Subgroups — #26110051

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243544	C₂	1	2	yes	maximal
2	#20243516	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×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093955	—	2	4	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688993	C₂²	1	2	yes	—
10	#15688886	—	2	2	no	—
11	#15688885	—	2	4	no	—
12	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₄₇×#340	—	2	2	no	—
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	#13093955	—	4	8	no	—
22	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
25	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₄⋊C₄×C₄₇	—	4	4	no	—
41	C₄⋊C₄×C₄₇	—	4	4	no	—
42	C₄⋊C₄×C₄₇	—	4	4	no	—
43	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄⋊Dic₄₇	—	4	4	no	—
48	C₄⋊Dic₄₇	—	4	4	no	—
49	C₄⋊Dic₄₇	—	4	4	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
54	C₂×C₄₇⋊C₈	SD₁₆	1	8	yes	—
55	C₂³×C₄₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	8	2	no	—
57	C₂×C₄×C₄₇	—	8	2	no	—
58	C₂×C₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
66	C₂×C₄×C₄₇	#135	1	2	yes	—
67	C₂×C₄×C₄₇	#135	1	4	yes	—
68	C₂×C₄×C₄₇	#135	1	2	yes	—
69	C₂×Dic₄₇	—	8	2	no	—
70	C₂×Dic₄₇	—	8	2	no	—
71	C₂×Dic₄₇	—	8	2	no	—
72	C₈×C₄₇	—	2	4	no	—
73	C₄₇⋊C₈	—	2	4	no	—
74	C₄₇⋊C₈	#106	1	8	yes	—
75	#8815	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	#384	1	1	yes	characteristic
77	C₂²×C₄₇	#492	1	2	yes	—
78	C₂²×C₄₇	#495	1	2	yes	—
79	C₂²×C₄₇	#495	1	1	yes	characteristic
80	C₂²×C₄₇	#495	1	1	yes	characteristic
81	C₄×C₄₇	—	16	2	no	—
82	C₄×C₄₇	C₂×#106	1	2	yes	—
83	C₄×C₄₇	#427	1	2	yes	—
84	C₄×C₄₇	#427	1	2	yes	—
85	C₄×C₄₇	#498	1	2	yes	—
86	C₄×C₄₇	#499	1	2	yes	—
87	C₄×C₄₇	#499	1	2	yes	—
88	Dic₄₇	—	16	2	no	—
89	#1844	—	47	2	no	—
90	#1834	—	47	1	no	—
91	#1834	—	47	2	no	—
92	#1834	D₄₇	1	1	yes	characteristic
93	C₂×C₈²	—	47	1	no	—
94	C₂×C₄₇	#1708	1	2	yes	—
95	C₂×C₄₇	#1708	1	2	yes	—
96	C₂×C₄₇	#1844	1	1	yes	characteristic
97	C₂×C₄₇	#1846	1	1	yes	characteristic
98	C₂×C₄₇	#1846	1	1	yes	characteristic
99	C₂×#106	—	94	4	no	—
100	C₂×C₄×C₈	—	47	1	no	—
101	C₂×C₄×C₈	—	47	1	no	—
102	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
103	#384	—	47	2	no	—
104	#384	C₂×D₄₇	1	2	yes	—
105	#340	—	2	2	no	—
106	#340	—	94	4	no	—
107	#340	—	94	2	no	—
108	C₈²	—	47	4	no	—
109	C₄₇	#8815	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	47	2	no	—
111	C₂²×C₈	—	47	2	no	—
112	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	94	2	no	—
117	C₂×C₄⋊C₄	—	94	2	no	—
118	C₂×C₄⋊C₄	—	94	2	no	—
119	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
120	#106	—	188	8	no	—
121	C₄×C₈	—	47	2	no	—
122	C₄×C₈	—	47	2	no	—
123	C₄×C₈	—	47	2	no	—
124	C₄×C₈	—	47	2	no	—
125	C₄×C₈	Dic₉₄	1	2	yes	—
126	C₄×C₈	C₄×D₄₇	1	2	yes	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	188	2	no	—
129	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	47	4	no	—
134	C₂×C₈	—	47	8	no	—
135	C₂×C₈	—	47	4	no	—
136	C₂×C₈	—	94	2	no	—
137	C₂×C₈	—	94	2	no	—
138	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	188	4	no	—
143	C₄⋊C₄	—	188	4	no	—
144	C₄⋊C₄	—	188	4	no	—
145	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₄²	Q₈×D₄₇	1	1	yes	characteristic
148	C₄²	Q₈×D₄₇	1	1	yes	characteristic
149	C₂³	#13094020	1	1	yes	characteristic
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	376	2	no	—
154	C₂×C₄	—	376	2	no	—
155	C₂×C₄	—	376	2	no	—
156	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#13094022	1	1	yes	characteristic
160	C₂×C₄	#13094023	1	1	yes	characteristic
161	C₂×C₄	#13094023	1	1	yes	characteristic
162	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
163	C₂×C₄	#13094043	1	2	yes	—
164	C₂×C₄	#13094044	1	4	yes	—
165	C₂×C₄	#13094047	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	47	8	no	—
168	C₈	—	94	4	no	—
169	C₂²	#15688992	1	1	yes	characteristic
170	C₂²	#15689494	1	2	yes	—
171	C₂²	#15689505	1	2	yes	—
172	C₂²	#15689506	1	1	yes	characteristic
173	C₂²	#15689508	1	1	yes	characteristic
174	C₄	—	16	2	no	—
175	C₄	—	752	2	no	—
176	C₄	#106×D₄₇	1	2	yes	—
177	C₄	#15689159	1	2	yes	—
178	C₄	#15689160	1	2	yes	—
179	C₄	#15689520	1	2	yes	—
180	C₄	#15689522	1	2	yes	—
181	C₄	#15689524	1	2	yes	—
182	C₂	#20242988	1	2	yes	—
183	C₂	#20242991	1	2	yes	—
184	C₂	#20243543	1	1	yes	characteristic
185	C₂	#20243550	1	1	yes	characteristic
186	C₂	#20243551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1834	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1834	D₄₇
hypocenter	C₄₇	#8815
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₄₇⋊C₈	C₄	semidirect
109. C₄₇	#8815	semidirect

Maximal subgroups

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

subgroup	index
#20243544	2
#20243516	2
#20243515	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1834	2
#8815	47