Subgroups — #26134286

Order 12032.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245148	C₂	1	2	yes	maximal
2	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20243267	C₂	1	1	yes	maximal, characteristic
4	#20241478	C₂	1	2	yes	maximal
5	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₄	1	1	yes	characteristic
9	#15689161	C₄	1	1	yes	characteristic
10	#15689078	C₂²	1	2	yes	—
11	Dic₄₇⋊C₁₆	—	2	4	no	—
12	C₄×C₄₇⋊C₁₆	—	2	2	no	—
13	#15688883	C₄	1	2	yes	—
14	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#364	—	2	2	no	—
17	C₂×#10625329	C₂×C₄	1	2	yes	—
18	#13093983	C₂×C₄	1	2	yes	—
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	1	no	—
22	C₂×C₄₇⋊C₁₆	—	2	1	no	—
23	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
24	C₂×C₄₇⋊C₁₆	Q₈	1	2	yes	—
25	#13093960	—	2	2	no	—
26	#13093958	C₈	1	2	yes	—
27	#13093956	C₈	1	2	yes	—
28	#13093934	—	2	2	no	—
29	C₈×Dic₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₄₇	—	2	2	no	—
34	M₃₂×C₄₇	—	2	2	no	—
35	C₂×C₁₆×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
44	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
50	C₄²×C₄₇	—	2	1	no	—
51	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²⋊Dic₄₇	—	2	2	no	—
53	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
54	C₄×Dic₄₇	—	2	2	no	—
55	#10625329	—	2	2	no	—
56	#10625329	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₁₆×C₄₇	—	4	2	no	—
59	C₄₇⋊C₁₆	—	2	2	no	—
60	C₄₇⋊C₁₆	—	2	2	no	—
61	C₂³×C₄₇	#131	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
70	C₂×Dic₄₇	—	2	4	no	—
71	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
72	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₄₇	#121	1	1	yes	characteristic
74	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₄₇	#125	1	1	yes	characteristic
76	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₄₇⋊C₈	—	4	2	no	—
78	#13161	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	#424	1	1	yes	characteristic
81	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₄₇	#446	1	1	yes	characteristic
83	C₄×C₄₇	—	2	4	no	—
84	C₄×C₄₇	#389	1	1	yes	characteristic
85	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₄₇	#436	1	1	yes	characteristic
87	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
88	Dic₄₇	—	4	4	no	—
89	C₂×#364	D₄₇	1	1	yes	characteristic
90	#2151	—	47	2	no	—
91	C₂×C₄×C₁₆	—	47	1	no	—
92	#1772	—	47	1	no	—
93	#1379	—	47	2	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	—	2	1	no	—
96	C₂×C₄₇	#1939	1	1	yes	characteristic
97	C₂×C₄₇	#2170	1	2	yes	—
98	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
99	C₂²×C₁₆	—	47	1	no	—
100	C₂²×C₁₆	—	47	1	no	—
101	C₂×#108	—	47	1	no	—
102	#427	—	47	1	no	—
103	#405	—	47	2	no	—
104	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
105	#364	—	2	2	no	—
106	C₄⋊C₁₆	—	94	4	no	—
107	C₄×C₁₆	—	94	2	no	—
108	#339	—	47	2	no	—
109	C₄₇	#13161	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	47	2	no	—
111	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	#117	—	47	2	no	—
114	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	47	2	no	—
119	C₂×C₁₆	—	47	2	no	—
120	C₂×C₁₆	—	94	1	no	—
121	C₂×C₁₆	—	94	1	no	—
122	C₂×C₁₆	—	94	1	no	—
123	C₂×C₁₆	—	94	1	no	—
124	#108	—	94	2	no	—
125	#107	—	47	2	no	—
126	#106	—	47	2	no	—
127	#97	—	94	2	no	—
128	C₄×C₈	—	94	2	no	—
129	C₄×C₈	Dic₉₄	1	2	yes	—
130	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
133	M₁₆	—	94	2	no	—
134	M₁₆	—	94	2	no	—
135	C₂×C₈	—	94	2	no	—
136	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
137	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
138	C₂×C₈	#10625324	1	2	yes	—
139	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
140	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
141	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
142	C₂×C₈	#10625352	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
144	C₂×C₈	#10625354	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₄⋊C₄	—	47	4	no	—
147	C₂²⋊C₄	—	94	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	94	2	no	—
150	C₄²	Dic₄₇⋊C₄	1	2	yes	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	94	2	no	—
153	C₁₆	—	94	2	no	—
154	C₂³	#13094032	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	94	4	no	—
158	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#13093986	1	1	yes	characteristic
160	C₂×C₄	#13093989	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
163	C₂×C₄	#13094032	1	1	yes	characteristic
164	C₈	—	188	2	no	—
165	C₈	Dic₄₇⋊C₈	1	2	yes	—
166	C₈	Dic₄₇⋊C₈	1	2	yes	—
167	C₈	#13093995	1	1	yes	characteristic
168	C₈	#13094000	1	1	yes	characteristic
169	C₈	#13094015	1	1	yes	characteristic
170	C₈	#13094023	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#15689151	1	1	yes	characteristic
173	C₂²	#15689188	1	1	yes	characteristic
174	C₂²	#15689250	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	188	4	no	—
177	C₄	#15689019	1	1	yes	characteristic
178	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
179	C₄	#15689200	1	1	yes	characteristic
180	C₄	#15689247	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#20244104	1	1	yes	characteristic
184	C₂	#20245224	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	#13161
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
109. C₄₇	#13161	semidirect

Maximal subgroups

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

subgroup	index
#20245148	2
C₂×C₄×C₄₇⋊C₁₆	2
#20243267	2
#20241478	2
C₂×C₄₇×#364	2
#13161	47