×C₄×C₄₇	#124	1	4	yes	—
61	C₂×C₄×C₄₇	#125	1	8	yes	—
62	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
65	C₂×Dic₄₇	—	4	8	no	—
66	C₂×Dic₄₇	—	4	16	no	—
67	C₈×C₄₇	—	2	8	no	—
68	#20762	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	—	2	8	no	—
70	C₂²×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	#382	1	4	yes	—
72	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂×#124	1	2	yes	—
74	C₂²×C₄₇	#575	1	4	yes	—
75	C₄×C₄₇	—	2	8	no	—
76	C₄×C₄₇	#382	1	4	yes	—
77	C₄×C₄₇	C₂×#117	1	2	yes	—
78	C₄×C₄₇	C₂×#125	1	2	yes	—
79	Dic₄₇	—	8	8	no	—
80	C₂²×#97	D₄₇	1	1	yes	characteristic
81	#2285	—	47	2	no	—
82	#1836	—	47	8	no	—
83	#1735	—	47	4	no	—
84	C₂×C₄₇	—	2	4	no	—
85	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
86	C₂×C₄₇	#2911	1	2	yes	—
87	C₂×C₄₇	#3135	1	4	yes	—
88	C₂³×C₈	C₂×D₄₇	1	2	yes	—
89	C₂×#117	—	47	4	no	—
90	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
91	C₂×#97	Dic₄₇	1	8	yes	complemented
92	C₂×#97	C₂×D₄₇	1	4	yes	—
93	#384	—	47	4	no	—
94	#382	—	47	4	no	—
95	#380	—	94	4	no	—
96	C₄×C₄⋊C₄	—	94	4	no	—
97	#340	—	47	16	no	—
98	C₄₇	#20762	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	D₁₈₈	1	2	yes	—
100	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
103	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
104	#117	—	94	16	no	—
105	C₂×C₄⋊C₄	—	47	8	no	—
106	C₂×C₄⋊C₄	—	94	8	no	—
107	C₂×C₂²⋊C₄	—	94	4	no	—
108	C₂×C₄²	—	94	4	no	—
109	C₂×C₄²	Dic₉₄	1	4	yes	—
110	C₂×C₄²	D₁₈₈	1	2	yes	—
111	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
112	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
113	#97	C₂×Dic₄₇	1	16	yes	—
114	#95	—	94	8	no	—
115	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
116	C₂²×C₄	—	4	4	no	—
117	C₂²×C₄	—	94	8	no	—
118	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
119	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
120	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
121	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
122	C₂²×C₄	#10625350	1	8	yes	—
123	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
124	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
128	C₄⋊C₄	—	94	16	no	—
129	C₄⋊C₄	—	188	16	no	—
130	C₂²⋊C₄	—	188	8	no	—
131	C₄²	—	188	8	no	—
132	C₄²	C₄⋊Dic₄₇	1	8	yes	—
133	C₄²	C₂×Dic₉₄	1	4	yes	—
134	C₄²	C₂²×Dic₄₇	1	4	yes	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
137	C₂³	#13094012	1	4	yes	—
138	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	188	8	no	—
142	C₂×C₄	—	188	16	no	—
143	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
145	C₂×C₄	#13093983	1	8	yes	—
146	C₂×C₄	#13094012	1	4	yes	—
147	C₂×C₄	#13094016	1	8	yes	—
148	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#10625350	1	4	yes	—
151	C₈	—	2	8	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	#15688989	1	4	yes	—
155	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
156	C₂²	C₂×#13094012	1	2	yes	—
157	C₂²	#15689923	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	376	8	no	—
160	C₄	#15688989	1	4	yes	—
161	C₄	C₂×#13093983	1	2	yes	—
162	C₄	C₂×#13094016	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×#15688989	1	1	yes	characteristic
165	C₂	#20251978	1	2	yes	—
166	C₂	#20253734	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#20762
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20243522	C₂	semidirect
13. C₂×#97×C₄₇	C₄	semidirect
91. C₂×#97	Dic₄₇	semidirect
98. C₄₇	#20762	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20243522	2
#20243105	2
C₂²×#97×C₄₇	2
#20762	47