Subgroups — #26132444

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#15688948	C₂	1	2	yes	maximal, complemented
3	#20243417	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1808	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	1	yes	characteristic
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	#15688983	—	2	1	no	—
12	#15688948	—	2	4	no	—
13	#15688886	—	2	1	no	—
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#380	—	2	1	no	—
17	C₄₇×#340	—	2	1	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₂²⋊Dic₄₇	—	4	1	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	2	yes	—
24	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
25	#13093958	—	2	2	no	—
26	#13093958	—	2	2	no	—
27	C₄₇⋊#95	—	4	1	no	—
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
37	#107×C₄₇	—	2	2	no	—
38	#107×C₄₇	—	2	2	no	—
39	#95×C₄₇	—	4	1	no	—
40	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₄₇	—	4	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	4	no	—
53	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₇	#41	1	2	yes	—
58	C₄⋊C₄×C₄₇	—	4	2	no	—
59	C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₂²⋊C₄×C₄₇	—	8	2	no	—
61	C₂²⋊Dic₄₇	—	8	2	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄₇⋊C₁₆	—	2	2	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	#107	1	2	yes	—
69	C₂³×C₄₇	D₁₆	1	2	yes	—
70	C₂³×C₄₇	Q₃₂	1	2	yes	—
71	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	8	2	no	—
77	C₂×C₄×C₄₇	—	8	2	no	—
78	C₂×C₄×C₄₇	—	8	2	no	—
79	C₂×C₄×C₄₇	#107	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#121	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#135	1	2	yes	—
84	C₂×Dic₄₇	—	8	2	no	—
85	C₂×Dic₄₇	—	8	2	no	—
86	C₂×Dic₄₇	—	8	2	no	—
87	C₈×C₄₇	—	2	2	no	—
88	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₄₇	#117	1	1	yes	characteristic
90	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
91	#12834	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	2	4	no	—
95	C₂²×C₄₇	—	2	4	no	—
96	C₂²×C₄₇	#366	1	4	yes	complemented
97	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
98	C₂²×C₄₇	#466	1	2	yes	—
99	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
100	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
101	C₂²×C₄₇	#508	1	2	yes	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	16	2	no	—
104	C₄×C₄₇	#380	1	1	yes	characteristic
105	C₄×C₄₇	#427	1	1	yes	characteristic
106	C₄×C₄₇	#467	1	2	yes	—
107	Dic₄₇	—	16	2	no	—
108	C₂³×C₁₆	—	47	1	no	—
109	C₂×#366	—	47	2	no	—
110	#1808	—	47	1	no	—
111	#1808	D₄₇	1	1	yes	characteristic
112	#1380	—	47	2	no	—
113	C₂×C₄₇	—	2	4	no	—
114	C₂×C₄₇	#1808	1	1	yes	characteristic
115	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
116	C₂×C₄₇	#2158	1	1	yes	characteristic
117	C₂×C₄₇	#2215	1	2	yes	—
118	C₂×C₄₇	#2216	1	2	yes	—
119	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₂²×C₁₆	—	47	1	no	—
121	C₂²×C₁₆	—	47	4	no	—
122	C₂²×C₁₆	—	47	1	no	—
123	C₂²×C₁₆	—	94	4	no	—
124	C₂×#107	—	47	2	no	—
125	C₂×#107	C₂×D₄₇	1	2	yes	—
126	#380	—	2	1	no	—
127	#380	—	94	1	no	—
128	#366	—	94	4	no	—
129	#340	—	2	1	no	—
130	#340	—	94	1	no	—
131	C₄₇	#12834	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	Dic₉₄	1	2	yes	—
136	C₂²×C₈	C₄×D₄₇	1	2	yes	—
137	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	94	2	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	188	1	no	—
143	C₂×C₁₆	—	47	2	no	—
144	C₂×C₁₆	—	47	4	no	—
145	C₂×C₁₆	—	47	2	no	—
146	C₂×C₁₆	—	94	8	no	—
147	C₂×C₁₆	—	94	2	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	94	2	no	—
151	#107	—	94	2	no	—
152	#95	—	4	1	no	—
153	#95	—	188	1	no	—
154	C₂⁴	#10625334	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	188	2	no	—
159	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
160	C₂²×C₄	#10625334	1	1	yes	characteristic
161	C₂²×C₄	#10625337	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
168	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₈	D₄×D₄₇	1	2	yes	—
171	C₂×C₈	#10625352	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	188	2	no	—
175	C₄⋊C₄	—	188	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	376	2	no	—
178	C₁₆	—	47	4	no	—
179	C₁₆	—	94	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#13093957	1	2	yes	—
183	C₂³	#13093965	1	2	yes	—
184	C₂³	#13093970	1	2	yes	—
185	C₂³	C₂×#10625334	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	376	2	no	—
194	C₂×C₄	—	376	2	no	—
195	C₂×C₄	—	376	2	no	—
196	C₂×C₄	#13093957	1	2	yes	—
197	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#13093996	1	2	yes	—
199	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
200	C₂×C₄	#13094044	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
203	C₈	#13093980	1	1	yes	characteristic
204	C₈	#13094001	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#15688948	1	4	yes	complemented
210	C₂²	C₂×#13093957	1	1	yes	characteristic
211	C₂²	#15689357	1	2	yes	—
212	C₂²	C₂×#13093965	1	1	yes	characteristic
213	C₂²	C₂×#13093970	1	1	yes	characteristic
214	C₂²	#15689553	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	752	2	no	—
218	C₄	#15688981	1	1	yes	characteristic
219	C₄	#15689159	1	1	yes	characteristic
220	C₄	#15689362	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#20243415	1	1	yes	characteristic
223	C₂	C₂×#15688948	1	1	yes	characteristic
224	C₂	#20245170	1	1	yes	characteristic
225	C₂	#20245472	1	2	yes	—
226	C₂	#20245477	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1808	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1808	D₄₇
hypocenter	C₄₇	#12834
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#15688948	C₂	semidirect
4. #20241479	C₂	semidirect
7. C₂²×C₄₇⋊C₁₆	C₄	semidirect
65. C₄₇⋊C₁₆	C₂²⋊C₄	semidirect
96. C₂²×C₄₇	#366	semidirect
131. C₄₇	#12834	semidirect
209. C₂²	#15688948	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×#15688948	2
#20243417	2
#20241479	2
C₄₇×#1808	2
#12834	47