Subgroups — #26371778

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20245675	C₂	1	4	yes	maximal, complemented
3	#20245671	C₂	1	4	yes	maximal, complemented
4	#20245170	C₂	1	4	yes	maximal
5	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
7	#15689680	—	2	1	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#15689528	C₂²	1	2	yes	—
10	#15689525	—	2	1	no	—
11	#15689525	C₂²	1	2	yes	—
12	#15689521	—	2	1	no	—
13	#15689161	—	2	1	no	—
14	#15689161	C₂²	1	2	yes	—
15	#15688950	C₂²	1	8	yes	—
16	#15688948	C₂²	1	8	yes	—
17	#15688899	C₂²	1	2	yes	—
18	#15688899	C₂²	1	2	yes	—
19	C₄₇×#533	—	2	1	no	—
20	C₄₇×#500	C₂²	1	2	yes	—
21	C₄₇×#499	—	2	1	no	—
22	C₄₇×#499	C₂²	1	2	yes	—
23	C₄₇×#498	—	2	1	no	—
24	C₄₇×#427	—	2	1	no	—
25	C₄₇×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	#13094024	—	2	2	no	—
28	#13094016	—	2	2	no	—
29	#13094003	—	4	2	no	—
30	C₄×Dic₉₄	—	4	2	no	—
31	#13093983	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	Q₈	1	8	yes	complemented
33	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
34	#13093958	—	2	2	no	—
35	#13093958	C₂³	1	4	yes	—
36	#13093956	—	2	2	no	—
37	#13093956	—	2	2	no	—
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₇	—	2	2	no	—
41	#125×C₄₇	—	2	2	no	—
42	C₂²⋊Q₈×C₄₇	—	4	2	no	—
43	C₄×Q₈×C₄₇	—	4	2	no	—
44	#117×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	#107×C₄₇	—	2	2	no	—
47	#107×C₄₇	C₂³	1	4	yes	—
48	#106×C₄₇	—	2	2	no	—
49	#106×C₄₇	—	2	2	no	—
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	D₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₉₄	—	4	2	no	—
58	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₂²⋊C₄×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	—	4	2	no	—
69	C₄²×C₄₇	D₈	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	Dic₄₇⋊C₄	—	4	4	no	—
76	C₄×Dic₄₇	—	4	2	no	—
77	C₄₇⋊C₁₆	C₂×Q₈	1	8	yes	—
78	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₄₇	—	8	4	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₄₇	—	4	4	no	—
89	Dic₉₄	—	8	4	no	—
90	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
91	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
93	C₈×C₄₇	C₂²×Q₈	1	2	yes	—
94	#33728	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	#500	1	2	yes	—
97	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	—	8	4	no	—
101	C₄×C₄₇	#500	1	2	yes	—
102	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₄₇	—	8	4	no	—
105	#3160	—	47	1	no	—
106	#3160	D₄₇	1	1	yes	characteristic
107	#2255	—	47	4	no	—
108	#2255	—	47	4	no	—
109	#2158	—	47	4	no	—
110	C₂×#346	—	47	1	no	—
111	C₂×C₄₇	—	2	2	no	—
112	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
113	C₂×C₄₇	#3414	1	2	yes	—
114	#533	—	2	1	no	—
115	#533	—	94	1	no	—
116	C₂²×C₁₆	—	47	2	no	—
117	#500	—	47	2	no	—
118	#500	C₂×D₄₇	1	2	yes	—
119	#499	—	2	1	no	—
120	#499	—	47	2	no	—
121	#499	—	94	1	no	—
122	#499	C₂×D₄₇	1	2	yes	—
123	#498	—	2	1	no	—
124	#498	—	94	1	no	—
125	#427	—	2	1	no	—
126	#427	—	47	2	no	—
127	#427	—	94	1	no	—
128	#427	C₂×D₄₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
130	#367	—	47	8	no	—
131	#366	—	47	8	no	—
132	#346	—	47	2	no	—
133	#346	—	47	2	no	—
134	C₄₇	#33728	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	94	2	no	—
139	#125	—	2	2	no	—
140	#125	—	94	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	188	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₄×Q₈	—	188	2	no	—
145	#117	—	2	2	no	—
146	#117	—	94	2	no	—
147	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂×C₁₆	—	47	8	no	—
149	C₂×C₁₆	—	47	4	no	—
150	#107	—	2	2	no	—
151	#107	—	47	4	no	—
152	#107	—	94	2	no	—
153	#107	C₂²×D₄₇	1	4	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	94	2	no	—
157	#106	—	94	2	no	—
158	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	188	2	no	—
164	C₂²×C₄	#10625334	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂×C₈	#10625337	1	4	yes	—
167	C₂×C₈	Q₈×D₄₇	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	94	4	no	—
176	C₄⋊C₄	—	188	4	no	—
177	C₄⋊C₄	—	188	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	188	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	188	2	no	—
182	C₄²	#10625334	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
185	C₁₆	—	47	8	no	—
186	C₂³	C₂×#10625334	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	376	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	188	4	no	—
192	C₂×C₄	#13094022	1	2	yes	—
193	C₂×C₄	C₂×#10625334	1	2	yes	—
194	C₂×C₄	C₂×#10625334	1	2	yes	—
195	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
196	C₂×C₄	C₂×#10625337	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₈	#13094022	1	2	yes	—
199	C₈	C₂×#10625337	1	2	yes	—
200	C₈	C₂×#10625337	1	2	yes	—
201	C₈	C₂×Q₈×D₄₇	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#15689526	1	2	yes	—
204	C₂²	C₂²×#10625334	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	376	4	no	—
209	C₄	#15689526	1	2	yes	—
210	C₄	C₂×#13094022	1	1	yes	characteristic
211	C₄	C₂²×#10625337	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#15689526	1	1	yes	characteristic
214	C₂	#20256522	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#33728
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #20245675	C₂	semidirect
3. #20245671	C₂	semidirect
32. C₂×C₄₇⋊C₁₆	Q₈	semidirect
134. C₄₇	#33728	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20245675	2
#20245671	2
#20245170	2
C₂×#15688899	2
C₄₇×#3160	2
#33728	47