Subgroups — #26134898

Order 12032.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 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	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	#20241495	C₂	1	4	yes	maximal
4	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
5	#15689680	—	2	1	no	—
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689528	C₄	1	2	yes	complemented
8	#15689525	—	2	1	no	—
9	#15689525	C₄	1	2	yes	complemented
10	#15689521	—	2	1	no	—
11	#15689161	—	2	1	no	—
12	#15689161	C₂²	1	2	yes	—
13	#15688899	—	2	1	no	—
14	C₄×C₄₇⋊C₁₆	—	2	1	no	—
15	C₂×#106×C₄₇	—	2	1	no	—
16	C₂×#106×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#384	—	2	1	no	—
19	#13094024	—	2	2	no	—
20	#13094016	—	2	2	no	—
21	#13094003	—	4	2	no	—
22	C₄×Dic₉₄	—	4	2	no	—
23	#13093983	—	2	2	no	—
24	#13093964	—	2	2	no	—
25	#13093964	Q₈	1	4	yes	complemented
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	#13093958	—	2	2	no	—
28	#13093958	C₂×C₄	1	4	yes	—
29	#13093956	—	2	2	no	—
30	#13093956	—	2	2	no	—
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	#106×C₄₇	—	2	4	no	—
37	#106×C₄₇	—	4	2	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	D₈	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₉₄	—	4	2	no	—
46	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₄₇	#41	1	2	yes	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄⋊C₄×C₄₇	—	4	2	no	—
56	C₄⋊C₄×C₄₇	—	8	4	no	—
57	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₄₇	SD₁₆	1	2	yes	—
60	C₂²⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	2	4	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	Dic₄₇⋊C₄	—	4	4	no	—
64	C₄×Dic₄₇	—	4	2	no	—
65	C₄₇⋊C₁₆	—	2	4	no	—
66	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	8	4	no	—
69	C₂×C₄×C₄₇	—	8	2	no	—
70	C₂×C₄×C₄₇	#102	1	2	yes	—
71	C₂×C₄×C₄₇	#102	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₄₇	—	4	4	no	—
79	Dic₉₄	—	8	4	no	—
80	C₈×C₄₇	#103	1	2	yes	—
81	C₈×C₄₇	#103	1	2	yes	—
82	C₈×C₄₇	C₄×Q₈	1	2	yes	—
83	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
84	#13287	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₇	#486	1	1	yes	characteristic
88	C₂²×C₄₇	#500	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	16	2	no	—
91	C₄×C₄₇	#360	1	4	yes	—
92	C₄×C₄₇	#391	1	1	yes	characteristic
93	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₄₇	#487	1	1	yes	characteristic
95	C₄×C₄₇	#498	1	1	yes	characteristic
96	Dic₄₇	—	8	4	no	—
97	#3160	—	47	1	no	—
98	C₄×M₃₂	—	47	1	no	—
99	#1844	D₄₇	1	1	yes	characteristic
100	#1386	—	47	4	no	—
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	—	2	1	no	—
103	C₂×C₄₇	#1990	1	1	yes	characteristic
104	C₂×C₄₇	C₂×#360	1	2	yes	—
105	#533	—	94	1	no	—
106	C₂×M₃₂	—	47	2	no	—
107	#500	—	47	2	no	—
108	#499	—	47	2	no	—
109	#499	—	94	1	no	—
110	#498	—	94	1	no	—
111	#427	—	47	2	no	—
112	#427	—	94	1	no	—
113	C₂×#106	—	2	1	no	—
114	C₂×#106	C₂×D₄₇	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	—	2	1	no	—
117	#346	—	94	1	no	—
118	C₄×C₁₆	—	94	1	no	—
119	C₄₇	#13287	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	C₄⋊Q₈	—	94	2	no	—
123	#125	—	94	2	no	—
124	C₂²⋊Q₈	—	188	2	no	—
125	C₄×Q₈	—	188	2	no	—
126	#117	—	94	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	2	no	—
129	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
130	M₃₂	—	47	4	no	—
131	M₃₂	—	94	2	no	—
132	C₂×C₁₆	—	94	2	no	—
133	#107	—	47	4	no	—
134	#107	—	94	2	no	—
135	#106	—	2	4	no	—
136	#106	—	4	2	no	—
137	#106	—	94	2	no	—
138	#106	—	94	2	no	—
139	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
140	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
141	C₄×C₈	D₁₈₈	1	1	yes	characteristic
142	C₄×C₈	D₁₈₈	1	1	yes	characteristic
143	C₂×Q₈	—	188	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	#10625334	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
149	C₂×C₈	#10625335	1	2	yes	—
150	C₂×C₈	#10625337	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
153	C₂×C₈	#10625352	1	2	yes	—
154	C₂×C₈	Q₈×D₄₇	1	2	yes	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	8	4	no	—
158	C₄⋊C₄	—	94	4	no	—
159	C₄⋊C₄	—	188	4	no	—
160	C₄⋊C₄	—	188	2	no	—
161	C₂²⋊C₄	—	188	2	no	—
162	C₄²	—	188	2	no	—
163	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
164	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
165	C₄²	#10625336	1	2	yes	—
166	C₁₆	—	94	4	no	—
167	C₂³	C₂×#10625334	1	1	yes	characteristic
168	Q₈	—	376	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	188	4	no	—
173	C₂×C₄	#13093944	1	2	yes	—
174	C₂×C₄	#13093944	1	2	yes	—
175	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#13094011	1	1	yes	characteristic
177	C₂×C₄	#13094022	1	1	yes	characteristic
178	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
179	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
180	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
181	C₈	#13093947	1	2	yes	—
182	C₈	#13093947	1	2	yes	—
183	C₈	#13093989	1	2	yes	—
184	C₈	#13094004	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#13093944	1	1	yes	characteristic
187	C₂²	#15689469	1	1	yes	characteristic
188	C₂²	#15689526	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	16	2	no	—
191	C₄	—	376	4	no	—
192	C₄	#15688932	1	4	yes	—
193	C₄	#15689028	1	1	yes	characteristic
194	C₄	C₂×#13093947	1	1	yes	characteristic
195	C₄	#15689473	1	1	yes	characteristic
196	C₄	#15689519	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20244348	1	1	yes	characteristic
200	C₂	C₂×#15688932	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#13287
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
7. #15689528	C₄	semidirect
9. #15689525	C₄	semidirect
25. #13093964	Q₈	semidirect
119. C₄₇	#13287	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
C₄×#13093964	2
#20241495	2
C₄₇×#1844	2
#13287	47