Subgroups — #26131602

Order 12032.

358 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
2	M₃₂×Dic₄₇	C₂	1	2	yes	maximal
3	#20243074	C₂	1	1	yes	maximal, characteristic
4	#20241469	C₂	1	2	yes	maximal
5	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	1	yes	characteristic
8	C₂×#13093934	C₄	1	1	yes	characteristic
9	C₂×#13093934	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688969	C₄	1	1	yes	characteristic
13	#15688944	C₄	1	4	yes	complemented
14	#15688898	—	2	2	no	—
15	C₁₆×Dic₄₇	—	2	2	no	—
16	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#364	C₄	1	4	yes	complemented
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	#13093964	—	2	2	no	—
24	#13093964	C₂×C₄	1	4	yes	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	#13093935	—	2	2	no	—
27	#13093934	C₈	1	4	yes	—
28	#13093934	C₈	1	4	yes	—
29	C₈×Dic₄₇	—	2	1	no	—
30	C₈×Dic₄₇	—	2	1	no	—
31	C₈×Dic₄₇	D₄	1	2	yes	—
32	C₈×Dic₄₇	Q₈	1	2	yes	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₄₇	—	2	2	no	—
38	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₄₇	—	2	2	no	—
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	—	2	1	no	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₄₇	—	2	1	no	—
45	C₂²×Dic₄₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₄₇	C₄²	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
53	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
55	C₄²×C₄₇	—	2	1	no	—
56	C₄²×C₄₇	C₄²	1	2	yes	—
57	C₄×Dic₄₇	—	2	1	no	—
58	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
59	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
60	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
61	C₂×C₄₇⋊C₈	—	2	1	no	—
62	C₂×C₄₇⋊C₈	—	2	1	no	—
63	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
64	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
66	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
67	C₁₆×C₄₇	—	2	4	no	—
68	C₄₇⋊C₁₆	—	2	4	no	—
69	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₄₇	—	2	2	no	—
79	C₂×Dic₄₇	—	4	2	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
82	C₈×C₄₇	C₄×C₈	1	4	yes	—
83	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
84	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
85	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
87	C₄₇⋊C₈	—	2	2	no	—
88	C₄₇⋊C₈	C₄⋊C₈	1	4	yes	—
89	#12656	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₄₇	#431	1	1	yes	characteristic
94	C₄×C₄₇	—	4	2	no	—
95	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₄₇	#431	1	1	yes	characteristic
99	Dic₄₇	—	2	4	no	—
100	Dic₄₇	—	4	2	no	—
101	C₂×#364	—	47	1	no	—
102	C₂×#364	D₄₇	1	1	yes	characteristic
103	C₄×M₃₂	—	47	2	no	—
104	#1726	—	47	1	no	—
105	#1376	—	47	2	no	—
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₄₇	#2153	1	2	yes	—
110	C₂×M₃₂	—	47	2	no	—
111	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
112	C₂×#97	—	47	1	no	—
113	C₂×#97	—	47	1	no	—
114	C₂×#97	—	47	1	no	—
115	C₂×C₄×C₈	—	47	1	no	—
116	C₂×C₄×C₈	—	47	1	no	—
117	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
118	#376	—	47	1	no	—
119	#364	—	47	4	no	—
120	#364	Dic₄₇	1	4	yes	complemented
121	#346	—	94	2	no	—
122	C₄×C₁₆	—	94	2	no	—
123	C₄₇	#12656	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	—	47	1	no	—
125	C₂²×C₈	—	47	1	no	—
126	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	C₂×C₄²	—	47	1	no	—
129	C₂×C₄²	—	47	1	no	—
130	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	47	4	no	—
133	M₃₂	—	94	2	no	—
134	M₃₂	C₂×Dic₄₇	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	94	2	no	—
137	#97	—	47	4	no	—
138	#97	—	47	4	no	—
139	#97	—	94	2	no	—
140	C₄×C₈	—	47	2	no	—
141	C₄×C₈	—	47	2	no	—
142	C₄×C₈	—	94	1	no	—
143	C₄×C₈	—	94	1	no	—
144	C₄×C₈	C₄×D₄₇	1	2	yes	—
145	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
146	#95	—	94	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	47	2	no	—
149	C₂²×C₄	—	94	1	no	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₈	—	47	2	no	—
152	C₂×C₈	—	47	2	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	2	no	—
155	C₂×C₈	—	94	1	no	—
156	C₂×C₈	—	94	1	no	—
157	C₂×C₈	C₈×D₄₇	1	2	yes	—
158	C₂×C₈	C₈×D₄₇	1	2	yes	—
159	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
162	C₂×C₈	#10625352	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
164	C₂×C₈	#10625354	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
166	C₄²	—	2	1	no	—
167	C₄²	—	47	2	no	—
168	C₄²	—	47	2	no	—
169	C₄²	—	47	2	no	—
170	C₄²	—	94	1	no	—
171	C₄²	C₄×Dic₄₇	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	94	4	no	—
174	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	47	4	no	—
178	C₂×C₄	—	94	2	no	—
179	C₂×C₄	—	188	2	no	—
180	C₂×C₄	—	188	2	no	—
181	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄	#13093982	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
187	C₈	—	47	4	no	—
188	C₈	—	94	2	no	—
189	C₈	C₈×Dic₄₇	1	4	yes	—
190	C₈	#13093986	1	1	yes	characteristic
191	C₈	D₄×Dic₄₇	1	1	yes	characteristic
192	C₈	#13093989	1	1	yes	characteristic
193	C₈	Q₈×Dic₄₇	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	M₁₆×Dic₄₇	1	1	yes	characteristic
196	C₂²	C₄⋊C₈×D₄₇	1	1	yes	characteristic
197	C₂²	#15689175	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	94	4	no	—
200	C₄	—	188	2	no	—
201	C₄	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
202	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
203	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
204	C₄	#15689175	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄⋊C₈×Dic₄₇	1	1	yes	characteristic
208	C₂	#20245152	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₂×M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	#12656
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #15688944	C₄	semidirect
18. C₄₇×#364	C₄	semidirect
120. #364	Dic₄₇	semidirect
123. C₄₇	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688944	2
M₃₂×Dic₄₇	2
#20243074	2
#20241469	2
C₂×C₄₇×#364	2
#12656	47