Subgroups — #26107404

Order 12032.

502 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×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
4	C₄₇⋊#1273	C₂	1	2	yes	maximal
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
10	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
11	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688877	C₄	1	4	yes	—
13	#15688844	—	2	2	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#322	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
23	Dic₄₇⋊C₈	—	2	8	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₈×Dic₄₇	—	2	4	no	—
28	C₈×Dic₄₇	C₈	1	8	yes	complemented
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₇	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
38	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
39	C₂²×Dic₄₇	C₄²	1	2	yes	—
40	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	2	4	no	—
44	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
45	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	2	yes	—
50	C₄×Dic₄₇	—	2	2	no	—
51	C₄×Dic₄₇	—	2	4	no	—
52	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
53	C₄×Dic₄₇	M₁₆	1	8	yes	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	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	C₂×C₈	1	4	yes	—
60	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
67	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₄₇	#131	1	4	yes	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
73	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
74	C₂×Dic₄₇	#97	1	4	yes	—
75	C₂×Dic₄₇	#97	1	4	yes	—
76	C₈×C₄₇	—	4	4	no	—
77	C₈×C₄₇	C₄⋊C₈	1	8	yes	complemented
78	C₄₇⋊C₈	—	2	4	no	—
79	C₄₇⋊C₈	—	4	4	no	—
80	#8260	—	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	C₂×#97	1	2	yes	—
84	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₄₇	#405	1	2	yes	—
86	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₄₇	#431	1	2	yes	—
88	C₄×C₄₇	#436	1	4	yes	—
89	C₄×C₄₇	#446	1	4	yes	—
90	Dic₄₇	—	2	2	no	—
91	Dic₄₇	—	2	4	no	—
92	Dic₄₇	#322	1	4	yes	complemented
93	C₄×C₄⋊C₈	—	47	2	no	—
94	C₄²×C₈	—	47	1	no	—
95	C₂×#322	—	47	1	no	—
96	C₂×#322	D₄₇	1	1	yes	characteristic
97	#1273	—	47	2	no	—
98	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
99	C₂×C₄₇	#1450	1	1	yes	characteristic
100	C₂×C₄₇	#1451	1	2	yes	—
101	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₄₇	#1770	1	2	yes	—
103	C₂×C₄⋊C₈	—	47	4	no	—
104	C₂×C₄×C₈	—	47	2	no	—
105	C₂×C₄×C₈	—	47	2	no	—
106	C₂×C₄×C₈	—	47	2	no	—
107	C₂×C₄×C₈	—	47	1	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	C₄³	—	47	1	no	—
111	#336	—	47	4	no	—
112	#322	—	2	2	no	—
113	#322	—	94	2	no	—
114	C₄₇	#8260	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	47	4	no	—
117	C₂²×C₈	C₄×D₄₇	1	4	yes	—
118	C₂²×C₈	C₄×D₄₇	1	2	yes	—
119	C₂×C₄²	—	47	1	no	—
120	C₂×C₄²	—	47	2	no	—
121	C₂×C₄²	—	47	2	no	—
122	C₂×C₄²	—	47	1	no	—
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	94	8	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	47	8	no	—
128	C₄×C₈	—	94	4	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	C₂²⋊D₄₇	1	2	yes	—
134	C₂²×C₄	—	47	1	no	—
135	C₂²×C₄	—	47	1	no	—
136	C₂²×C₄	—	47	2	no	—
137	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	47	4	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	—	94	2	no	—
145	C₂×C₈	—	94	4	no	—
146	C₂×C₈	—	94	4	no	—
147	C₂×C₈	—	94	4	no	—
148	C₂×C₈	C₈×D₄₇	1	4	yes	—
149	C₂×C₈	#10625324	1	4	yes	—
150	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
151	C₄²	—	47	4	no	—
152	C₄²	—	47	8	no	—
153	C₄²	—	94	2	no	—
154	C₄²	—	94	4	no	—
155	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
156	C₄²	#10625350	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂³	C₄²×D₄₇	1	1	yes	characteristic
159	C₂×C₄	—	47	2	no	—
160	C₂×C₄	—	47	4	no	—
161	C₂×C₄	—	47	2	no	—
162	C₂×C₄	—	47	4	no	—
163	C₂×C₄	—	94	2	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	—	94	2	no	—
166	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#13093981	1	1	yes	characteristic
168	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
171	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
172	C₂×C₄	C₂×#10625324	1	2	yes	—
173	C₂×C₄	#13094032	1	4	yes	—
174	C₈	—	4	4	no	—
175	C₈	—	94	4	no	—
176	C₈	—	188	4	no	—
177	C₈	Dic₄₇⋊C₈	1	8	yes	complemented
178	C₂²	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
179	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
180	C₂²	#97×D₄₇	1	2	yes	—
181	C₂²	C₄×#10625324	1	1	yes	characteristic
182	C₂²	#15689080	1	2	yes	—
183	C₄	—	47	4	no	—
184	C₄	—	94	2	no	—
185	C₄	—	94	4	no	—
186	C₄	C₂×Dic₄₇⋊C₈	1	2	yes	—
187	C₄	#15689174	1	2	yes	—
188	C₄	#15689199	1	4	yes	—
189	C₄	#15689251	1	4	yes	—
190	C₂	#322×D₄₇	1	1	yes	characteristic
191	C₂	#20241673	1	1	yes	characteristic
192	C₂	#20241674	1	2	yes	—
193	C₂	C₄×Dic₄₇⋊C₈	1	1	yes	characteristic
194	C₂	#20243256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₄₇×#322	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8260
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
28. C₈×Dic₄₇	C₈	semidirect
77. C₈×C₄₇	C₄⋊C₈	semidirect
92. Dic₄₇	#322	semidirect
114. C₄₇	#8260	semidirect
177. C₈	Dic₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₇⋊C₈	2
C₄×C₈×Dic₄₇	2
C₂×#15688844	2
C₄₇⋊#1273	2
C₂×C₄₇×#322	2
#8260	47