Subgroups — #26105663

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	C₄₇⋊#1273	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	4	no	—
5	#15689176	—	2	4	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
9	C₂³×C₈×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
15	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
17	C₄×C₄₇⋊C₈	—	2	4	no	—
18	C₄×C₄₇⋊C₈	—	2	4	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	—	2	4	no	—
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	C₈	1	8	yes	complemented
24	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	D₄	1	4	yes	—
29	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	—	2	4	no	—
32	C₄×C₈×C₄₇	—	2	4	no	—
33	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
36	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₄₇	—	2	16	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	—
49	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
50	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×D₄	1	2	yes	—
55	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄₇⋊C₈	—	2	8	no	—
57	C₂×C₄₇⋊C₈	—	2	16	no	—
58	C₂³×C₄₇	—	2	4	no	—
59	C₂³×C₄₇	#95	1	2	yes	—
60	C₂³×C₄₇	C₄×C₈	1	2	yes	—
61	C₂³×C₄₇	#97	1	2	yes	—
62	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	8	no	—
65	C₂×C₄×C₄₇	—	2	8	no	—
66	C₂×C₄×C₄₇	#95	1	4	yes	—
67	C₂×C₄×C₄₇	#95	1	2	yes	—
68	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₄₇	#97	1	2	yes	—
70	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
71	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₄₇	#131	1	8	yes	—
79	C₈×C₄₇	—	2	8	no	—
80	C₄₇⋊C₈	—	4	8	no	—
81	#7849	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	—	2	4	no	—
83	C₂²×C₄₇	—	2	4	no	—
84	C₂²×C₄₇	—	2	4	no	—
85	C₂²×C₄₇	#336	1	4	yes	—
86	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
89	C₂²×C₄₇	#405	1	4	yes	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	#336	1	4	yes	—
92	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
93	C₄×C₄₇	#408	1	4	yes	—
94	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₄₇	#424	1	4	yes	—
96	C₂×#431	—	47	2	no	—
97	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
98	#1273	—	47	4	no	—
99	C₂×C₄₇	—	2	4	no	—
100	C₂×C₄₇	#1450	1	2	yes	—
101	C₂×C₄₇	#1452	1	2	yes	—
102	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
103	C₂×C₄₇	#1729	1	2	yes	—
104	C₂³×C₈	Dic₄₇	1	2	yes	—
105	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	#431	—	94	4	no	—
107	#431	—	94	4	no	—
108	C₂×C₄⋊C₈	—	47	4	no	—
109	C₂×C₂²⋊C₈	—	47	4	no	—
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	—	47	4	no	—
112	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
114	C₄₇	#7849	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₄	C₄×D₄₇	1	2	yes	—
116	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	47	8	no	—
120	C₂²×C₈	C₄₇⋊C₈	1	8	yes	complemented
121	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
122	C₂×C₄²	Dic₉₄	1	2	yes	—
123	C₂×C₄²	C₄×D₄₇	1	2	yes	—
124	C₂×C₄²	D₁₈₈	1	2	yes	—
125	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₈	—	94	8	no	—
129	C₂²⋊C₈	—	94	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	94	4	no	—
133	C₄×C₈	—	94	4	no	—
134	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
137	C₂²×C₄	#10625329	1	4	yes	—
138	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
139	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
140	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
142	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
143	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
146	C₂×C₈	—	2	16	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	94	8	no	—
150	C₂×C₈	—	94	16	no	—
151	C₂×C₈	C₂×C₄₇⋊C₈	1	8	yes	—
152	C₄²	Dic₄₇⋊C₄	1	4	yes	—
153	C₄²	C₄⋊Dic₄₇	1	2	yes	—
154	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
155	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
156	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
157	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
158	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
159	C₂³	—	2	4	no	—
160	C₂³	C₄₇⋊#95	1	2	yes	—
161	C₂³	C₄×C₄₇⋊C₈	1	2	yes	—
162	C₂³	#13093935	1	2	yes	—
163	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	C₄₇⋊#95	1	4	yes	—
168	C₂×C₄	C₄₇⋊#95	1	2	yes	—
169	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
170	C₂×C₄	#13093935	1	2	yes	—
171	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
172	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
173	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
178	C₂×C₄	C₂×#10625329	1	2	yes	—
179	C₂×C₄	#13094030	1	8	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	188	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	#15688878	1	4	yes	—
186	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
187	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
188	C₂²	C₂×#13093935	1	1	yes	characteristic
189	C₂²	#15689078	1	4	yes	—
190	C₄	—	2	4	no	—
191	C₄	#15688878	1	4	yes	—
192	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
193	C₄	#15689089	1	4	yes	—
194	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
195	C₄	#15689149	1	4	yes	—
196	C₂	—	2	4	no	—
197	C₂	#20241671	1	2	yes	—
198	C₂	#20241680	1	2	yes	—
199	C₂	C₂×#15688878	1	1	yes	characteristic
200	C₂	#20243084	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#7849
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄₇⋊#1273	C₂	semidirect
23. C₂²×C₈×C₄₇	C₈	semidirect
114. C₄₇	#7849	semidirect
120. C₂²×C₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₄₇⋊#1273	2
C₂²×C₄×C₈×C₄₇	2
#7849	47