Subgroups — #26107412

Order 12032.

494 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×#15688844	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₇⋊#1273	C₂	1	2	yes	maximal
5	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	C₄	1	4	yes	—
10	#15688844	—	2	2	no	—
11	C₈×C₄₇⋊C₈	—	2	2	no	—
12	C₂×#97×C₄₇	C₄	1	2	yes	—
13	C₂×#97×C₄₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₄×C₄₇⋊C₈	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	#97×C₄₇	—	2	4	no	—
32	#97×C₄₇	C₈	1	8	yes	complemented
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	D₄	1	2	yes	—
35	C₄×C₈×C₄₇	Q₈	1	2	yes	—
36	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
44	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
47	C₄²×C₄₇	—	2	4	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	—	2	4	no	—
50	C₄²×C₄₇	C₂×C₈	1	4	yes	—
51	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₄₇	#41	1	2	yes	—
54	C₂×C₄₇⋊C₈	—	2	4	no	—
55	C₂×C₄₇⋊C₈	—	2	4	no	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	—	2	4	no	—
58	C₂×C₄₇⋊C₈	—	2	4	no	—
59	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	—
60	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
68	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
74	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₄₇	#131	1	4	yes	—
76	C₈×C₄₇	—	2	4	no	—
77	C₈×C₄₇	C₄⋊C₈	1	8	yes	complemented
78	C₄₇⋊C₈	—	2	8	no	—
79	C₄₇⋊C₈	—	4	4	no	—
80	#8262	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
84	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₄₇	#405	1	2	yes	—
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	—	4	1	no	—
88	C₄×C₄₇	—	4	1	no	—
89	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
90	C₄×C₄₇	#431	1	2	yes	—
91	C₄×C₄₇	#435	1	4	yes	—
92	C₄×C₄₇	#446	1	4	yes	—
93	C₄×C₄⋊C₈	—	47	2	no	—
94	C₄×#97	D₄₇	1	1	yes	characteristic
95	C₂×#322	—	47	1	no	—
96	C₂×C₈²	—	47	1	no	—
97	#1273	—	47	2	no	—
98	C₂×C₄₇	C₈×M₁₆	1	2	yes	—
99	C₂×C₄₇	#1453	1	1	yes	characteristic
100	C₂×C₄₇	#1455	1	1	yes	characteristic
101	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₄₇	#1771	1	2	yes	—
103	C₂×C₄⋊C₈	—	47	4	no	—
104	C₂×#97	Dic₄₇	1	2	yes	—
105	C₂×#97	Dic₄₇	1	2	yes	—
106	C₂×C₄×C₈	—	47	2	no	—
107	C₂×C₄×C₈	—	47	2	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	C₄³	C₂×D₄₇	1	1	yes	characteristic
111	#336	—	47	4	no	—
112	#322	—	94	2	no	—
113	C₈²	—	94	2	no	—
114	C₄₇	#8262	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	47	4	no	—
116	C₂²×C₈	—	47	4	no	—
117	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
118	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
119	C₂×C₄²	C₄×D₄₇	1	2	yes	—
120	C₂×C₄²	C₄×D₄₇	1	2	yes	—
121	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	94	8	no	—
125	#97	—	2	4	no	—
126	#97	C₄₇⋊C₈	1	8	yes	complemented
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	94	2	no	—
129	C₄×C₈	—	94	2	no	—
130	C₄×C₈	—	94	2	no	—
131	C₄×C₈	—	94	2	no	—
132	C₄×C₈	Dic₉₄	1	2	yes	—
133	C₄×C₈	D₁₈₈	1	2	yes	—
134	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
135	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
136	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	47	8	no	—
142	C₂×C₈	—	94	4	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	—	94	4	no	—
145	C₂×C₈	—	94	4	no	—
146	C₂×C₈	—	94	4	no	—
147	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
148	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
149	C₂×C₈	#10625329	1	4	yes	—
150	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
155	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
156	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
157	C₄²	#10625350	1	2	yes	—
158	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
167	C₂×C₄	#13093983	1	1	yes	characteristic
168	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
169	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
170	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
171	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
172	C₂×C₄	C₂×#10625329	1	2	yes	—
173	C₂×C₄	#13094032	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	94	8	no	—
176	C₈	—	188	4	no	—
177	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	complemented
178	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
179	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
180	C₂²	M₁₆×Dic₄₇	1	2	yes	—
181	C₂²	C₄×#10625329	1	1	yes	characteristic
182	C₂²	#15689081	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
187	C₄	#15689176	1	2	yes	—
188	C₄	#15689196	1	4	yes	—
189	C₄	#15689251	1	4	yes	—
190	C₂	M₁₆×C₄₇⋊C₈	1	2	yes	—
191	C₂	#20241685	1	1	yes	characteristic
192	C₂	#20241691	1	1	yes	characteristic
193	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
194	C₂	#20243260	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#8262
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
32. #97×C₄₇	C₈	semidirect
77. C₈×C₄₇	C₄⋊C₈	semidirect
114. C₄₇	#8262	semidirect
126. #97	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×#15688844	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇⋊#1273	2
C₄×#97×C₄₇	2
#8262	47