Subgroups — #26116290

Order 12032.

494 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	#20243505	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal
4	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688878	—	2	2	no	—
9	C₂×#125×C₄₇	C₄	1	1	yes	characteristic
10	C₄₇×#389	C₄	1	2	yes	—
11	C₄₇×#387	—	2	2	no	—
12	C₄₇×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄×C₄₇⋊C₈	—	2	8	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
22	#125×C₄₇	—	2	2	no	—
23	#125×C₄₇	C₈	1	4	yes	complemented
24	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	—	2	2	no	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#95×C₄₇	—	2	2	no	—
35	#95×C₄₇	—	2	2	no	—
36	#95×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₄₇	—	2	8	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₄⋊C₄×C₄₇	—	4	4	no	—
48	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
49	C₄²×C₄₇	—	2	4	no	—
50	C₄²×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	—	4	4	no	—
53	C₄²×C₄₇	C₂×C₈	1	2	yes	—
54	C₄²×C₄₇	M₁₆	1	2	yes	—
55	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
56	C₄²×C₄₇	#41	1	2	yes	—
57	C₂×C₄₇⋊C₈	—	2	8	no	—
58	C₂×C₄₇⋊C₈	—	2	4	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	4	2	no	—
69	C₂×C₄×C₄₇	—	4	2	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₄₇	#104	1	4	yes	—
75	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#131	1	2	yes	—
82	C₂×C₄×C₄₇	#135	1	4	yes	—
83	C₄₇⋊C₈	—	2	8	no	—
84	C₄₇⋊C₈	—	8	2	no	—
85	#9871	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	#391	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
88	C₂²×C₄₇	#408	1	1	yes	characteristic
89	C₂²×C₄₇	#416	1	2	yes	—
90	C₂²×C₄₇	C₂×#104	1	2	yes	—
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	4	4	no	—
95	C₄×C₄₇	—	8	2	no	—
96	C₄×C₄₇	C₈×Q₈	1	2	yes	—
97	C₄×C₄₇	#446	1	2	yes	—
98	C₄×C₄₇	#488	1	2	yes	—
99	C₄×C₄₇	#499	1	2	yes	—
100	C₄×#125	D₄₇	1	1	yes	characteristic
101	#1831	—	47	1	no	—
102	C₄²×C₈	—	47	1	no	—
103	#1279	—	47	4	no	—
104	C₂×C₄₇	#1476	1	2	yes	—
105	C₂×C₄₇	#1494	1	2	yes	—
106	C₂×C₄₇	#1984	1	1	yes	characteristic
107	C₂×C₄₇	#1991	1	1	yes	characteristic
108	C₂×C₄₇	#1994	1	1	yes	characteristic
109	C₂×#125	Dic₄₇	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	47	2	no	—
111	C₂×C₄×C₈	—	47	2	no	—
112	C₂×C₄×C₈	—	94	2	no	—
113	#389	Dic₄₇	1	2	yes	—
114	#387	—	2	2	no	—
115	#382	Dic₄₇	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	2	2	no	—
117	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	94	2	no	—
120	C₄₇	#9871	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	94	2	no	—
122	C₂²×C₈	—	94	2	no	—
123	#125	—	2	2	no	—
124	#125	C₄₇⋊C₈	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	C₄⋊C₈	—	94	4	no	—
136	C₄×C₈	—	47	4	no	—
137	C₄×C₈	—	94	8	no	—
138	C₄×C₈	—	94	2	no	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	94	8	no	—
150	C₂×C₈	—	94	4	no	—
151	C₂×C₈	—	188	2	no	—
152	C₂×C₈	—	188	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
163	C₄²	#10625329	1	2	yes	—
164	C₄²	Q₈×D₄₇	1	2	yes	—
165	C₄²	#10625354	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
179	C₂×C₄	#13093951	1	4	yes	—
180	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
181	C₂×C₄	#13094000	1	2	yes	—
182	C₂×C₄	#13094013	1	1	yes	characteristic
183	C₂×C₄	#13094022	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
186	C₂×C₄	#13094031	1	2	yes	—
187	C₂×C₄	#13094044	1	4	yes	—
188	C₈	—	94	8	no	—
189	C₈	—	376	2	no	—
190	C₂²	#15689030	1	1	yes	characteristic
191	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
192	C₂²	#15689090	1	1	yes	characteristic
193	C₂²	#15689119	1	2	yes	—
194	C₂²	C₂×#13093951	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₄	Q₈×C₄₇⋊C₈	1	2	yes	—
201	C₄	#15689254	1	2	yes	—
202	C₄	#15689480	1	2	yes	—
203	C₄	#15689522	1	2	yes	—
204	C₂	#20241770	1	2	yes	—
205	C₂	#20241848	1	2	yes	—
206	C₂	#20244325	1	1	yes	characteristic
207	C₂	#20244354	1	1	yes	characteristic
208	C₂	#20244366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#125×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#125	D₄₇
hypocenter	C₄₇	#9871
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
23. #125×C₄₇	C₈	semidirect
120. C₄₇	#9871	semidirect
124. #125	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
C₄²×C₄₇⋊C₈	2
#20241229	2
C₄×#125×C₄₇	2
#9871	47