Subgroups — #26107527

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	#20241228	C₂	1	2	yes	maximal
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093956	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₈×Dic₄₇	—	2	2	no	—
9	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#15688886	C₄	1	2	yes	—
11	#15688877	—	2	2	no	—
12	#15688876	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#322	C₄	1	4	yes	complemented
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₄₇	—	2	2	no	—
20	#13093956	C₈	1	8	yes	complemented
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
23	C₈×Dic₄₇	—	4	4	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₄₇	—	2	1	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₄₇	—	2	1	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₇	Q₈	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	C₄²	1	2	yes	—
46	C₂×C₈×C₄₇	C₄²	1	2	yes	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
49	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₄₇	#41	1	1	yes	characteristic
53	C₄⋊Dic₄₇	—	2	4	no	—
54	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	—
55	C₄×Dic₄₇	—	4	2	no	—
56	C₄×Dic₄₇	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	2	4	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₄₇	#131	1	2	yes	—
70	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₄₇	#137	1	2	yes	—
72	C₂×Dic₄₇	—	2	8	no	—
73	C₂×Dic₄₇	—	4	4	no	—
74	C₂×Dic₄₇	—	4	4	no	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	—	2	4	no	—
77	C₈×C₄₇	—	4	2	no	—
78	C₈×C₄₇	C₄×C₈	1	4	yes	complemented
79	C₄₇⋊C₈	—	4	4	no	—
80	#8285	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄₇	C₈×D₄	1	2	yes	—
83	C₂²×C₄₇	#440	1	2	yes	—
84	C₂²×C₄₇	#444	1	2	yes	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₄₇	#405	1	1	yes	characteristic
89	C₄×C₄₇	#428	1	2	yes	—
90	C₄×C₄₇	#441	1	1	yes	characteristic
91	C₄×C₄₇	#442	1	1	yes	characteristic
92	C₄×C₄₇	C₈×Q₈	1	2	yes	—
93	Dic₄₇	—	4	8	no	—
94	C₄×#106	—	47	1	no	—
95	C₈×C₄⋊C₄	—	47	2	no	—
96	C₂×#335	—	47	1	no	—
97	C₂×#322	D₄₇	1	1	yes	characteristic
98	#1279	—	47	2	no	—
99	C₂×C₄₇	SD₁₆⋊C₈	1	4	yes	—
100	C₂×C₄₇	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄₇	#1773	1	1	yes	characteristic
102	C₂×C₄₇	#1776	1	1	yes	characteristic
103	C₂×#106	—	47	2	no	—
104	C₂×C₄⋊C₈	—	47	2	no	—
105	C₂×C₄×C₈	—	94	2	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	47	2	no	—
110	#340	—	47	2	no	—
111	#336	—	94	2	no	—
112	#335	—	47	4	no	—
113	#322	Dic₄₇	1	4	yes	complemented
114	C₄₇	#8285	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₂²×C₈	D₁₈₈	1	2	yes	—
120	C₂×C₄⋊C₄	—	47	4	no	—
121	C₂×C₄²	—	94	2	no	—
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	#106	—	47	8	no	—
124	C₄⋊C₈	—	47	4	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	188	4	no	—
128	C₄×C₈	Dic₉₄	1	2	yes	—
129	C₄×C₈	C₄×D₄₇	1	2	yes	—
130	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
131	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
132	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
133	#95	—	94	2	no	—
134	C₂²×C₄	—	94	4	no	—
135	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂²×C₄	#10625350	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	—	188	2	no	—
145	C₂×C₈	—	188	2	no	—
146	C₂×C₈	C₈×D₄₇	1	4	yes	—
147	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
148	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
150	C₄⋊C₄	—	47	8	no	—
151	C₄⋊C₄	—	94	4	no	—
152	C₄²	—	188	2	no	—
153	C₄²	—	188	2	no	—
154	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
156	C₄²	#10625350	1	1	yes	characteristic
157	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
158	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	94	8	no	—
161	C₂×C₄	—	188	4	no	—
162	C₂×C₄	—	188	4	no	—
163	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
165	C₂×C₄	#13093983	1	1	yes	characteristic
166	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
167	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
170	C₂×C₄	#13094030	1	2	yes	—
171	C₂×C₄	#13094052	1	2	yes	—
172	C₂×C₄	#13094057	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	188	4	no	—
177	C₈	C₈×Dic₄₇	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
179	C₂²	C₈×D₁₈₈	1	2	yes	—
180	C₂²	#15689225	1	2	yes	—
181	C₂²	#15689246	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	188	8	no	—
185	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
186	C₄	#15689078	1	1	yes	characteristic
187	C₄	#15689165	1	2	yes	—
188	C₄	#15689231	1	1	yes	characteristic
189	C₄	#15689236	1	1	yes	characteristic
190	C₄	C₈×Dic₉₄	1	2	yes	—
191	C₂	#20242202	1	4	yes	—
192	C₂	C₈×C₄⋊Dic₄₇	1	1	yes	characteristic
193	C₂	#20243272	1	1	yes	characteristic
194	C₂	#20243285	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8285
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #15688876	C₄	semidirect
16. C₄₇×#322	C₄	semidirect
20. #13093956	C₈	semidirect
78. C₈×C₄₇	C₄×C₈	semidirect
113. #322	Dic₄₇	semidirect
114. C₄₇	#8285	semidirect
177. C₈	C₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093956	2
C₈×C₄⋊Dic₄₇	2
C₂×#15688876	2
#20241228	2
C₂×C₄₇×#322	2
#8285	47