Subgroups — #26182107

Order 12032.

1162 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252469	C₂	1	2	yes	maximal, complemented
2	#20241712	C₂	1	6	yes	maximal, complemented
3	#20241702	C₂	1	6	yes	maximal
4	C₂×C₄₇×#519	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄₇⋊C₈	—	2	8	no	—
6	#15689176	C₂²	1	12	yes	—
7	#15688849	C₂²	1	12	yes	—
8	#15688847	C₂²	1	4	yes	—
9	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
10	C₄₇×#519	—	2	8	no	—
11	C₂×C₄×Q₈×C₄₇	C₄	1	3	yes	—
12	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
14	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	C₄	1	3	yes	—
16	C₂×#117×C₄₇	C₂²	1	3	yes	—
17	C₄⋊(C₄₇⋊C₈)	—	2	12	no	—
18	C₂²⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
19	C₄×C₄₇⋊C₈	—	2	12	no	—
20	C₂²×Q₈×C₄₇	—	2	4	no	—
21	C₂²×Q₈×C₄₇	—	2	4	no	—
22	C₂²×Q₈×C₄₇	C₈	1	8	yes	complemented
23	C₂²×Q₈×C₄₇	C₂×C₄	1	3	yes	—
24	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
26	C₄×Q₈×C₄₇	—	2	4	no	—
27	C₄×Q₈×C₄₇	—	2	12	no	—
28	C₄×Q₈×C₄₇	—	2	12	no	—
29	C₄×Q₈×C₄₇	D₄	1	8	yes	—
30	#117×C₄₇	—	2	12	no	—
31	#117×C₄₇	—	2	12	no	—
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
36	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
37	C₂×C₂²⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×Q₈×C₄₇	—	2	32	no	—
43	C₂×Q₈×C₄₇	—	2	6	no	—
44	C₂×Q₈×C₄₇	—	2	6	no	—
45	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
46	C₂×Q₈×C₄₇	C₂×C₈	1	12	yes	—
47	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
48	C₂×Q₈×C₄₇	M₁₆	1	4	yes	—
49	C₂²×C₄×C₄₇	—	2	6	no	—
50	C₂²×C₄×C₄₇	—	2	6	no	—
51	C₂²×C₄×C₄₇	C₂×C₈	1	12	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	—	2	6	no	—
59	C₄⋊C₄×C₄₇	—	2	6	no	—
60	C₄⋊C₄×C₄₇	—	2	6	no	—
61	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	12	yes	—
62	C₂²⋊C₄×C₄₇	—	2	6	no	—
63	C₂²⋊C₄×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	12	no	—
65	C₄²×C₄₇	—	2	6	no	—
66	C₄²×C₄₇	C₂×D₄	1	12	yes	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂³×C₄₇	—	2	1	no	—
69	C₂³×C₄₇	—	2	1	no	—
70	C₂³×C₄₇	—	4	1	no	—
71	C₂³×C₄₇	—	4	1	no	—
72	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
73	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	Q₈×C₄₇	—	2	24	no	—
75	Q₈×C₄₇	C₂²⋊C₈	1	16	yes	—
76	C₂×C₄×C₄₇	—	2	3	no	—
77	C₂×C₄×C₄₇	—	2	3	no	—
78	C₂×C₄×C₄₇	—	2	24	no	—
79	C₂×C₄×C₄₇	—	2	3	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	4	3	no	—
82	C₂×C₄×C₄₇	—	4	1	no	—
83	C₂×C₄×C₄₇	—	4	3	no	—
84	C₂×C₄×C₄₇	—	4	1	no	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
87	C₂×C₄×C₄₇	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₄₇	C₂²×C₈	1	6	yes	—
89	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
90	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
91	C₄₇⋊C₈	—	4	8	no	—
92	#17352	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	1	no	—
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	—	4	2	no	—
97	C₂²×C₄₇	—	4	2	no	—
98	C₂²×C₄₇	—	4	1	no	—
99	C₂²×C₄₇	—	4	1	no	—
100	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
102	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
103	C₄×C₄₇	—	2	6	no	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	6	no	—
106	C₄×C₄₇	—	4	3	no	—
107	C₄×C₄₇	—	4	1	no	—
108	C₄×C₄₇	C₂×C₂²⋊C₈	1	12	yes	—
109	C₂×#519	D₄₇	1	1	yes	characteristic
110	#2982	—	47	2	no	—
111	#1461	—	47	6	no	—
112	#1458	—	47	6	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	—	2	1	no	—
115	C₂×C₄₇	—	4	2	no	—
116	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄₇	#2884	1	1	yes	characteristic
118	C₂×C₄₇	#2889	1	1	yes	characteristic
119	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
120	#519	—	2	8	no	—
121	C₂×C₄×Q₈	Dic₄₇	1	3	yes	—
122	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
124	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
125	C₂×#117	Dic₄₇	1	3	yes	—
126	C₂×#117	C₂×D₄₇	1	3	yes	—
127	C₈×Q₈	—	94	8	no	—
128	#431	—	47	12	no	—
129	#324	—	47	12	no	—
130	#323	—	47	4	no	—
131	C₄₇	#17352	1	1	yes	characteristic, complemented, Sylow
132	C₂²×Q₈	—	2	4	no	—
133	C₂²×Q₈	—	2	4	no	—
134	C₂²×Q₈	C₄₇⋊C₈	1	8	yes	complemented
135	C₂²×Q₈	C₂×Dic₄₇	1	3	yes	—
136	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	2	12	no	—
140	C₄×Q₈	—	2	12	no	—
141	C₄×Q₈	C₂²⋊D₄₇	1	8	yes	—
142	#117	—	2	12	no	—
143	#117	—	2	12	no	—
144	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
147	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
148	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	3	yes	—
149	C₂×C₂²⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₄₇	1	6	yes	—
151	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
152	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
153	C₄⋊C₈	—	94	12	no	—
154	C₂²⋊C₈	—	47	8	no	—
155	C₄×C₈	—	94	12	no	—
156	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂×Q₈	—	2	32	no	—
158	C₂×Q₈	—	2	6	no	—
159	C₂×Q₈	—	2	6	no	—
160	C₂×Q₈	C₂×C₄₇⋊C₈	1	12	yes	—
161	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
162	C₂×Q₈	#10625329	1	4	yes	—
163	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	C₂×C₄₇⋊C₈	1	12	yes	—
167	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
168	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
169	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	C₂×C₈	—	94	8	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	6	no	—
176	C₄⋊C₄	—	2	6	no	—
177	C₄⋊C₄	C₂²⋊Dic₄₇	1	12	yes	—
178	C₂²⋊C₄	—	2	6	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	12	no	—
181	C₄²	—	2	6	no	—
182	C₄²	C₂×C₂²⋊D₄₇	1	12	yes	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
188	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
189	Q₈	—	2	24	no	—
190	Q₈	C₂²⋊(C₄₇⋊C₈)	1	16	yes	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	3	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
201	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
202	C₂×C₄	C₂²×C₄₇⋊C₈	1	6	yes	—
203	C₂×C₄	C₂²×C₄₇⋊C₈	1	6	yes	—
204	C₂×C₄	C₂×#10625329	1	6	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
206	C₈	—	188	8	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
215	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
216	C₂²	C₂²×#10625329	1	1	yes	characteristic
217	C₄	—	2	6	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	6	no	—
220	C₄	—	4	3	no	—
221	C₄	—	4	1	no	—
222	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
227	C₂	#20251852	1	1	yes	characteristic
228	C₂	#20251871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#519	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#519	D₄₇
hypocenter	C₄₇	#17352
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
1. #20252469	C₂	semidirect
2. #20241712	C₂	semidirect
22. C₂²×Q₈×C₄₇	C₈	semidirect
131. C₄₇	#17352	semidirect
134. C₂²×Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20252469	2
#20241712	2
#20241702	2
C₂×C₄₇×#519	2
#17352	47