Subgroups — #26109945

Order 12032.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	#20241223	C₂	1	4	yes	maximal
4	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15688878	—	2	2	no	—
9	C₂×#97×C₄₇	C₄	1	2	yes	—
10	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	2	yes	—
24	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	#97×C₄₇	—	2	4	no	—
27	#97×C₄₇	—	2	2	no	—
28	#97×C₄₇	C₈	1	4	yes	complemented
29	C₄×C₈×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	4	2	no	—
46	C₄²×C₄₇	C₂×C₈	1	2	yes	—
47	C₄²×C₄₇	M₁₆	1	2	yes	—
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₄²×C₄₇	#41	1	2	yes	—
50	C₂×C₄₇⋊C₈	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	4	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₄₇	#117	1	2	yes	—
64	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#125	1	2	yes	—
67	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₄₇	#137	1	2	yes	—
72	C₈×C₄₇	—	2	4	no	—
73	C₈×C₄₇	—	2	4	no	—
74	C₄₇⋊C₈	—	8	2	no	—
75	C₄₇⋊C₈	—	8	2	no	—
76	#8786	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	#382	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₄₇	#423	1	1	yes	characteristic
80	C₂²×C₄₇	#428	1	2	yes	—
81	C₂²×C₄₇	#430	1	2	yes	—
82	C₄×C₄₇	—	2	4	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	4	2	no	—
85	C₄×C₄₇	#431	1	2	yes	—
86	C₄×C₄₇	#432	1	2	yes	—
87	C₄×C₄₇	#489	1	2	yes	—
88	C₄×C₄₇	#490	1	1	yes	characteristic
89	C₄×C₄₇	#491	1	1	yes	characteristic
90	#1842	—	47	1	no	—
91	#1831	—	47	1	no	—
92	C₄×#97	D₄₇	1	1	yes	characteristic
93	#1277	—	47	4	no	—
94	C₂×C₄₇	#1565	1	2	yes	—
95	C₂×C₄₇	#1569	1	2	yes	—
96	C₂×C₄₇	#1831	1	1	yes	characteristic
97	C₂×C₄₇	#1836	1	1	yes	characteristic
98	C₂×C₄₇	#1838	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	47	2	no	—
100	C₂×C₄⋊C₈	—	47	2	no	—
101	C₂×C₄⋊C₈	—	94	2	no	—
102	C₂×#97	Dic₄₇	1	2	yes	—
103	C₂×#97	Dic₄₇	1	1	yes	characteristic
104	C₂×#97	Dic₄₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
106	C₄³	C₂×D₄₇	1	1	yes	characteristic
107	#336	—	94	2	no	—
108	C₄₇	#8786	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	94	2	no	—
110	C₂²×C₈	—	94	2	no	—
111	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
112	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
113	C₂×C₄²	Dic₉₄	1	2	yes	—
114	C₂×C₄²	D₁₈₈	1	2	yes	—
115	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	C₄⋊C₈	—	94	4	no	—
119	C₄⋊C₈	—	94	4	no	—
120	C₄⋊C₈	—	188	4	no	—
121	#97	—	2	4	no	—
122	#97	—	2	2	no	—
123	#97	C₄₇⋊C₈	1	4	yes	complemented
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
127	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
128	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	188	2	no	—
136	C₂×C₈	—	188	2	no	—
137	C₂×C₈	—	188	4	no	—
138	C₂×C₈	—	188	2	no	—
139	C₂×C₈	—	188	2	no	—
140	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
141	C₂×C₈	#10625329	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
147	C₄²	#10625329	1	2	yes	—
148	C₄²	#10625350	1	2	yes	—
149	C₄²	#10625350	1	2	yes	—
150	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
158	C₂×C₄	#13093983	1	2	yes	—
159	C₂×C₄	#13094012	1	1	yes	characteristic
160	C₂×C₄	#13094012	1	1	yes	characteristic
161	C₂×C₄	#13094016	1	2	yes	—
162	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₂×#10625329	1	2	yes	—
164	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
165	C₂×C₄	#13094052	1	2	yes	—
166	C₂×C₄	#13094057	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	2	4	no	—
169	C₈	—	376	2	no	—
170	C₈	—	376	2	no	—
171	C₂²	#15688989	1	1	yes	characteristic
172	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
173	C₂²	#15689147	1	1	yes	characteristic
174	C₂²	#15689165	1	2	yes	—
175	C₂²	#15689172	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#15689176	1	2	yes	—
180	C₄	#15689180	1	2	yes	—
181	C₄	#15689483	1	2	yes	—
182	C₄	#15689488	1	1	yes	characteristic
183	C₄	#15689492	1	1	yes	characteristic
184	C₂	#20242140	1	2	yes	—
185	C₂	#20242155	1	2	yes	—
186	C₂	#20243505	1	1	yes	characteristic
187	C₂	#20243522	1	1	yes	characteristic
188	C₂	#20243529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#8786
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
28. #97×C₄₇	C₈	semidirect
108. C₄₇	#8786	semidirect
123. #97	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243505	2
#20241223	2
C₄×#97×C₄₇	2
#8786	47