Subgroups — #26115309

Order 12032.

570 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	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688872	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	#15688872	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	4	yes	—
12	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#384	—	2	1	no	—
14	C₄₇×#384	—	2	1	no	—
15	C₄₇×#384	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
25	C₄⋊Q₈×C₄₇	—	2	4	no	—
26	C₄⋊Q₈×C₄₇	—	2	4	no	—
27	C₄⋊Q₈×C₄₇	—	4	8	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₄₇	—	2	8	no	—
38	C₂×Q₈×C₄₇	—	4	8	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₄₇	—	4	8	no	—
46	C₄⋊C₄×C₄₇	—	4	8	no	—
47	C₄⋊C₄×C₄₇	—	4	4	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄²×C₄₇	—	2	4	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	SD₁₆	1	2	yes	—
53	C₄²×C₄₇	SD₁₆	1	4	yes	—
54	C₄²×C₄₇	Q₁₆	1	4	yes	—
55	C₄²×C₄₇	Q₁₆	1	2	yes	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₂×C₄₇⋊C₈	—	2	4	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	4	4	no	—
63	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	Q₈×C₄₇	—	8	8	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	—	4	8	no	—
70	C₂×C₄×C₄₇	#100	1	4	yes	—
71	C₂×C₄×C₄₇	#103	1	4	yes	—
72	C₂×C₄×C₄₇	#103	1	4	yes	—
73	C₂×C₄×C₄₇	#103	1	4	yes	—
74	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₄₇	#121	1	2	yes	—
76	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₄₇	#137	1	2	yes	—
83	C₄₇⋊C₈	—	4	4	no	—
84	C₄₇⋊C₈	—	8	2	no	—
85	#9691	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	#390	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#100	1	2	yes	—
88	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×#103	1	2	yes	—
90	C₂²×C₄₇	#419	1	1	yes	characteristic
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	8	4	no	—
94	C₄×C₄₇	#333	1	8	yes	complemented
95	C₄×C₄₇	#461	1	2	yes	—
96	C₄×C₄₇	#462	1	2	yes	—
97	C₄×C₄₇	#487	1	2	yes	—
98	C₄×C₄₇	#492	1	1	yes	characteristic
99	C₄×C₄₇	#494	1	1	yes	characteristic
100	#2868	D₄₇	1	1	yes	characteristic
101	#1842	—	47	1	no	—
102	C₄×C₄⋊C₈	—	47	1	no	—
103	C₂×#333	—	47	4	no	—
104	C₂×C₄₇	C₂×#333	1	2	yes	—
105	C₂×C₄₇	#1548	1	2	yes	—
106	C₂×C₄₇	#1956	1	1	yes	characteristic
107	C₂×C₄₇	#1965	1	1	yes	characteristic
108	C₂×C₄₇	#1981	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	Dic₄₇	1	4	yes	—
110	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
111	C₂×C₄⋊C₈	—	47	2	no	—
112	C₂×C₄⋊C₈	—	47	2	no	—
113	C₂×C₄⋊C₈	—	94	2	no	—
114	C₂×C₄×C₈	—	94	1	no	—
115	#384	—	2	1	no	—
116	#384	—	2	1	no	—
117	#384	—	2	2	no	—
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	94	1	no	—
120	#333	—	94	8	no	—
121	C₄₇	#9691	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
123	C₂²×C₈	—	94	2	no	—
124	C₂²×C₈	—	94	2	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	8	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊C₈	—	47	4	no	—
138	C₄⋊C₈	—	94	4	no	—
139	C₄⋊C₈	—	94	2	no	—
140	C₄⋊C₈	—	188	4	no	—
141	C₄×C₈	—	188	2	no	—
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	4	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
146	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂×C₈	—	94	4	no	—
151	C₂×C₈	—	188	2	no	—
152	C₂×C₈	—	188	4	no	—
153	C₂×C₈	—	188	2	no	—
154	C₂×C₈	—	188	2	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	#10625336	1	2	yes	—
163	C₄²	#10625336	1	4	yes	—
164	C₄²	#10625337	1	4	yes	—
165	C₄²	#10625337	1	2	yes	—
166	C₄²	D₄×D₄₇	1	2	yes	—
167	C₄²	#10625352	1	2	yes	—
168	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	#13093941	1	4	yes	—
176	C₂×C₄	#13093948	1	4	yes	—
177	C₂×C₄	#13093948	1	4	yes	—
178	C₂×C₄	#13093948	1	4	yes	—
179	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
180	C₂×C₄	#13093995	1	2	yes	—
181	C₂×C₄	#13094011	1	1	yes	characteristic
182	C₂×C₄	#13094020	1	1	yes	characteristic
183	C₂×C₄	C₂×#10625336	1	2	yes	—
184	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625337	1	2	yes	—
186	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
187	C₂×C₄	#13094055	1	2	yes	—
188	C₈	—	376	2	no	—
189	C₈	—	188	4	no	—
190	C₂²	#15689026	1	1	yes	characteristic
191	C₂²	C₂×#13093941	1	2	yes	—
192	C₂²	C₂×#13093948	1	1	yes	characteristic
193	C₂²	C₂×#13093948	1	2	yes	—
194	C₂²	#15689132	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	8	4	no	—
198	C₄	#15688872	1	8	yes	complemented
199	C₄	#15689325	1	2	yes	—
200	C₄	#15689330	1	2	yes	—
201	C₄	#15689473	1	2	yes	—
202	C₄	#15689497	1	1	yes	characteristic
203	C₄	#15689502	1	1	yes	characteristic
204	C₂	C₂×#15688872	1	2	yes	—
205	C₂	#20242069	1	2	yes	—
206	C₂	#20244191	1	1	yes	characteristic
207	C₂	#20244229	1	1	yes	characteristic
208	C₂	#20244312	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	#9691
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
94. C₄×C₄₇	#333	semidirect
121. C₄₇	#9691	semidirect
198. C₄	#15688872	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×#15688872	2
C₄₇×#2868	2
#9691	47