Subgroups — #26103191

Order 12032.

262 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×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1770	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	complemented
8	#15688899	—	2	1	no	—
9	#15688899	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
14	C₄₇×#336	—	2	1	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	4	1	no	—
21	C₂²×C₈×C₄₇	—	2	1	no	—
22	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₇	C₈	1	2	yes	—
25	C₂×C₄²×C₄₇	—	2	1	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	1	no	—
28	C₄⋊C₈×C₄₇	C₈	1	2	yes	—
29	#97×C₄₇	—	4	1	no	—
30	#97×C₄₇	—	4	1	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
35	#95×C₄₇	—	2	1	no	—
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₄₇	—	2	1	no	—
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	4	1	no	—
44	C₂×C₈×C₄₇	—	4	1	no	—
45	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
50	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄²×C₄₇	—	4	1	no	—
55	C₄²×C₄₇	—	4	1	no	—
56	C₄²×C₄₇	C₄²	1	1	yes	characteristic
57	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
60	C₄₇⋊C₁₆	—	2	4	no	—
61	C₄₇⋊C₁₆	—	4	2	no	—
62	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₄₇	—	2	1	no	—
79	C₈×C₄₇	—	2	1	no	—
80	C₈×C₄₇	—	4	2	no	—
81	C₈×C₄₇	#102	1	1	yes	characteristic
82	C₈×C₄₇	#103	1	1	yes	characteristic
83	C₈×C₄₇	#106	1	1	yes	characteristic
84	C₈×C₄₇	#107	1	1	yes	characteristic
85	#7170	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
88	C₂²×C₄₇	#336	1	1	yes	characteristic
89	C₂²×C₄₇	#349	1	2	yes	—
90	C₂²×C₄₇	#350	1	2	yes	—
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	1	no	—
93	C₄×C₄₇	—	2	1	no	—
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	8	2	no	—
96	C₄×C₄₇	D₄⋊C₈	1	1	yes	characteristic
97	C₄×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
98	C₄×C₄₇	#339	1	1	yes	characteristic
99	C₄×C₄₇	#340	1	1	yes	characteristic
100	C₂×C₄⋊C₁₆	—	47	1	no	—
101	C₂×#346	—	47	1	no	—
102	#1770	D₄₇	1	1	yes	characteristic
103	C₂×C₄₇	#1279	1	1	yes	characteristic
104	C₂×C₄₇	#1330	1	2	yes	—
105	C₂×C₄₇	#1338	1	2	yes	—
106	C₂×C₄₇	#1377	1	1	yes	characteristic
107	C₂×C₄₇	#1379	1	1	yes	characteristic
108	C₂²×C₁₆	—	94	1	no	—
109	C₂²×C₁₆	—	94	1	no	—
110	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
111	C₂×#97	—	2	1	no	—
112	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
113	C₄×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
114	C₄⋊C₁₆	—	47	2	no	—
115	C₄⋊C₁₆	—	94	1	no	—
116	#346	—	47	2	no	—
117	#346	—	94	1	no	—
118	#336	—	2	1	no	—
119	C₄₇	#7170	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₄₇⋊C₈	1	2	yes	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂×C₁₆	—	94	2	no	—
127	C₂×C₁₆	—	94	2	no	—
128	C₂×C₁₆	—	94	2	no	—
129	C₂×C₁₆	—	94	2	no	—
130	C₂×C₁₆	—	188	1	no	—
131	C₂×C₁₆	—	188	1	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	C₄₇⋊C₈	1	2	yes	—
134	#97	—	4	1	no	—
135	#97	—	4	1	no	—
136	C₄×C₈	Dic₉₄	1	1	yes	characteristic
137	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
138	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
139	C₄×C₈	D₁₈₈	1	1	yes	characteristic
140	#95	—	2	1	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
143	C₂²×C₄	#10625329	1	1	yes	characteristic
144	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
151	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
152	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
153	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₈	#10625334	1	1	yes	characteristic
155	C₂×C₈	#10625335	1	1	yes	characteristic
156	C₂×C₈	#10625336	1	1	yes	characteristic
157	C₂×C₈	#10625337	1	1	yes	characteristic
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
162	C₄²	#10625329	1	1	yes	characteristic
163	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
164	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
165	C₁₆	—	94	4	no	—
166	C₁₆	—	188	2	no	—
167	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
174	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
175	C₂×C₄	#13093935	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
177	C₂×C₄	#13093945	1	1	yes	characteristic
178	C₂×C₄	#13093948	1	1	yes	characteristic
179	C₂×C₄	#13093951	1	1	yes	characteristic
180	C₂×C₄	#13093951	1	1	yes	characteristic
181	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
182	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#13093944	1	1	yes	characteristic
187	C₈	#13093947	1	1	yes	characteristic
188	C₈	#13093955	1	1	yes	characteristic
189	C₈	#13093957	1	1	yes	characteristic
190	C₂²	D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
191	C₂²	Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
192	C₂²	#15688878	1	1	yes	characteristic
193	C₂²	#15688905	1	2	yes	—
194	C₂²	#15688907	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
201	C₄	Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
202	C₄	#15688882	1	1	yes	characteristic
203	C₄	#15688885	1	1	yes	characteristic
204	C₂	#20241229	1	1	yes	characteristic
205	C₂	#20241346	1	2	yes	—
206	C₂	#20241370	1	2	yes	—
207	C₂	#20241472	1	1	yes	characteristic
208	C₂	#20241477	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1770	C₂
socle, prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1770	D₄₇
hypocenter	C₄₇	#7170
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
7. C₄⋊(C₄₇⋊C₁₆)	C₄	semidirect
9. #15688899	C₄	semidirect
119. C₄₇	#7170	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688899	2
C₄₇×#1770	2
#7170	47