Subgroups — #26131364

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688899	C₂	1	2	yes	maximal, complemented
4	#20241466	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	complemented
12	#15688899	—	2	4	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₄₇	C₄	1	1	yes	characteristic
17	C₄₇×#336	C₄	1	2	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	8	no	—
22	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₄₇	C₈	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈×C₄₇	C₈	1	4	yes	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	#95×C₄₇	C₈	1	2	yes	—
40	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	C₄²	1	2	yes	—
55	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
56	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₇	#41	1	2	yes	—
60	C₂²⋊C₄×C₄₇	—	2	4	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄₇⋊C₁₆	—	2	2	no	—
64	C₄₇⋊C₁₆	—	2	4	no	—
65	C₄₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	—	2	2	no	—
68	C₂³×C₄₇	C₄×C₈	1	2	yes	—
69	C₂³×C₄₇	M₃₂	1	4	yes	—
70	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#131	1	2	yes	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	—	2	4	no	—
85	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₄₇	#117	1	1	yes	characteristic
87	C₈×C₄₇	C₄×D₄	1	2	yes	—
88	#12611	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	8	no	—
91	C₂²×C₄₇	—	2	2	no	—
92	C₂²×C₄₇	#346	1	4	yes	complemented
93	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₄₇	C₈×D₄	1	2	yes	—
95	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
96	C₂²×C₄₇	#504	1	2	yes	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	4	4	no	—
99	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₄₇	#405	1	1	yes	characteristic
101	C₄×C₄₇	C₈×D₄	1	2	yes	—
102	C₂³×C₁₆	—	47	1	no	—
103	C₂×C₂²⋊C₁₆	—	47	1	no	—
104	C₂×#346	—	47	2	no	—
105	C₈×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
106	#1374	—	47	2	no	—
107	C₂×C₄₇	—	2	4	no	—
108	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
110	C₂×C₄₇	#2108	1	1	yes	characteristic
111	C₂×C₄₇	#2168	1	4	yes	—
112	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
113	C₂²×C₁₆	—	47	1	no	—
114	C₂²×C₁₆	—	47	4	no	—
115	C₂²×C₁₆	—	47	2	no	—
116	C₂²×C₁₆	—	47	1	no	—
117	C₂²×C₁₆	—	94	4	no	—
118	C₂×C₂²⋊C₈	Dic₄₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
120	C₄×C₂²⋊C₄	Dic₄₇	1	1	yes	characteristic
121	C₂²⋊C₁₆	—	47	4	no	—
122	#346	—	94	4	no	—
123	#336	Dic₄₇	1	2	yes	—
124	C₄₇	#12611	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₄×D₄₇	1	2	yes	—
129	C₂²×C₈	C₄×D₄₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	2	yes	—
134	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
135	C₂×C₁₆	—	47	2	no	—
136	C₂×C₁₆	—	47	4	no	—
137	C₂×C₁₆	—	47	4	no	—
138	C₂×C₁₆	—	47	2	no	—
139	C₂×C₁₆	—	94	8	no	—
140	C₂×C₁₆	—	94	2	no	—
141	C₂×C₁₆	—	94	2	no	—
142	C₂×C₁₆	—	94	2	no	—
143	C₂²⋊C₈	C₄₇⋊C₈	1	4	yes	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	C₄₇⋊C₈	1	2	yes	—
147	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
151	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
153	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
162	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₈	D₄×D₄₇	1	2	yes	—
166	C₂×C₈	#10625352	1	2	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₁₆	—	47	4	no	—
171	C₁₆	—	94	2	no	—
172	C₁₆	—	94	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₄×C₄₇⋊C₈	1	2	yes	—
176	C₂³	#13093964	1	4	yes	—
177	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
188	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
189	C₂×C₄	#13094031	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	2	4	no	—
192	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
193	C₈	#13093980	1	1	yes	characteristic
194	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	2	no	—
198	C₂²	#15688899	1	4	yes	complemented
199	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
200	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
201	C₂²	C₂×#13093964	1	2	yes	—
202	C₂²	#15689540	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
206	C₄	#15689079	1	1	yes	characteristic
207	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₂²⋊C₄×C₄₇⋊C₈	1	1	yes	characteristic
210	C₂	C₂×#15688899	1	1	yes	characteristic
211	C₂	#20244985	1	1	yes	characteristic
212	C₂	#20245214	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₈×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₈×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#12611
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#15688899	C₂	semidirect
4. #20241466	C₂	semidirect
7. C₂²×C₄₇⋊C₁₆	C₄	semidirect
11. C₂²⋊(C₄₇⋊C₁₆)	C₄	semidirect
65. C₄₇⋊C₁₆	C₂²⋊C₄	semidirect
92. C₂²×C₄₇	#346	semidirect
124. C₄₇	#12611	semidirect
198. C₂²	#15688899	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×#15688899	2
#20241466	2
C₈×C₂²⋊C₄×C₄₇	2
#12611	47