Subgroups — #26103223

Order 12032.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
2	#20243267	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	1	no	—
5	C₂×#13093960	C₄	1	1	yes	characteristic
6	#15689161	C₄	1	1	yes	characteristic
7	#15689078	—	2	1	no	—
8	#15688944	—	2	1	no	—
9	#15688944	C₄	1	2	yes	complemented
10	#15688883	—	2	1	no	—
11	C₄₇×#431	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#322	C₄	1	2	yes	complemented
14	C₈²×C₄₇	—	2	1	no	—
15	C₂×#10625329	—	2	1	no	—
16	#13093983	—	2	1	no	—
17	#13093964	—	2	2	no	—
18	#13093964	—	4	1	no	—
19	C₂×C₄₇⋊C₁₆	—	4	1	no	—
20	#13093960	—	2	1	no	—
21	#13093960	C₈	1	2	yes	complemented
22	#13093958	—	2	1	no	—
23	#13093956	C₈	1	2	yes	complemented
24	#13093934	—	4	1	no	—
25	C₈×Dic₄₇	—	4	1	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₂²⋊C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	1	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
48	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
51	C₄²×C₄₇	—	2	1	no	—
52	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²⋊Dic₄₇	—	4	1	no	—
55	C₄⋊Dic₄₇	—	2	2	no	—
56	C₄×Dic₄₇	—	4	1	no	—
57	#10625329	—	2	2	no	—
58	#10625329	—	4	1	no	—
59	C₂×C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₁₆	—	4	2	no	—
61	C₂³×C₄₇	#104	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
71	C₂×Dic₄₇	—	4	2	no	—
72	C₈×C₄₇	—	4	4	no	—
73	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
74	C₈×C₄₇	#97	1	1	yes	characteristic
75	C₈×C₄₇	#102	1	1	yes	characteristic
76	C₈×C₄₇	#103	1	1	yes	characteristic
77	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₄₇	#106	1	1	yes	characteristic
80	C₈×C₄₇	#107	1	1	yes	characteristic
81	C₄₇⋊C₈	—	4	2	no	—
82	#7181	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	4	1	no	—
84	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₄₇	#339	1	1	yes	characteristic
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	4	1	no	—
89	C₄×C₄₇	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₄₇	#336	1	1	yes	characteristic
92	C₄×C₄₇	#340	1	1	yes	characteristic
93	Dic₄₇	—	8	2	no	—
94	C₂×#364	—	47	1	no	—
95	#1772	—	47	1	no	—
96	#1450	D₄₇	1	1	yes	characteristic
97	C₂×C₄₇	—	2	1	no	—
98	C₂×C₄₇	—	2	1	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	#1279	1	1	yes	characteristic
101	C₂×M₃₂	—	94	1	no	—
102	#431	—	2	1	no	—
103	C₂×#108	—	47	1	no	—
104	#427	—	47	1	no	—
105	#405	—	94	1	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	#364	—	47	2	no	—
108	#364	—	94	1	no	—
109	#339	—	94	1	no	—
110	#322	Dic₄₇	1	2	yes	complemented
111	C₈²	—	2	1	no	—
112	C₄₇	#7181	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	94	1	no	—
114	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
115	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
116	#117	—	94	1	no	—
117	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
118	M₃₂	—	94	2	no	—
119	M₃₂	—	188	1	no	—
120	C₂×C₁₆	—	188	1	no	—
121	#108	—	47	2	no	—
122	#108	—	94	1	no	—
123	#107	—	94	1	no	—
124	#106	—	47	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	#97	—	188	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	188	1	no	—
130	C₄×C₈	Dic₉₄	1	1	yes	characteristic
131	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
136	M₁₆	—	94	2	no	—
137	M₁₆	—	188	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	188	1	no	—
140	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
141	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
142	C₂×C₈	#10625324	1	1	yes	characteristic
143	C₂×C₈	#10625324	1	1	yes	characteristic
144	C₂×C₈	#10625325	1	1	yes	characteristic
145	C₂×C₈	#10625325	1	1	yes	characteristic
146	C₂×C₈	D₃₇₆	1	1	yes	characteristic
147	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
149	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
151	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
152	C₄⋊C₄	—	94	2	no	—
153	C₂²⋊C₄	—	188	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	188	1	no	—
156	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
157	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
158	C₁₆	—	188	2	no	—
159	C₂³	#13093952	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	188	2	no	—
164	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
165	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
166	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
167	C₂×C₄	#13093946	1	1	yes	characteristic
168	C₂×C₄	#13093949	1	1	yes	characteristic
169	C₂×C₄	#13093952	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	188	2	no	—
172	C₈	C₈×Dic₄₇	1	1	yes	characteristic
173	C₈	#13093934	1	1	yes	characteristic
174	C₈	#13093946	1	1	yes	characteristic
175	C₈	#13093949	1	1	yes	characteristic
176	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
177	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
178	C₈	#13093956	1	1	yes	characteristic
179	C₈	#13093958	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	#15688852	1	1	yes	characteristic
182	C₂²	#15688855	1	1	yes	characteristic
183	C₂²	#15688883	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	376	2	no	—
187	C₄	#15688852	1	1	yes	characteristic
188	C₄	#15688855	1	1	yes	characteristic
189	C₄	#15688877	1	1	yes	characteristic
190	C₄	#15688886	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#20241228	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13093946
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂×C₄₇	#1279
prime radical	C₂³×C₄₇	#104
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#7181
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
9. #15688944	C₄	semidirect
13. C₄₇×#322	C₄	semidirect
21. #13093960	C₈	semidirect
23. #13093956	C₈	semidirect
110. #322	Dic₄₇	semidirect
112. C₄₇	#7181	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688944	2
#20243267	2
C₄₇×#1450	2
#7181	47