Subgroups — #26134308

Order 12032.

326 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×#15688949	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688949	C₂	1	1	yes	maximal, characteristic
3	#20244121	C₂	1	2	yes	maximal
4	#20241480	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	#15689021	—	2	2	no	—
9	#15688949	—	2	2	no	—
10	#15688949	C₄	1	4	yes	complemented
11	#15688886	—	2	2	no	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₇	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	2	2	no	—
18	C₂×Dic₄₇⋊C₄	—	4	2	no	—
19	#13093958	—	2	2	no	—
20	#13093958	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	C₂×C₄	1	4	yes	—
23	C₄₇⋊#95	—	4	2	no	—
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	—	2	1	no	—
28	C₂×C₁₆×C₄₇	—	2	1	no	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
31	C₂×C₁₆×C₄₇	Q₈	1	2	yes	—
32	C₂×C₁₆×C₄₇	Q₈	1	2	yes	—
33	C₄×C₈×C₄₇	—	2	1	no	—
34	C₄×C₈×C₄₇	—	2	1	no	—
35	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	2	no	—
39	C₂²×Dic₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	2	1	no	—
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
48	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
49	C₄²×C₄₇	—	2	1	no	—
50	C₄²×C₄₇	—	2	1	no	—
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	C₄⋊Dic₄₇	—	4	2	no	—
55	Dic₄₇⋊C₄	—	8	4	no	—
56	C₁₆×C₄₇	—	2	2	no	—
57	C₁₆×C₄₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
65	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
71	C₂×Dic₄₇	—	4	4	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₈×C₄₇	—	2	2	no	—
78	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₄₇	#125	1	1	yes	characteristic
82	#13165	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
84	C₂²×C₄₇	#466	1	1	yes	characteristic
85	C₂²×C₄₇	#500	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
88	C₂²×C₄₇	#508	1	1	yes	characteristic
89	C₂²×C₄₇	#508	1	1	yes	characteristic
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	#389	1	1	yes	characteristic
93	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
94	C₄×C₄₇	#467	1	1	yes	characteristic
95	C₄×C₄₇	#499	1	1	yes	characteristic
96	Dic₄₇	—	8	4	no	—
97	Dic₄₇	—	16	2	no	—
98	C₂×#366	—	47	1	no	—
99	C₂×#366	—	47	1	no	—
100	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
101	#1942	—	47	2	no	—
102	#1380	—	47	2	no	—
103	C₂×C₄₇	#1942	1	1	yes	characteristic
104	C₂×C₄₇	C₄×D₁₆	1	1	yes	characteristic
105	C₂×C₄₇	C₄×Q₃₂	1	1	yes	characteristic
106	C₂×C₄₇	#2215	1	1	yes	characteristic
107	C₂×C₄₇	#2216	1	1	yes	characteristic
108	C₂×C₄₇	#2252	1	1	yes	characteristic
109	C₂×C₄₇	#2253	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
111	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
112	C₂×#107	—	47	2	no	—
113	C₂×#107	—	47	2	no	—
114	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
115	#389	—	94	2	no	—
116	#366	—	47	4	no	—
117	#366	—	94	2	no	—
118	C₄×C₁₆	—	2	2	no	—
119	#340	—	94	2	no	—
120	C₄₇	#13165	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	94	2	no	—
124	C₂×C₄⋊C₄	—	94	2	no	—
125	C₂×C₄⋊C₄	—	188	2	no	—
126	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	Dic₉₄	1	2	yes	—
130	C₂×C₁₆	Dic₉₄	1	2	yes	—
131	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
133	#107	—	47	4	no	—
134	#107	—	94	2	no	—
135	#107	—	94	2	no	—
136	#107	—	94	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	#95	—	188	2	no	—
140	C₂²×C₄	—	188	2	no	—
141	C₂²×C₄	—	188	2	no	—
142	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
143	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Dic₁₈₈	1	2	yes	—
148	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₈	#10625350	1	1	yes	characteristic
152	C₂×C₈	#10625350	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
154	C₄⋊C₄	—	94	4	no	—
155	C₄⋊C₄	—	376	4	no	—
156	C₄⋊C₄	—	188	2	no	—
157	C₄⋊C₄	—	188	2	no	—
158	C₄⋊C₄	—	188	2	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	Dic₄₇⋊C₄	1	4	yes	complemented
163	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	376	2	no	—
169	C₂×C₄	—	376	2	no	—
170	C₂×C₄	—	376	2	no	—
171	C₂×C₄	—	376	2	no	—
172	C₂×C₄	—	376	2	no	—
173	C₂×C₄	—	188	4	no	—
174	C₂×C₄	D₇₅₂	1	2	yes	—
175	C₂×C₄	Dic₃₇₆	1	2	yes	—
176	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
177	C₂×C₄	#13093998	1	1	yes	characteristic
178	C₂×C₄	#13094024	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
180	C₂×C₄	#13094046	1	1	yes	characteristic
181	C₂×C₄	#13094046	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
184	C₈	C₄×Dic₉₄	1	1	yes	characteristic
185	C₈	#13094003	1	1	yes	characteristic
186	C₈	#13094016	1	1	yes	characteristic
187	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
188	C₂²	#15689359	1	1	yes	characteristic
189	C₂²	#15689528	1	1	yes	characteristic
190	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
191	C₂²	C₂×Dic₃₇₆	1	1	yes	characteristic
192	C₂²	#15689555	1	1	yes	characteristic
193	C₂²	#15689555	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	376	4	no	—
197	C₄	—	752	2	no	—
198	C₄	#15689021	1	1	yes	characteristic
199	C₄	C₄×Dic₁₈₈	1	1	yes	characteristic
200	C₄	#15689364	1	1	yes	characteristic
201	C₄	#15689525	1	1	yes	characteristic
202	C₂	#20244121	1	1	yes	characteristic
203	C₂	C₄×D₇₅₂	1	1	yes	characteristic
204	C₂	C₄×Dic₃₇₆	1	1	yes	characteristic
205	C₂	#20245474	1	1	yes	characteristic
206	C₂	#20245479	1	1	yes	characteristic
207	C₂	#20245663	1	1	yes	characteristic
208	C₂	#20245667	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#13165
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #15688949	C₄	semidirect
57. C₁₆×C₄₇	C₄⋊C₄	semidirect
120. C₄₇	#13165	semidirect
162. C₁₆	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688949	2
#20244121	2
#20241480	2
C₂×C₄×C₁₆×C₄₇	2
#13165	47