Subgroups — #26103140

Order 12032.

262 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
9	C₈×C₄₇⋊C₈	—	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	C₂²×C₄₇⋊C₈	—	2	1	no	—
16	C₂²×C₄₇⋊C₈	C₈	1	2	yes	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	4	1	no	—
19	C₂×C₄₇⋊C₁₆	—	4	1	no	—
20	C₄×C₄₇⋊C₈	—	2	1	no	—
21	C₄×C₄₇⋊C₈	—	2	1	no	—
22	C₄×C₄₇⋊C₈	—	2	1	no	—
23	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
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	4	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	2	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	C₁₆	1	4	yes	complemented
56	C₁₆×C₄₇	—	4	2	no	—
57	C₄₇⋊C₁₆	—	4	2	no	—
58	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
72	C₈×C₄₇	—	2	1	no	—
73	C₈×C₄₇	—	2	1	no	—
74	C₈×C₄₇	#102	1	1	yes	characteristic
75	C₈×C₄₇	#102	1	1	yes	characteristic
76	C₈×C₄₇	#103	1	1	yes	characteristic
77	C₈×C₄₇	#103	1	1	yes	characteristic
78	C₄₇⋊C₈	—	2	4	no	—
79	C₄₇⋊C₈	—	2	4	no	—
80	#7151	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	#334	1	1	yes	characteristic
82	C₂²×C₄₇	#335	1	1	yes	characteristic
83	C₂²×C₄₇	#336	1	1	yes	characteristic
84	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
85	C₂²×C₄₇	#364	1	2	yes	—
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	#334	1	1	yes	characteristic
89	C₄×C₄₇	#335	1	1	yes	characteristic
90	C₄×C₄₇	#340	1	1	yes	characteristic
91	C₄×C₄₇	#341	1	1	yes	characteristic
92	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
93	C₄×C₄₇	#346	1	1	yes	characteristic
94	C₄×C₄₇	C₂²⋊C₁₆	1	2	yes	—
95	C₂×C₄⋊C₁₆	—	47	1	no	—
96	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
97	C₂×C₈²	—	47	1	no	—
98	C₂×C₄₇	#1277	1	1	yes	characteristic
99	C₂×C₄₇	#1367	1	2	yes	—
100	C₂×C₄₇	#1370	1	2	yes	—
101	C₂×C₄₇	#1374	1	1	yes	characteristic
102	C₂×C₄₇	#1376	1	1	yes	characteristic
103	C₂²×C₁₆	—	2	1	no	—
104	C₂²×C₁₆	—	94	1	no	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₂×C₄×C₈	—	47	1	no	—
107	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
108	C₄⋊C₁₆	—	2	1	no	—
109	C₄⋊C₁₆	—	47	2	no	—
110	C₄⋊C₁₆	—	94	1	no	—
111	C₄⋊C₁₆	Dic₄₇	1	2	yes	—
112	C₈²	—	94	1	no	—
113	C₈²	—	94	1	no	—
114	C₄₇	#7151	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	94	1	no	—
117	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	4	1	no	—
122	C₂×C₁₆	—	4	1	no	—
123	C₂×C₁₆	—	94	2	no	—
124	C₂×C₁₆	—	188	1	no	—
125	C₂×C₁₆	—	188	1	no	—
126	C₄×C₈	—	47	2	no	—
127	C₄×C₈	—	94	1	no	—
128	C₄×C₈	—	94	1	no	—
129	C₄×C₈	—	94	1	no	—
130	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
131	C₄×C₈	D₁₈₈	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	C₈×D₄₇	1	1	yes	characteristic
135	C₂²×C₄	#10625324	1	1	yes	characteristic
136	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	47	4	no	—
140	C₂×C₈	—	94	4	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
146	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	#10625334	1	1	yes	characteristic
148	C₂×C₈	#10625335	1	1	yes	characteristic
149	C₂×C₈	#10625336	1	1	yes	characteristic
150	C₂×C₈	#10625337	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
152	C₄²	C₈×D₄₇	1	1	yes	characteristic
153	C₄²	#10625324	1	1	yes	characteristic
154	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
155	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₁₆	—	188	2	no	—
158	C₂³	Dic₄₇⋊C₈	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
161	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
162	C₂×C₄	#13093934	1	1	yes	characteristic
163	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
164	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
165	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
166	C₂×C₄	#13093955	1	1	yes	characteristic
167	C₂×C₄	#13093957	1	1	yes	characteristic
168	C₂×C₄	#13093959	1	1	yes	characteristic
169	C₂×C₄	#13093959	1	1	yes	characteristic
170	C₂×C₄	C₁₆×D₄₇	1	2	yes	—
171	C₂×C₄	#13093963	1	2	yes	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	94	4	no	—
175	C₈	—	94	4	no	—
176	C₈	#13093944	1	1	yes	characteristic
177	C₈	#13093945	1	1	yes	characteristic
178	C₈	#13093947	1	1	yes	characteristic
179	C₈	#13093948	1	1	yes	characteristic
180	C₂²	#15688873	1	1	yes	characteristic
181	C₂²	#15688875	1	1	yes	characteristic
182	C₂²	#15688877	1	1	yes	characteristic
183	C₂²	Dic₄₇⋊C₁₆	1	2	yes	—
184	C₂²	#15688943	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	#15688873	1	1	yes	characteristic
188	C₄	#15688875	1	1	yes	characteristic
189	C₄	#15688885	1	1	yes	characteristic
190	C₄	#15688887	1	1	yes	characteristic
191	C₄	C₁₆×Dic₄₇	1	1	yes	characteristic
192	C₄	#15688898	1	1	yes	characteristic
193	C₄	(C₂×D₄₇)⋊C₁₆	1	2	yes	—
194	C₂	#20241222	1	1	yes	characteristic
195	C₂	#20241451	1	2	yes	—
196	C₂	#20241457	1	2	yes	—
197	C₂	#20241465	1	1	yes	characteristic
198	C₂	#20241469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#7151
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
55. C₂×C₄₇⋊C₈	C₁₆	semidirect
114. C₄₇	#7151	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×C₄⋊C₁₆	2
#7151	47