Subgroups — #26115322

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688869	C₂	1	4	yes	maximal
4	C₄₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688869	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#384	—	2	2	no	—
15	C₄₇×#384	C₄	1	4	yes	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
23	C₄×C₄₇⋊C₈	—	4	2	no	—
24	C₂²×Q₈×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	4	no	—
26	C₄⋊Q₈×C₄₇	—	2	4	no	—
27	C₄⋊Q₈×C₄₇	—	4	4	no	—
28	C₄⋊Q₈×C₄₇	—	4	2	no	—
29	C₄⋊Q₈×C₄₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₇	—	4	4	no	—
40	C₂×Q₈×C₄₇	—	4	2	no	—
41	C₂×Q₈×C₄₇	—	4	2	no	—
42	C₂×Q₈×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₄₇	—	2	8	no	—
50	C₄⋊C₄×C₄₇	—	4	8	no	—
51	C₄⋊C₄×C₄₇	—	4	8	no	—
52	C₄⋊C₄×C₄₇	—	4	2	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	D₈	1	2	yes	—
57	C₄²×C₄₇	SD₁₆	1	2	yes	—
58	C₄²×C₄₇	SD₁₆	1	4	yes	—
59	C₄²×C₄₇	Q₁₆	1	4	yes	—
60	C₄²×C₄₇	C₂×D₄	1	2	yes	—
61	C₄²×C₄₇	#41	1	2	yes	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	4	4	no	—
67	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₄₇	—	8	4	no	—
69	Q₈×C₄₇	—	8	4	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	4	4	no	—
74	C₂×C₄×C₄₇	—	4	4	no	—
75	C₂×C₄×C₄₇	—	4	4	no	—
76	C₂×C₄×C₄₇	#101	1	4	yes	—
77	C₂×C₄×C₄₇	#102	1	4	yes	—
78	C₂×C₄×C₄₇	#103	1	4	yes	—
79	C₂×C₄×C₄₇	#103	1	4	yes	—
80	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₇	#121	1	2	yes	—
82	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₄₇⋊C₈	—	4	4	no	—
90	C₄₇⋊C₈	—	8	2	no	—
91	#9694	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	#390	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#101	1	2	yes	—
94	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#103	1	2	yes	—
96	C₂²×C₄₇	#418	1	1	yes	characteristic
97	C₄×C₄₇	—	2	4	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	8	4	no	—
100	C₄×C₄₇	#332	1	8	yes	complemented
101	C₄×C₄₇	#459	1	2	yes	—
102	C₄×C₄₇	#460	1	2	yes	—
103	C₄×C₄₇	#487	1	2	yes	—
104	C₄×C₄₇	#492	1	1	yes	characteristic
105	C₄×C₄₇	#494	1	1	yes	characteristic
106	#2868	D₄₇	1	1	yes	characteristic
107	#1842	—	47	1	no	—
108	C₄×C₄⋊C₈	—	47	1	no	—
109	C₂×#332	—	47	4	no	—
110	C₂×C₄₇	C₂×#332	1	2	yes	—
111	C₂×C₄₇	#1549	1	2	yes	—
112	C₂×C₄₇	#1956	1	1	yes	characteristic
113	C₂×C₄₇	#1966	1	1	yes	characteristic
114	C₂×C₄₇	#1982	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	—	2	1	no	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
118	C₂×C₄⋊C₈	—	47	2	no	—
119	C₂×C₄⋊C₈	—	47	2	no	—
120	C₂×C₄⋊C₈	—	94	2	no	—
121	C₂×C₄×C₈	—	94	1	no	—
122	#384	—	2	2	no	—
123	#384	Dic₄₇	1	4	yes	—
124	C₄³	C₂×D₄₇	1	1	yes	characteristic
125	#336	—	94	1	no	—
126	#332	—	94	8	no	—
127	C₄₇	#9694	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×C₈	—	94	2	no	—
130	C₂²×C₈	—	94	2	no	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	—	4	4	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
140	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
141	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
144	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
145	C₄⋊C₈	—	47	4	no	—
146	C₄⋊C₈	—	94	4	no	—
147	C₄⋊C₈	—	94	2	no	—
148	C₄⋊C₈	—	188	4	no	—
149	C₄×C₈	—	188	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂×C₈	—	94	4	no	—
161	C₂×C₈	—	188	2	no	—
162	C₂×C₈	—	188	4	no	—
163	C₂×C₈	—	188	2	no	—
164	C₂×C₈	—	188	2	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	#10625334	1	2	yes	—
173	C₄²	#10625335	1	2	yes	—
174	C₄²	#10625336	1	4	yes	—
175	C₄²	#10625337	1	4	yes	—
176	C₄²	D₄×D₄₇	1	2	yes	—
177	C₄²	#10625352	1	2	yes	—
178	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	#13093943	1	4	yes	—
188	C₂×C₄	#13093945	1	4	yes	—
189	C₂×C₄	#13093948	1	4	yes	—
190	C₂×C₄	#13093948	1	4	yes	—
191	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
192	C₂×C₄	#13093995	1	2	yes	—
193	C₂×C₄	#13094011	1	1	yes	characteristic
194	C₂×C₄	#13094020	1	1	yes	characteristic
195	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
196	C₂×C₄	C₂×#10625336	1	2	yes	—
197	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625337	1	2	yes	—
199	C₂×C₄	#13094050	1	2	yes	—
200	C₈	—	376	2	no	—
201	C₈	—	188	4	no	—
202	C₂²	#15689026	1	1	yes	characteristic
203	C₂²	C₂×#13093943	1	2	yes	—
204	C₂²	C₂×#13093945	1	1	yes	characteristic
205	C₂²	C₂×#13093948	1	2	yes	—
206	C₂²	#15689128	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	8	4	no	—
210	C₄	#15688869	1	8	yes	complemented
211	C₄	#15689315	1	2	yes	—
212	C₄	#15689320	1	2	yes	—
213	C₄	#15689473	1	2	yes	—
214	C₄	#15689497	1	1	yes	characteristic
215	C₄	#15689502	1	1	yes	characteristic
216	C₂	C₂×#15688869	1	2	yes	—
217	C₂	#20242073	1	2	yes	—
218	C₂	#20244191	1	1	yes	characteristic
219	C₂	#20244233	1	1	yes	characteristic
220	C₂	#20244316	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2868	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2868	D₄₇
hypocenter	C₄₇	#9694
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
100. C₄×C₄₇	#332	semidirect
127. C₄₇	#9694	semidirect
210. C₄	#15688869	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688869	2
C₄₇×#2868	2
#9694	47