Subgroups — #26368866

Order 12032.

806 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	C₂×#15689525	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689522	C₂	1	2	yes	maximal
3	#20251748	C₂	1	2	yes	maximal
4	#20243555	C₂	1	4	yes	maximal
5	#20243553	C₂	1	4	yes	maximal
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#499	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094016	C₂²	1	2	yes	—
9	#15689525	—	2	4	no	—
10	#15689522	—	2	8	no	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093957	C₂²	1	4	yes	—
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×#13093955	C₂²	1	4	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	#15689018	—	2	8	no	—
19	#15688992	C₂²	1	4	yes	—
20	#15688988	C₂²	1	4	yes	—
21	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×#125×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#499	—	2	4	no	—
24	C₂×#107×C₄₇	C₂²	1	2	yes	—
25	C₂×#106×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
28	#13094016	—	2	4	no	—
29	#13094016	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×Dic₄₇⋊C₄	—	2	4	no	—
32	C₂×Dic₄₇⋊C₄	—	2	8	no	—
33	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
35	#13093958	—	2	4	no	—
36	#13093957	—	2	8	no	—
37	#13093956	—	2	4	no	—
38	#13093955	—	2	8	no	—
39	#13093934	Q₈	1	8	yes	—
40	#13093934	Q₈	1	8	yes	—
41	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
42	C₄₇⋊#95	—	2	8	no	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	#125×C₄₇	—	2	4	no	—
46	#125×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#107×C₄₇	—	2	4	no	—
51	#106×C₄₇	—	2	4	no	—
52	C₄×C₈×C₄₇	D₄	1	4	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	4	yes	—
58	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₄⋊C₄×C₄₇	—	4	8	no	—
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	Dic₄₇⋊C₄	—	4	16	no	—
70	Dic₄₇⋊C₄	—	4	8	no	—
71	C₄×Dic₄₇	—	4	4	no	—
72	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
73	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
74	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
75	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
76	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
77	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	4	8	no	—
81	C₂×C₄×C₄₇	#127	1	4	yes	—
82	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
86	C₂×Dic₄₇	—	2	8	no	—
87	C₂×Dic₄₇	—	4	4	no	—
88	C₂×Dic₄₇	—	4	8	no	—
89	C₂×Dic₄₇	C₄⋊Q₈	1	8	yes	—
90	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
91	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
92	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	—
93	#33315	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
96	C₂²×C₄₇	#575	1	4	yes	—
97	C₄×C₄₇	—	2	4	no	—
98	C₄×C₄₇	—	8	4	no	—
99	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
100	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	Dic₄₇	—	2	8	no	—
103	Dic₄₇	—	8	4	no	—
104	C₂×#499	—	47	1	no	—
105	C₂×#499	—	47	2	no	—
106	C₂×#499	D₄₇	1	1	yes	characteristic
107	#2859	—	47	2	no	—
108	#1847	—	47	4	no	—
109	#1847	—	47	4	no	—
110	C₄×#97	—	47	1	no	—
111	C₂×C₄₇	#2868	1	1	yes	characteristic
112	C₂×C₄₇	#3150	1	2	yes	—
113	C₂×C₄₇	#3166	1	4	yes	—
114	C₂×#125	—	47	2	no	—
115	C₂×#125	C₂×D₄₇	1	2	yes	—
116	#499	—	2	4	no	—
117	#499	—	94	8	no	—
118	#499	—	94	4	no	—
119	C₂×#107	—	47	2	no	—
120	C₂×#107	—	47	4	no	—
121	C₂×#107	C₂×D₄₇	1	2	yes	—
122	C₂×#106	—	47	2	no	—
123	C₂×#106	—	47	4	no	—
124	C₂×#106	C₂×D₄₇	1	2	yes	—
125	C₂×#97	—	47	2	no	—
126	C₂×#97	—	47	2	no	—
127	C₂×C₄×C₈	—	47	1	no	—
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	#389	—	94	8	no	—
130	#384	—	47	4	no	—
131	#382	—	47	4	no	—
132	C₄³	—	47	1	no	—
133	C₄₇	#33315	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	—	47	2	no	—
135	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
137	#125	—	2	4	no	—
138	#125	—	2	4	no	—
139	#125	—	94	4	no	—
140	#125	—	94	4	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	—	47	4	no	—
143	C₂×C₄⋊C₄	—	94	8	no	—
144	C₂×C₄⋊C₄	—	94	4	no	—
145	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
146	C₂×C₄²	—	47	2	no	—
147	C₂×C₄²	—	47	4	no	—
148	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
149	#107	—	2	4	no	—
150	#107	—	94	8	no	—
151	#107	—	94	4	no	—
152	#106	—	2	4	no	—
153	#106	—	94	8	no	—
154	#106	—	94	4	no	—
155	#97	—	47	8	no	—
156	#97	—	47	8	no	—
157	C₄×C₈	—	47	4	no	—
158	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
159	#95	—	94	8	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	47	4	no	—
162	C₂²×C₄	—	94	4	no	—
163	C₂²×C₄	D₄×D₄₇	1	2	yes	—
164	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
165	C₂×C₈	—	47	4	no	—
166	C₂×C₈	—	47	4	no	—
167	C₂×C₈	—	47	4	no	—
168	C₂×C₈	Q₈×D₄₇	1	4	yes	—
169	C₂×C₈	Q₈×D₄₇	1	4	yes	—
170	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
171	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄⋊C₄	—	94	4	no	—
176	C₄⋊C₄	—	94	4	no	—
177	C₄⋊C₄	—	188	16	no	—
178	C₄⋊C₄	—	188	8	no	—
179	C₄²	—	47	4	no	—
180	C₄²	—	47	4	no	—
181	C₄²	—	188	4	no	—
182	C₄²	D₄×D₄₇	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
184	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	—	47	8	no	—
189	C₂×C₄	—	94	8	no	—
190	C₂×C₄	—	188	4	no	—
191	C₂×C₄	—	188	8	no	—
192	C₂×C₄	#13094020	1	4	yes	—
193	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
196	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
197	C₈	—	47	8	no	—
198	C₈	#13094022	1	4	yes	—
199	C₈	#13094022	1	4	yes	—
200	C₂²	C₂×#13094020	1	1	yes	characteristic
201	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
202	C₂²	#15689920	1	4	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	8	4	no	—
205	C₄	—	94	8	no	—
206	C₄	—	376	4	no	—
207	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
208	C₄	C₂×#13094022	1	1	yes	characteristic
209	C₄	C₂×#13094022	1	1	yes	characteristic
210	C₂	#20251784	1	1	yes	characteristic
211	C₂	#20253831	1	2	yes	—
212	C₂	#20253936	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#499	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#499	D₄₇
hypocenter	C₄₇	#33315
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
133. C₄₇	#33315	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689525	2
C₂×#15689522	2
#20251748	2
#20243555	2
#20243553	2
C₄×#13093934	2
C₂×C₄₇×#499	2
#33315	47