Subgroups — #26109944

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	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20241222	C₂	1	4	yes	maximal
4	C₄₇×#1831	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×#13093935	C₄	1	2	yes	—
8	C₂×#13093935	C₄	1	1	yes	characteristic
9	C₂×#13093935	C₄	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
12	C₄³×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#336	—	2	2	no	—
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
19	#13093935	—	2	2	no	—
20	#13093935	—	2	4	no	—
21	#13093935	C₈	1	4	yes	complemented
22	C₄×C₄₇⋊C₈	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	4	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	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	C₂×C₈	1	2	yes	—
44	C₄²×C₄₇	M₁₆	1	2	yes	—
45	C₄²×C₄₇	#41	1	2	yes	—
46	C₄²×C₄₇	#41	1	2	yes	—
47	C₂×C₄₇⋊C₈	—	2	2	no	—
48	C₂×C₄₇⋊C₈	—	2	2	no	—
49	C₂×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	4	no	—
53	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
54	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
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	—	8	2	no	—
73	C₄₇⋊C₈	—	2	4	no	—
74	C₄₇⋊C₈	—	2	4	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	D₄₇	1	1	yes	characteristic
92	C₄×#97	—	47	1	no	—
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	—	94	2	no	—
101	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
102	C₂×#97	—	47	1	no	—
103	C₂×#97	—	47	2	no	—
104	C₂×#97	—	47	1	no	—
105	C₂×C₄×C₈	—	47	2	no	—
106	C₄³	C₂×D₄₇	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	C₄₇	#8786	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	47	2	no	—
111	C₂²×C₈	—	47	2	no	—
112	C₂²×C₈	—	94	2	no	—
113	C₂×C₄²	Dic₉₄	1	2	yes	—
114	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	94	4	no	—
120	C₄⋊C₈	—	188	4	no	—
121	#97	—	47	4	no	—
122	#97	—	94	4	no	—
123	#97	—	94	2	no	—
124	C₄×C₈	—	94	2	no	—
125	C₄×C₈	—	94	2	no	—
126	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
127	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
128	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	47	4	no	—
135	C₂×C₈	—	47	4	no	—
136	C₂×C₈	—	94	2	no	—
137	C₂×C₈	—	94	2	no	—
138	C₂×C₈	—	94	2	no	—
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	188	2	no	—
141	C₂×C₈	—	188	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₈×D₄₇	1	2	yes	—
147	C₄²	#10625324	1	2	yes	—
148	C₄²	#10625352	1	2	yes	—
149	C₄²	#10625354	1	2	yes	—
150	C₂³	C₂×Dic₄₇⋊C₄	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	Dic₄₇⋊C₈	1	4	yes	—
158	C₂×C₄	#13093982	1	2	yes	—
159	C₂×C₄	#13094011	1	1	yes	characteristic
160	C₂×C₄	#13094013	1	1	yes	characteristic
161	C₂×C₄	#13094015	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
164	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
165	C₂×C₄	#13094049	1	2	yes	—
166	C₂×C₄	#13094054	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	94	4	no	—
169	C₈	—	94	4	no	—
170	C₈	—	376	2	no	—
171	C₂²	#15688988	1	1	yes	characteristic
172	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
173	C₂²	#15689146	1	1	yes	characteristic
174	C₂²	#15689166	1	2	yes	—
175	C₂²	#15689171	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#15689175	1	2	yes	—
180	C₄	#15689179	1	2	yes	—
181	C₄	#15689482	1	2	yes	—
182	C₄	#15689489	1	1	yes	characteristic
183	C₄	#15689493	1	1	yes	characteristic
184	C₂	#20242141	1	2	yes	—
185	C₂	#20242156	1	2	yes	—
186	C₂	#20243504	1	1	yes	characteristic
187	C₂	#20243521	1	1	yes	characteristic
188	C₂	#20243528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#8786
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
21. #13093935	C₈	semidirect
108. C₄₇	#8786	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄×#13093935	2
#20241222	2
C₄₇×#1831	2
#8786	47