Subgroups — #26116288

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	C₄×#13094016	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241227	C₂	1	4	yes	maximal
4	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	C₄	1	1	yes	characteristic
6	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689021	C₄	1	2	yes	—
9	#15689007	—	2	2	no	—
10	#15688989	C₄	1	1	yes	characteristic
11	C₄×Dic₄₇⋊C₄	—	2	2	no	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	#13094016	—	2	2	no	—
18	#13094016	C₈	1	4	yes	complemented
19	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₄₇⋊C₄	—	2	2	no	—
22	C₂×Dic₄₇⋊C₄	—	2	2	no	—
23	C₂×C₄×Dic₄₇	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	8	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
27	C₄₇⋊#95	—	2	2	no	—
28	C₄₇⋊#95	—	2	2	no	—
29	C₄₇⋊#95	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₇	—	2	2	no	—
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	4	4	no	—
46	C₄²×C₄₇	—	2	4	no	—
47	C₄²×C₄₇	—	2	2	no	—
48	C₄²×C₄₇	—	2	4	no	—
49	C₄²×C₄₇	C₂×C₈	1	2	yes	—
50	C₄²×C₄₇	M₁₆	1	2	yes	—
51	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₄⋊Dic₄₇	—	2	2	no	—
54	C₄⋊Dic₄₇	—	2	4	no	—
55	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
56	Dic₄₇⋊C₄	—	2	8	no	—
57	Dic₄₇⋊C₄	—	4	4	no	—
58	C₄×Dic₄₇	—	4	4	no	—
59	C₂×C₄₇⋊C₈	—	2	8	no	—
60	C₂×C₄₇⋊C₈	—	2	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	C₂²⋊C₈	1	4	yes	—
68	C₂×C₄×C₄₇	#104	1	4	yes	—
69	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#131	1	2	yes	—
76	C₂×C₄×C₄₇	#135	1	4	yes	—
77	C₂×Dic₄₇	—	2	4	no	—
78	C₂×Dic₄₇	—	4	2	no	—
79	C₂×Dic₄₇	—	4	2	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	C₈×C₄₇	—	8	2	no	—
84	C₄₇⋊C₈	—	2	8	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	C₈×Q₈	1	2	yes	—
95	C₄×C₄₇	#446	1	2	yes	—
96	C₄×C₄₇	#488	1	2	yes	—
97	C₄×C₄₇	#499	1	2	yes	—
98	Dic₄₇	—	4	4	no	—
99	Dic₄₇	—	8	2	no	—
100	C₄×#125	—	47	1	no	—
101	#1831	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
110	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
111	C₂×C₄×C₈	—	47	2	no	—
112	C₂×C₄×C₈	—	94	2	no	—
113	#389	—	47	2	no	—
114	#387	—	94	2	no	—
115	#382	—	47	1	no	—
116	C₄×C₄⋊C₄	—	47	2	no	—
117	C₄×C₄⋊C₄	—	94	2	no	—
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	C₄₇	#9871	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	94	2	no	—
123	#125	—	47	4	no	—
124	#125	—	94	2	no	—
125	C₂×C₄⋊C₄	—	47	2	no	—
126	C₂×C₄⋊C₄	—	47	2	no	—
127	C₂×C₄⋊C₄	—	94	2	no	—
128	C₂×C₄⋊C₄	—	94	2	no	—
129	C₂×C₄²	—	94	2	no	—
130	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄²	D₁₈₈	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄×C₈	—	47	4	no	—
137	C₄×C₈	—	94	8	no	—
138	C₄×C₈	—	94	2	no	—
139	#95	—	94	2	no	—
140	#95	—	94	2	no	—
141	#95	—	94	2	no	—
142	C₂²×C₄	—	94	2	no	—
143	C₂²×C₄	—	94	2	no	—
144	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
145	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	94	8	no	—
152	C₂×C₈	—	94	4	no	—
153	C₄⋊C₄	—	47	4	no	—
154	C₄⋊C₄	—	94	8	no	—
155	C₄⋊C₄	—	94	2	no	—
156	C₄⋊C₄	—	94	4	no	—
157	C₄⋊C₄	—	188	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	188	4	no	—
162	C₄²	C₈×D₄₇	1	2	yes	—
163	C₄²	#10625324	1	2	yes	—
164	C₄²	#10625352	1	2	yes	—
165	C₄²	Q₈×D₄₇	1	2	yes	—
166	C₂³	C₂×(C₂×D₄₇)⋊C₄	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	—	94	4	no	—
173	C₂×C₄	—	188	2	no	—
174	C₂×C₄	—	188	2	no	—
175	C₂×C₄	—	188	2	no	—
176	C₂×C₄	—	188	2	no	—
177	C₂×C₄	—	188	2	no	—
178	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
179	C₂×C₄	#13093950	1	4	yes	—
180	C₂×C₄	#13093989	1	2	yes	—
181	C₂×C₄	#13094004	1	2	yes	—
182	C₂×C₄	#13094011	1	1	yes	characteristic
183	C₂×C₄	#13094022	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
186	C₂×C₄	#13094032	1	2	yes	—
187	C₂×C₄	#13094044	1	4	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	94	8	no	—
190	C₂²	#15689028	1	1	yes	characteristic
191	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
192	C₂²	#15689091	1	1	yes	characteristic
193	C₂²	#15689117	1	2	yes	—
194	C₂²	C₂×#13093950	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	376	2	no	—
199	C₄	—	188	4	no	—
200	C₄	#15689247	1	2	yes	—
201	C₄	#15689250	1	2	yes	—
202	C₄	#15689477	1	2	yes	—
203	C₄	#15689522	1	2	yes	—
204	C₂	#20241768	1	2	yes	—
205	C₂	#20241846	1	2	yes	—
206	C₂	#20244323	1	1	yes	characteristic
207	C₂	#20244352	1	1	yes	characteristic
208	C₂	#20244364	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	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#9871
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
18. #13094016	C₈	semidirect
120. C₄₇	#9871	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094016	2
C₄²×C₄₇⋊C₈	2
#20241227	2
C₄₇×#1831	2
#9871	47