Subgroups — #26103146

Order 12032.

218 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	#20245144	C₂	1	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	1	no	—
5	#15689176	C₄	1	1	yes	characteristic
6	#15689176	C₄	1	1	yes	characteristic
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
9	#15688844	—	2	1	no	—
10	C₈×C₄₇⋊C₈	—	2	1	no	—
11	C₂²×C₁₆×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×C₄⋊C₁₆	—	2	1	no	—
14	C₄₇×C₄⋊C₁₆	C₄	1	2	yes	—
15	#13093964	—	4	1	no	—
16	#13093964	—	4	1	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₈)	C₈	1	2	yes	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₄₇⋊C₈)	C₈	1	2	yes	—
22	C₄×C₄₇⋊C₈	—	2	1	no	—
23	C₄×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	—	2	2	no	—
28	C₂×C₁₆×C₄₇	—	4	1	no	—
29	C₂×C₁₆×C₄₇	—	4	1	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	—	2	1	no	—
39	C₂×C₈×C₄₇	C₂²⋊C₄	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	D₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
47	C₄²×C₄₇	C₄²	1	1	yes	characteristic
48	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₁₆×C₄₇	—	4	2	no	—
54	C₄₇⋊C₁₆	—	4	2	no	—
55	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	#102	1	1	yes	characteristic
71	C₈×C₄₇	#102	1	1	yes	characteristic
72	C₈×C₄₇	#103	1	1	yes	characteristic
73	C₈×C₄₇	#103	1	1	yes	characteristic
74	C₄₇⋊C₈	—	4	2	no	—
75	C₄₇⋊C₈	—	4	2	no	—
76	#7153	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	#334	1	1	yes	characteristic
80	C₂²×C₄₇	#335	1	1	yes	characteristic
81	C₂²×C₄₇	#336	1	1	yes	characteristic
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	#334	1	1	yes	characteristic
86	C₄×C₄₇	#335	1	1	yes	characteristic
87	C₄×C₄₇	#340	1	1	yes	characteristic
88	C₄×C₄₇	#341	1	1	yes	characteristic
89	C₄×C₄₇	#350	1	2	yes	—
90	#2150	—	47	1	no	—
91	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
92	#1450	—	47	1	no	—
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	#1277	1	1	yes	characteristic
96	C₂×C₄₇	#1378	1	1	yes	characteristic
97	C₂×C₄₇	#1379	1	1	yes	characteristic
98	C₂×M₃₂	—	94	1	no	—
99	C₂²×C₁₆	—	2	1	no	—
100	#431	—	47	1	no	—
101	#431	—	47	1	no	—
102	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
103	C₄⋊C₁₆	—	2	1	no	—
104	C₄⋊C₁₆	—	47	2	no	—
105	C₄⋊C₁₆	—	94	1	no	—
106	C₄⋊C₁₆	Dic₄₇	1	2	yes	—
107	#322	—	94	1	no	—
108	C₈²	—	94	1	no	—
109	C₄₇	#7153	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
113	M₃₂	—	188	1	no	—
114	M₃₂	—	188	1	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₂×C₁₆	—	4	1	no	—
118	C₂×C₁₆	—	94	2	no	—
119	C₄⋊C₈	—	47	2	no	—
120	C₄⋊C₈	—	94	1	no	—
121	C₂²⋊C₈	—	47	2	no	—
122	C₂²⋊C₈	—	94	1	no	—
123	C₄×C₈	—	94	1	no	—
124	C₄×C₈	—	94	1	no	—
125	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₄×C₈	D₁₈₈	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	#10625324	1	1	yes	characteristic
131	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	94	2	no	—
135	C₂×C₈	—	94	2	no	—
136	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
137	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
138	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#10625334	1	1	yes	characteristic
140	C₂×C₈	#10625335	1	1	yes	characteristic
141	C₂×C₈	#10625336	1	1	yes	characteristic
142	C₂×C₈	#10625337	1	1	yes	characteristic
143	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₄²	C₈×D₄₇	1	1	yes	characteristic
145	C₄²	#10625324	1	1	yes	characteristic
146	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
147	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₁₆	—	4	2	no	—
149	C₁₆	—	188	2	no	—
150	C₂³	Dic₄₇⋊C₈	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
154	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
155	C₂×C₄	#13093934	1	1	yes	characteristic
156	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
157	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
158	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
159	C₂×C₄	#13093955	1	1	yes	characteristic
160	C₂×C₄	#13093957	1	1	yes	characteristic
161	C₂×C₄	#13093959	1	1	yes	characteristic
162	C₂×C₄	#13093959	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	188	2	no	—
166	C₈	—	188	2	no	—
167	C₈	#13093944	1	1	yes	characteristic
168	C₈	#13093945	1	1	yes	characteristic
169	C₈	#13093947	1	1	yes	characteristic
170	C₈	#13093948	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#15688873	1	1	yes	characteristic
174	C₂²	#15688875	1	1	yes	characteristic
175	C₂²	#15688877	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#15688873	1	1	yes	characteristic
180	C₄	#15688875	1	1	yes	characteristic
181	C₄	#15688885	1	1	yes	characteristic
182	C₄	#15688887	1	1	yes	characteristic
183	C₄	#15688906	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#20241222	1	1	yes	characteristic
187	C₂	#20241475	1	1	yes	characteristic
188	C₂	#20241476	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂²×C₄₇	#336
prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#7153
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
109. C₄₇	#7153	semidirect

Maximal subgroups

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

subgroup	index
#20245144	2
#20241671	2
C₂×C₄₇×C₄⋊C₁₆	2
#7153	47