Subgroups — #26105651

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243134	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#15688878	C₄	1	4	yes	—
13	#15688878	C₄	1	4	yes	—
14	#15688845	C₄	1	4	yes	complemented
15	#15688845	C₄	1	4	yes	complemented
16	#15688844	C₄	1	4	yes	complemented
17	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#336	C₄	1	4	yes	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	C₄²	1	4	yes	—
51	C₂×C₈×C₄₇	C₄²	1	4	yes	—
52	C₄²×C₄₇	—	2	4	no	—
53	C₄²×C₄₇	—	2	4	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄₇⋊C₈	—	2	4	no	—
60	C₂×C₄₇⋊C₈	—	2	4	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
64	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
65	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
66	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
67	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#131	1	4	yes	—
78	C₂×C₄×C₄₇	#131	1	4	yes	—
79	C₂×C₄×C₄₇	#131	1	4	yes	—
80	C₈×C₄₇	—	2	4	no	—
81	C₈×C₄₇	—	2	4	no	—
82	C₄₇⋊C₈	—	2	4	no	—
83	C₄₇⋊C₈	—	2	4	no	—
84	C₄₇⋊C₈	—	2	4	no	—
85	C₄₇⋊C₈	—	2	4	no	—
86	#7845	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₄³	1	1	yes	characteristic
88	C₂²×C₄₇	#405	1	2	yes	—
89	C₂²×C₄₇	#405	1	2	yes	—
90	C₂²×C₄₇	#405	1	2	yes	—
91	C₄×C₄₇	—	4	4	no	—
92	C₄×C₄₇	#405	1	2	yes	—
93	C₄×C₄₇	#405	1	2	yes	—
94	C₄×C₄₇	#405	1	2	yes	—
95	C₄×C₄₇	#405	1	2	yes	—
96	C₄×C₄₇	#405	1	2	yes	—
97	C₄×C₄₇	#405	1	2	yes	—
98	#1744	—	47	1	no	—
99	#1744	—	47	1	no	—
100	#1744	D₄₇	1	1	yes	characteristic
101	C₂×#322	—	47	1	no	—
102	C₂×#322	—	47	1	no	—
103	C₂×#322	—	47	1	no	—
104	C₂×C₈²	—	47	1	no	—
105	C₂×C₄₇	#1454	1	4	yes	—
106	C₂×C₄₇	#1726	1	1	yes	characteristic
107	C₂×C₄₇	#1726	1	1	yes	characteristic
108	C₂×C₄₇	#1726	1	1	yes	characteristic
109	C₂×C₄×C₈	—	47	1	no	—
110	C₂×C₄×C₈	—	47	1	no	—
111	C₂×C₄×C₈	—	47	1	no	—
112	C₂×C₄×C₈	—	47	1	no	—
113	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
115	C₄³	C₂×D₄₇	1	1	yes	characteristic
116	#336	—	47	4	no	—
117	#336	—	47	4	no	—
118	#336	Dic₄₇	1	4	yes	—
119	#322	—	47	4	no	—
120	#322	—	47	4	no	—
121	#322	—	47	4	no	—
122	C₈²	—	47	4	no	—
123	C₄₇	#7845	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	47	2	no	—
125	C₂²×C₈	—	47	2	no	—
126	C₂²×C₈	—	47	2	no	—
127	C₂²×C₈	—	47	2	no	—
128	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
129	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
130	C₂×C₄²	C₄×D₄₇	1	2	yes	—
131	C₂×C₄²	C₄×D₄₇	1	2	yes	—
132	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄×C₈	—	47	2	no	—
135	C₄×C₈	—	47	2	no	—
136	C₄×C₈	—	47	2	no	—
137	C₄×C₈	—	47	2	no	—
138	C₄×C₈	—	47	2	no	—
139	C₄×C₈	—	47	2	no	—
140	C₄×C₈	—	47	2	no	—
141	C₄×C₈	—	47	2	no	—
142	C₄×C₈	C₄×D₄₇	1	2	yes	—
143	C₄×C₈	C₄×D₄₇	1	2	yes	—
144	C₄×C₈	C₄×D₄₇	1	2	yes	—
145	C₄×C₈	C₄×D₄₇	1	2	yes	—
146	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
147	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	47	4	no	—
153	C₂×C₈	—	47	4	no	—
154	C₂×C₈	—	47	4	no	—
155	C₂×C₈	—	47	4	no	—
156	C₂×C₈	—	94	4	no	—
157	C₂×C₈	—	94	4	no	—
158	C₂×C₈	—	94	4	no	—
159	C₂×C₈	—	94	4	no	—
160	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
161	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
166	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
167	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
169	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
175	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
176	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	#13094030	1	4	yes	—
180	C₂×C₄	#13094031	1	4	yes	—
181	C₂×C₄	#13094032	1	4	yes	—
182	C₈	—	2	4	no	—
183	C₈	—	2	4	no	—
184	C₈	—	94	4	no	—
185	C₈	—	94	4	no	—
186	C₈	—	94	4	no	—
187	C₈	—	94	4	no	—
188	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
189	C₂²	#15689078	1	2	yes	—
190	C₂²	#15689079	1	2	yes	—
191	C₂²	#15689081	1	2	yes	—
192	C₄	—	4	4	no	—
193	C₄	#15689077	1	2	yes	—
194	C₄	#15689077	1	2	yes	—
195	C₄	#15689079	1	2	yes	—
196	C₄	#15689079	1	2	yes	—
197	C₄	#15689080	1	2	yes	—
198	C₄	#15689080	1	2	yes	—
199	C₂	#20241688	1	4	yes	—
200	C₂	#20243074	1	1	yes	characteristic
201	C₂	#20243075	1	1	yes	characteristic
202	C₂	#20243076	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#7845
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
14. #15688845	C₄	semidirect
15. #15688845	C₄	semidirect
16. #15688844	C₄	semidirect
17. C₈×C₄₇⋊C₈	C₄	semidirect
123. C₄₇	#7845	semidirect

Maximal subgroups

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

subgroup	index
#20243134	2
C₂×#15688845	2
C₂×#15688844	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1744	2
#7845	47