Subgroups — #26116108

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244350	C₂	1	2	yes	maximal
2	#20244350	C₂	1	2	yes	maximal
3	#20243516	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	#15688993	C₂²	1	2	yes	—
9	#15688886	—	2	2	no	—
10	#15688885	—	2	4	no	—
11	#15688885	—	2	4	no	—
12	#15688845	—	2	2	no	—
13	C₂²×Q₁₆×C₄₇	—	2	2	no	—
14	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
15	C₄₇×#494	—	2	2	no	—
16	C₄₇×#494	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	1	no	—
24	C₄×C₄₇⋊C₈	—	2	1	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂×Q₁₆×C₄₇	—	2	8	no	—
29	C₂×Q₁₆×C₄₇	—	4	8	no	—
30	C₂²×C₈×C₄₇	D₄	1	2	yes	—
31	C₄⋊Q₈×C₄₇	—	2	2	no	—
32	C₄⋊Q₈×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	2	no	—
34	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂×Q₈×C₄₇	—	2	4	no	—
41	C₂×Q₈×C₄₇	—	4	2	no	—
42	C₂×Q₈×C₄₇	—	4	8	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₄₇	—	4	2	no	—
48	Q₁₆×C₄₇	—	4	16	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	#41	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	1	yes	characteristic
59	C₄⋊Dic₄₇	—	4	4	no	—
60	C₄⋊Dic₄₇	—	4	4	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	4	no	—
66	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	complemented
67	C₂³×C₄₇	#127	1	1	yes	characteristic
68	Q₈×C₄₇	—	4	8	no	—
69	Q₈×C₄₇	—	8	4	no	—
70	C₂×C₄×C₄₇	—	4	4	no	—
71	C₂×C₄×C₄₇	—	8	4	no	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₄₇	#135	1	2	yes	—
80	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₄₇	#137	1	4	yes	—
82	C₂×Dic₄₇	—	8	2	no	—
83	C₂×Dic₄₇	—	8	2	no	—
84	C₂×Dic₄₇	—	8	2	no	—
85	C₈×C₄₇	—	2	4	no	—
86	C₄₇⋊C₈	—	2	4	no	—
87	C₄₇⋊C₈	—	4	2	no	—
88	#9835	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	#390	1	1	yes	characteristic
90	C₂²×C₄₇	#494	1	1	yes	characteristic
91	C₂²×C₄₇	#495	1	1	yes	characteristic
92	C₂²×C₄₇	#496	1	2	yes	—
93	C₂²×C₄₇	#497	1	2	yes	—
94	C₄×C₄₇	—	8	4	no	—
95	C₄×C₄₇	#418	1	2	yes	—
96	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
97	C₄×C₄₇	#441	1	2	yes	—
98	C₄×C₄₇	#467	1	2	yes	—
99	C₄×C₄₇	#470	1	2	yes	—
100	C₄×C₄₇	#490	1	2	yes	—
101	Dic₄₇	—	16	2	no	—
102	C₂×#494	D₄₇	1	1	yes	characteristic
103	#1990	—	47	2	no	—
104	#1990	—	47	2	no	—
105	#1834	—	47	1	no	—
106	C₂×#322	—	47	1	no	—
107	C₂×C₄₇	#1720	1	2	yes	—
108	C₂×C₄₇	#1723	1	2	yes	—
109	C₂×C₄₇	#1961	1	1	yes	characteristic
110	C₂×C₄₇	#1969	1	1	yes	characteristic
111	C₂×C₄₇	#1971	1	1	yes	characteristic
112	C₂²×Q₁₆	—	2	2	no	—
113	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
114	#494	—	2	2	no	—
115	#494	Dic₄₇	1	4	yes	complemented
116	C₂×C₄×C₈	—	47	1	no	—
117	C₂×C₄×C₈	—	47	1	no	—
118	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
119	#384	—	47	2	no	—
120	#340	—	94	4	no	—
121	#340	—	94	4	no	—
122	#340	—	94	2	no	—
123	#322	—	94	2	no	—
124	C₄₇	#9835	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	2	2	no	—
126	C₂×Q₁₆	—	2	8	no	—
127	C₂×Q₁₆	—	4	8	no	—
128	C₂²×C₈	—	47	2	no	—
129	C₂²×C₈	—	47	2	no	—
130	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄⋊C₄	—	94	2	no	—
139	C₂×C₄⋊C₄	—	94	2	no	—
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	C₄×C₈	—	94	1	no	—
142	C₄×C₈	—	94	1	no	—
143	C₄×C₈	—	94	1	no	—
144	C₄×C₈	—	94	1	no	—
145	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
146	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	188	2	no	—
152	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	Q₁₆	—	4	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	47	4	no	—
158	C₂×C₈	—	94	2	no	—
159	C₂×C₈	—	94	2	no	—
160	C₂×C₈	—	94	2	no	—
161	C₂×C₈	—	94	4	no	—
162	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	188	4	no	—
168	C₄⋊C₄	—	188	4	no	—
169	C₄⋊C₄	—	188	4	no	—
170	C₄²	#10625352	1	1	yes	characteristic
171	C₄²	#10625352	1	1	yes	characteristic
172	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂³	#13094020	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	376	2	no	—
180	C₂×C₄	—	376	2	no	—
181	C₂×C₄	—	376	2	no	—
182	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄	#13093995	1	1	yes	characteristic
186	C₂×C₄	#13093995	1	1	yes	characteristic
187	C₂×C₄	#13094011	1	1	yes	characteristic
188	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
189	C₂×C₄	#13094047	1	2	yes	—
190	C₂×C₄	#13094050	1	4	yes	—
191	C₂×C₄	#13094056	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	94	4	no	—
194	C₈	—	188	2	no	—
195	C₂²	#15689026	1	1	yes	characteristic
196	C₂²	#15689501	1	1	yes	characteristic
197	C₂²	#15689508	1	1	yes	characteristic
198	C₂²	#15689510	1	2	yes	—
199	C₂²	#15689516	1	2	yes	—
200	C₄	—	8	4	no	—
201	C₄	—	752	2	no	—
202	C₄	#15689128	1	2	yes	—
203	C₄	Q₁₆×Dic₄₇	1	2	yes	—
204	C₄	#15689230	1	2	yes	—
205	C₄	#15689361	1	2	yes	—
206	C₄	#15689378	1	2	yes	—
207	C₄	#15689487	1	2	yes	—
208	C₂	#20243051	1	2	yes	—
209	C₂	#20243063	1	2	yes	—
210	C₂	#20244213	1	1	yes	characteristic
211	C₂	#20244250	1	1	yes	characteristic
212	C₂	#20244260	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#9835
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
16. C₄₇×#494	C₄	semidirect
66. C₂×C₄₇⋊C₈	Q₁₆	semidirect
115. #494	Dic₄₇	semidirect
124. C₄₇	#9835	semidirect

Maximal subgroups

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

subgroup	index
#20244350	2
#20243516	2
C₂×#15688845	2
C₂×C₄₇×#494	2
#9835	47