Subgroups — #26107529

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⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
4	#20241229	C₂	1	2	yes	maximal
5	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688878	—	2	2	no	—
11	#15688876	C₄	1	4	yes	complemented
12	#15688844	C₄	1	4	yes	complemented
13	C₂×#106×C₄₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#340	C₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
22	C₄×C₄₇⋊C₈	—	2	1	no	—
23	C₄×C₄₇⋊C₈	—	4	4	no	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#106×C₄₇	C₈	1	8	yes	complemented
33	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	#95×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	C₄²	1	2	yes	—
42	C₂×C₈×C₄₇	C₄²	1	2	yes	—
43	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
44	C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
46	C₄²×C₄₇	—	4	2	no	—
47	C₄²×C₄₇	—	4	2	no	—
48	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	2	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	4	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
60	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	8	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	4	4	no	—
65	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₄₇	#131	1	2	yes	—
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₄₇	#137	1	2	yes	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	C₄×C₈	1	4	yes	complemented
77	C₄₇⋊C₈	—	2	4	no	—
78	C₄₇⋊C₈	—	4	2	no	—
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	—	4	8	no	—
88	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₄₇	#405	1	1	yes	characteristic
90	C₄×C₄₇	#428	1	2	yes	—
91	C₄×C₄₇	#441	1	1	yes	characteristic
92	C₄×C₄₇	#442	1	1	yes	characteristic
93	C₄×C₄₇	C₈×Q₈	1	2	yes	—
94	C₄×#106	D₄₇	1	1	yes	characteristic
95	C₈×C₄⋊C₄	—	47	2	no	—
96	C₂×#335	—	47	1	no	—
97	C₂×#322	—	47	1	no	—
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	Dic₄₇	1	2	yes	—
104	C₂×C₄⋊C₈	—	47	2	no	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₂×C₄×C₈	—	47	1	no	—
107	C₂×C₄×C₈	—	94	2	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
110	#340	Dic₄₇	1	2	yes	—
111	#336	—	94	2	no	—
112	#335	—	47	4	no	—
113	#322	—	47	4	no	—
114	C₄₇	#8285	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	—	94	1	no	—
118	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
121	C₂×C₄²	—	2	2	no	—
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	#106	C₄₇⋊C₈	1	8	yes	complemented
124	C₄⋊C₈	—	47	4	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄×C₈	—	47	2	no	—
127	C₄×C₈	—	47	2	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	—	94	1	no	—
130	C₄×C₈	—	188	4	no	—
131	C₄×C₈	C₄×D₄₇	1	2	yes	—
132	C₄×C₈	C₄×D₄₇	1	2	yes	—
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	#10625352	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	47	4	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	—	188	2	no	—
145	C₂×C₈	—	188	2	no	—
146	C₂×C₈	—	188	2	no	—
147	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
148	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
149	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
156	C₄²	Q₈×D₄₇	1	1	yes	characteristic
157	C₄²	#10625354	1	1	yes	characteristic
158	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	#13093982	1	1	yes	characteristic
166	C₂×C₄	#13093986	1	1	yes	characteristic
167	C₂×C₄	#13093989	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
170	C₂×C₄	#13094031	1	2	yes	—
171	C₂×C₄	#13094051	1	2	yes	—
172	C₂×C₄	#13094053	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	94	4	no	—
175	C₈	—	188	2	no	—
176	C₈	—	188	4	no	—
177	C₈	C₄×C₄₇⋊C₈	1	4	yes	complemented
178	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
179	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
180	C₂²	#15689224	1	2	yes	—
181	C₂²	#15689245	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	8	no	—
185	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
186	C₄	#15689079	1	1	yes	characteristic
187	C₄	#15689162	1	2	yes	—
188	C₄	#15689229	1	1	yes	characteristic
189	C₄	#15689234	1	1	yes	characteristic
190	C₄	Q₈×C₄₇⋊C₈	1	2	yes	—
191	C₂	SD₁₆⋊(C₄₇⋊C₈)	1	4	yes	—
192	C₂	C₄⋊C₄×C₄₇⋊C₈	1	1	yes	characteristic
193	C₂	#20243274	1	1	yes	characteristic
194	C₂	#20243288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#8285
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. #15688876	C₄	semidirect
12. #15688844	C₄	semidirect
32. #106×C₄₇	C₈	semidirect
76. C₈×C₄₇	C₄×C₈	semidirect
114. C₄₇	#8285	semidirect
123. #106	C₄₇⋊C₈	semidirect
177. C₈	C₄×C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×C₄₇⋊C₈	2
C₂×#15688876	2
C₂×#15688844	2
#20241229	2
C₄×#106×C₄₇	2
#8285	47