Subgroups — #26224741

Order 12032.

1062 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689230	C₂	1	4	yes	maximal
2	C₄⋊Q₈×Dic₄₇	C₂	1	2	yes	maximal
3	#20244260	C₂	1	4	yes	maximal
4	#20244191	C₂	1	2	yes	maximal
5	#20243545	C₂	1	1	yes	maximal, characteristic
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
9	#15689230	—	2	16	no	—
10	C₂×#13093956	C₂²	1	4	yes	—
11	C₂×#13093948	C₂²	1	8	yes	—
12	C₂×#13093934	C₂²	1	2	yes	—
13	C₂×#13093934	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	#15689030	C₂²	1	4	yes	—
16	#15689022	—	2	4	no	—
17	#15688993	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₄₇	—	2	4	no	—
19	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
22	C₄₇×#494	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
25	Q₈×Dic₄₇	—	2	16	no	—
26	C₂×C₄⋊Dic₄₇	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	C₂×C₄×Dic₄₇	—	2	4	no	—
30	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
33	#13093956	—	2	8	no	—
34	#13093948	—	2	16	no	—
35	#13093934	—	2	4	no	—
36	#13093934	D₄	1	8	yes	—
37	C₄×C₄₇⋊C₈	—	2	2	no	—
38	C₄₇⋊#95	—	2	8	no	—
39	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
40	C₂×Q₁₆×C₄₇	—	2	8	no	—
41	C₂×Q₁₆×C₄₇	—	2	8	no	—
42	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₄₇	—	2	4	no	—
45	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×Q₈×C₄₇	—	2	4	no	—
50	C₂×Q₈×C₄₇	—	2	8	no	—
51	C₂×Q₈×C₄₇	—	2	8	no	—
52	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
58	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
59	Q₁₆×C₄₇	—	2	32	no	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₄₇	#41	1	4	yes	—
63	C₄⋊C₄×C₄₇	—	2	16	no	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	#41	1	2	yes	—
66	C₄⋊Dic₄₇	—	2	8	no	—
67	C₄⋊Dic₄₇	—	4	8	no	—
68	C₄⋊Dic₄₇	—	4	8	no	—
69	C₄×Dic₄₇	—	2	8	no	—
70	C₄×Dic₄₇	—	2	2	no	—
71	C₄×Dic₄₇	—	4	8	no	—
72	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
76	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₄₇	—	2	16	no	—
78	Q₈×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₄₇	#137	1	8	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
88	C₂×Dic₄₇	—	2	4	no	—
89	C₂×Dic₄₇	—	2	2	no	—
90	C₂×Dic₄₇	—	2	4	no	—
91	C₂×Dic₄₇	—	4	4	no	—
92	C₂×Dic₄₇	—	4	8	no	—
93	C₂×Dic₄₇	#127	1	4	yes	—
94	C₈×C₄₇	C₄×D₄	1	8	yes	—
95	C₄₇⋊C₈	—	2	4	no	—
96	#21691	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
99	C₂²×C₄₇	#532	1	1	yes	characteristic
100	C₂²×C₄₇	C₂×#137	1	4	yes	—
101	C₄×C₄₇	—	4	8	no	—
102	C₄×C₄₇	#441	1	8	yes	complemented
103	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
104	C₄×C₄₇	#532	1	2	yes	—
105	Dic₄₇	—	2	4	no	—
106	Dic₄₇	—	4	2	no	—
107	Dic₄₇	—	8	4	no	—
108	C₂×#494	D₄₇	1	1	yes	characteristic
109	C₂×#441	—	47	4	no	—
110	C₄×C₄⋊Q₈	—	47	2	no	—
111	#1971	—	47	4	no	—
112	#1956	—	47	2	no	—
113	#1844	—	47	1	no	—
114	C₄×#97	—	47	1	no	—
115	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
116	C₂×C₄₇	C₂×#441	1	2	yes	—
117	C₂×C₄₇	#3152	1	2	yes	—
118	C₂×C₄₇	#3185	1	2	yes	—
119	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
121	C₂×C₄×Q₈	—	47	4	no	—
122	#494	Dic₄₇	1	8	yes	complemented
123	#441	—	94	16	no	—
124	C₂×#106	—	47	4	no	—
125	C₂×#103	—	47	8	no	—
126	C₂×#97	—	47	2	no	—
127	C₂×#97	—	47	2	no	—
128	C₂×C₄×C₈	—	47	1	no	—
129	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
130	#391	—	47	4	no	—
131	#389	—	94	4	no	—
132	#384	—	47	2	no	—
133	C₄×C₄⋊C₄	—	94	4	no	—
134	C₄³	—	47	1	no	—
135	C₄₇	#21691	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
137	C₂×Q₁₆	—	2	8	no	—
138	C₂×Q₁₆	—	2	8	no	—
139	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	—
140	C₂²×C₈	—	47	2	no	—
141	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
144	C₄×Q₈	—	94	16	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	47	4	no	—
147	C₂×C₄⋊C₄	—	94	4	no	—
148	C₂×C₄⋊C₄	—	94	4	no	—
149	C₂×C₄²	—	47	1	no	—
150	C₂×C₄²	—	47	4	no	—
151	C₂×C₄²	—	47	1	no	—
152	C₂×C₄²	—	94	4	no	—
153	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
154	#106	—	94	8	no	—
155	#103	—	94	16	no	—
156	#97	—	47	8	no	—
157	#97	—	94	4	no	—
158	C₄×C₈	—	94	2	no	—
159	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
160	#95	—	94	8	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	47	2	no	—
167	C₂²×C₄	—	47	2	no	—
168	C₂²×C₄	—	94	4	no	—
169	C₂²×C₄	D₄×D₄₇	1	2	yes	—
170	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
171	Q₁₆	—	2	32	no	—
172	C₂×C₈	—	47	4	no	—
173	C₂×C₈	—	94	2	no	—
174	C₂×C₈	—	94	2	no	—
175	C₂×C₈	D₄×D₄₇	1	4	yes	—
176	C₂×C₈	#10625352	1	4	yes	—
177	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	94	8	no	—
180	C₄⋊C₄	—	188	8	no	—
181	C₄⋊C₄	—	188	8	no	—
182	C₄²	—	47	4	no	—
183	C₄²	—	94	8	no	—
184	C₄²	—	94	2	no	—
185	C₄²	—	188	8	no	—
186	C₄²	#10625352	1	2	yes	—
187	C₄²	C₂²×Dic₄₇	1	2	yes	—
188	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	4	8	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	47	4	no	—
195	C₂×C₄	—	94	4	no	—
196	C₂×C₄	—	94	4	no	—
197	C₂×C₄	—	94	2	no	—
198	C₂×C₄	—	188	4	no	—
199	C₂×C₄	—	188	8	no	—
200	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
201	C₂×C₄	#13094056	1	8	yes	—
202	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
204	C₂×C₄	C₂×#10625352	1	2	yes	—
205	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
206	C₈	—	94	4	no	—
207	C₈	D₄×Dic₄₇	1	8	yes	—
208	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
209	C₂²	#127×D₄₇	1	1	yes	characteristic
210	C₂²	#15689671	1	1	yes	characteristic
211	C₂²	C₂×#13094056	1	4	yes	—
212	C₄	—	4	8	no	—
213	C₄	—	94	4	no	—
214	C₄	—	376	4	no	—
215	C₄	—	188	2	no	—
216	C₄	#15689230	1	8	yes	complemented
217	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
218	C₄	#15689671	1	2	yes	—
219	C₂	#127×Dic₄₇	1	1	yes	characteristic
220	C₂	C₂×#15689230	1	2	yes	—
221	C₂	#20253846	1	2	yes	—
222	C₂	#20254131	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#21691
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
22. C₄₇×#494	C₄	semidirect
102. C₄×C₄₇	#441	semidirect
122. #494	Dic₄₇	semidirect
135. C₄₇	#21691	semidirect
216. C₄	#15689230	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689230	2
C₄⋊Q₈×Dic₄₇	2
#20244260	2
#20244191	2
#20243545	2
C₄×#13093934	2
C₂×C₄₇×#494	2
#21691	47