Subgroups — #26110032

Order 12032.

402 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243539	C₂	1	2	yes	maximal
2	#20243133	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688875	C₂	1	2	yes	maximal
5	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₄₇⋊C₈	—	2	4	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688877	C₄	1	2	yes	—
11	#15688877	C₄	1	2	yes	—
12	#15688876	—	2	2	no	—
13	#15688875	—	2	4	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#335	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	Dic₄₇⋊C₈	—	4	8	no	—
25	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	C₄×C₈×C₄₇	Q₈	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	4	4	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₄₇	—	2	2	no	—
50	C₄×Dic₄₇	—	2	2	no	—
51	C₄×Dic₄₇	—	4	2	no	—
52	C₄×Dic₄₇	D₈	1	4	yes	—
53	C₄×Dic₄₇	Q₁₆	1	4	yes	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	4	no	—
58	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	—	2	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	#127	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	#131	1	2	yes	—
68	C₂×C₄×C₄₇	#131	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	—	2	4	no	—
73	C₂×Dic₄₇	#107	1	8	yes	—
74	C₈×C₄₇	—	2	2	no	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	—	8	2	no	—
77	C₄₇⋊C₈	—	2	4	no	—
78	C₄₇⋊C₈	—	8	2	no	—
79	#8810	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	#384	1	1	yes	characteristic
81	C₂²×C₄₇	#424	1	2	yes	—
82	C₂²×C₄₇	C₂×#107	1	2	yes	—
83	C₂²×C₄₇	#430	1	2	yes	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	#424	1	2	yes	—
87	C₄×C₄₇	#424	1	1	yes	characteristic
88	C₄×C₄₇	#424	1	1	yes	characteristic
89	C₄×C₄₇	#493	1	1	yes	characteristic
90	C₄×C₄₇	#494	1	1	yes	characteristic
91	C₄×C₄₇	#500	1	2	yes	—
92	Dic₄₇	—	2	8	no	—
93	#1842	—	47	2	no	—
94	#1744	—	47	1	no	—
95	C₂×#335	—	47	2	no	—
96	C₂×#335	—	47	1	no	—
97	C₂×#335	D₄₇	1	1	yes	characteristic
98	C₂×C₄₇	#1571	1	4	yes	—
99	C₂×C₄₇	#1843	1	1	yes	characteristic
100	C₂×C₄₇	#1845	1	1	yes	characteristic
101	C₂×C₄₇	#1849	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	47	2	no	—
103	C₂×C₄⋊C₈	—	94	4	no	—
104	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	C₄³	—	47	1	no	—
108	#336	—	47	2	no	—
109	#336	—	47	2	no	—
110	#335	—	2	2	no	—
111	#335	—	94	4	no	—
112	#335	—	94	2	no	—
113	C₄₇	#8810	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₂×C₄²	—	47	2	no	—
120	C₂×C₄²	—	47	2	no	—
121	C₂×C₄²	—	47	2	no	—
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄⋊C₈	—	94	2	no	—
127	C₄⋊C₈	—	188	8	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	—	47	2	no	—
130	C₄×C₈	Dic₉₄	1	2	yes	—
131	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
132	C₂²×C₄	—	47	4	no	—
133	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	47	4	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	4	no	—
145	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
146	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
147	C₄²	—	47	4	no	—
148	C₄²	—	47	4	no	—
149	C₄²	—	94	2	no	—
150	C₄²	—	94	2	no	—
151	C₄²	—	188	2	no	—
152	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
153	C₄²	D₄×D₄₇	1	1	yes	characteristic
154	C₄²	Q₈×D₄₇	1	1	yes	characteristic
155	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
156	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	47	8	no	—
159	C₂×C₄	—	94	4	no	—
160	C₂×C₄	—	94	4	no	—
161	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#13094020	1	1	yes	characteristic
164	C₂×C₄	#13094022	1	1	yes	characteristic
165	C₂×C₄	#13094023	1	1	yes	characteristic
166	C₂×C₄	#13094023	1	1	yes	characteristic
167	C₂×C₄	#13094030	1	2	yes	—
168	C₂×C₄	#13094032	1	2	yes	—
169	C₂×C₄	D₈×D₄₇	1	2	yes	—
170	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	94	4	no	—
175	C₈	—	376	2	no	—
176	C₂²	#15688992	1	1	yes	characteristic
177	C₂²	#15689148	1	2	yes	—
178	C₂²	#107×D₄₇	1	2	yes	—
179	C₂²	#15689170	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	94	8	no	—
183	C₄	#15689148	1	2	yes	—
184	C₄	#15689149	1	1	yes	characteristic
185	C₄	#15689151	1	1	yes	characteristic
186	C₄	#15689498	1	1	yes	characteristic
187	C₄	#15689501	1	1	yes	characteristic
188	C₄	#15689527	1	2	yes	—
189	C₂	#20242161	1	4	yes	—
190	C₂	#20243541	1	1	yes	characteristic
191	C₂	#20243546	1	1	yes	characteristic
192	C₂	#20243561	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#8810
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
113. C₄₇	#8810	semidirect

Maximal subgroups

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

subgroup	index
#20243539	2
#20243133	2
C₂×#15688876	2
C₂×#15688875	2
C₂×C₄₇×#335	2
#8810	47