Subgroups — #26110047

Order 12032.

454 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243547	C₂	1	2	yes	maximal
2	#20243545	C₂	1	1	yes	maximal, characteristic
3	#20243515	C₂	1	2	yes	maximal
4	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093957	—	2	4	no	—
7	C₂×#13093956	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688993	C₂²	1	2	yes	—
11	#15688885	—	2	4	no	—
12	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
13	C₂×#106×C₄₇	—	2	2	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#384	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	#13093957	—	4	8	no	—
22	#13093956	—	4	4	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
26	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#106×C₄₇	—	4	4	no	—
33	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₇	Q₈	1	2	yes	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₄₇	D₈	1	4	yes	—
42	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
43	C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₄⋊C₄×C₄₇	—	4	4	no	—
45	C₄⋊C₄×C₄₇	—	4	4	no	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₄⋊Dic₄₇	—	4	4	no	—
52	C₄⋊Dic₄₇	—	4	4	no	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
55	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
56	C₂×C₄₇⋊C₈	SD₁₆	1	8	yes	—
57	C₂³×C₄₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	—	8	2	no	—
60	C₂×C₄×C₄₇	—	8	2	no	—
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	#135	1	4	yes	—
71	C₂×Dic₄₇	—	8	2	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₈×C₄₇	#107	1	8	yes	complemented
75	C₄₇⋊C₈	—	2	4	no	—
76	C₄₇⋊C₈	#106	1	8	yes	complemented
77	#8814	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	#384	1	1	yes	characteristic
79	C₂²×C₄₇	#492	1	2	yes	—
80	C₂²×C₄₇	#493	1	1	yes	characteristic
81	C₂²×C₄₇	#494	1	1	yes	characteristic
82	C₂²×C₄₇	#495	1	2	yes	—
83	C₄×C₄₇	—	16	2	no	—
84	C₄×C₄₇	C₂×#106	1	2	yes	—
85	C₄×C₄₇	C₂×#107	1	2	yes	—
86	C₄×C₄₇	#427	1	2	yes	—
87	C₄×C₄₇	#498	1	2	yes	—
88	C₄×C₄₇	#499	1	2	yes	—
89	C₄×C₄₇	#500	1	2	yes	—
90	Dic₄₇	—	16	2	no	—
91	#1845	—	47	2	no	—
92	#1844	—	47	1	no	—
93	#1844	D₄₇	1	1	yes	characteristic
94	#1834	—	47	2	no	—
95	C₂×C₈²	—	47	1	no	—
96	C₂×C₄₇	#1706	1	2	yes	—
97	C₂×C₄₇	#1707	1	2	yes	—
98	C₂×C₄₇	#1844	1	1	yes	characteristic
99	C₂×C₄₇	#1845	1	1	yes	characteristic
100	C₂×C₄₇	#1846	1	1	yes	characteristic
101	C₂×#107	—	94	4	no	—
102	C₂×#106	—	2	2	no	—
103	C₂×#106	—	94	2	no	—
104	C₂×C₄×C₈	—	47	1	no	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	#384	—	47	2	no	—
108	#384	C₂×D₄₇	1	2	yes	—
109	#340	—	94	4	no	—
110	C₈²	—	47	4	no	—
111	C₄₇	#8814	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	47	2	no	—
113	C₂²×C₈	—	47	2	no	—
114	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	94	2	no	—
119	C₂×C₄⋊C₄	—	94	2	no	—
120	C₂×C₄⋊C₄	—	94	2	no	—
121	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	#107	—	188	8	no	—
123	#106	—	4	4	no	—
124	#106	—	188	4	no	—
125	C₄×C₈	—	47	2	no	—
126	C₄×C₈	—	47	2	no	—
127	C₄×C₈	—	47	2	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	Dic₉₄	1	2	yes	—
130	C₄×C₈	C₄×D₄₇	1	2	yes	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	188	2	no	—
133	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂×C₈	—	47	4	no	—
137	C₂×C₈	—	47	8	no	—
138	C₂×C₈	—	47	4	no	—
139	C₂×C₈	—	94	4	no	—
140	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
141	C₂×C₈	#10625334	1	4	yes	—
142	C₂×C₈	#10625337	1	4	yes	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	188	4	no	—
147	C₄⋊C₄	—	188	4	no	—
148	C₄⋊C₄	—	188	4	no	—
149	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₄²	Q₈×D₄₇	1	1	yes	characteristic
152	C₄²	Q₈×D₄₇	1	1	yes	characteristic
153	C₂³	#13094020	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	376	2	no	—
158	C₂×C₄	—	376	2	no	—
159	C₂×C₄	—	376	2	no	—
160	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#13094022	1	1	yes	characteristic
164	C₂×C₄	#13094023	1	1	yes	characteristic
165	C₂×C₄	#13094023	1	1	yes	characteristic
166	C₂×C₄	C₂×#10625334	1	2	yes	—
167	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
168	C₂×C₄	C₂×#10625337	1	2	yes	—
169	C₂×C₄	#13094045	1	4	yes	—
170	C₈	—	47	8	no	—
171	C₈	—	94	4	no	—
172	C₈	#13093957	1	8	yes	complemented
173	C₂²	#15688992	1	1	yes	characteristic
174	C₂²	#15689494	1	2	yes	—
175	C₂²	#15689499	1	1	yes	characteristic
176	C₂²	#15689502	1	1	yes	characteristic
177	C₂²	#15689504	1	2	yes	—
178	C₄	—	16	2	no	—
179	C₄	—	752	2	no	—
180	C₄	#106×D₄₇	1	2	yes	—
181	C₄	C₂×#13093957	1	2	yes	—
182	C₄	#15689158	1	2	yes	—
183	C₄	#15689520	1	2	yes	—
184	C₄	#15689523	1	2	yes	—
185	C₄	#15689526	1	2	yes	—
186	C₂	#20242978	1	2	yes	—
187	C₂	#20242983	1	2	yes	—
188	C₂	#20243543	1	1	yes	characteristic
189	C₂	#20243547	1	1	yes	characteristic
190	C₂	#20243549	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#8814
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
12. C₈×C₄₇⋊C₈	C₄	semidirect
74. C₈×C₄₇	#107	semidirect
76. C₄₇⋊C₈	#106	semidirect
111. C₄₇	#8814	semidirect
172. C₈	#13093957	semidirect

Maximal subgroups

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

subgroup	index
#20243547	2
#20243545	2
#20243515	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1844	2
#8814	47