Subgroups — #26134294

Order 12032.

290 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
2	#20245147	C₂	1	2	yes	maximal
3	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
4	#20241477	C₂	1	2	yes	maximal
5	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#15688944	—	2	2	no	—
10	C₄⋊(C₄₇⋊C₁₆)	—	2	4	no	—
11	#15688899	—	2	2	no	—
12	C₂×#108×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#427	C₄	1	1	yes	characteristic
14	C₄₇×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#339	C₄	1	2	yes	—
17	#13093964	—	2	2	no	—
18	#13093964	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	1	no	—
22	C₂×C₄₇⋊C₁₆	—	2	1	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₄₇⋊C₁₆	Q₈	1	2	yes	—
26	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	#117×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	#108×C₄₇	—	2	2	no	—
32	#107×C₄₇	C₈	1	2	yes	—
33	#106×C₄₇	C₈	1	2	yes	—
34	#97×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₄₇	—	2	2	no	—
41	M₁₆×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	1	no	—
57	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
58	C₄₇⋊C₁₆	—	2	2	no	—
59	C₄₇⋊C₁₆	—	2	2	no	—
60	C₄₇⋊C₁₆	—	4	2	no	—
61	C₂³×C₄₇	#131	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
71	C₈×C₄₇	—	4	2	no	—
72	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
74	C₈×C₄₇	#121	1	1	yes	characteristic
75	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₄₇	#125	1	1	yes	characteristic
77	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
78	#13162	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	#424	1	1	yes	characteristic
81	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₄₇	#446	1	1	yes	characteristic
83	C₄×C₄₇	—	4	2	no	—
84	C₄×C₄₇	—	4	4	no	—
85	C₄×C₄₇	#389	1	1	yes	characteristic
86	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₄₇	#436	1	1	yes	characteristic
88	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
89	C₂×#364	—	47	1	no	—
90	#2151	—	47	2	no	—
91	C₂×#346	—	47	1	no	—
92	#1772	D₄₇	1	1	yes	characteristic
93	#1379	—	47	2	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	—	2	1	no	—
96	C₂×C₄₇	#1939	1	1	yes	characteristic
97	C₂×C₄₇	#2171	1	2	yes	—
98	C₂×M₃₂	—	47	2	no	—
99	C₂²×C₁₆	—	47	1	no	—
100	C₂²×C₁₆	—	47	1	no	—
101	C₂×#108	Dic₄₇	1	1	yes	characteristic
102	#427	Dic₄₇	1	1	yes	characteristic
103	#405	C₂×D₄₇	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
105	#364	—	94	2	no	—
106	C₄⋊C₁₆	—	94	4	no	—
107	#346	—	94	2	no	—
108	#339	Dic₄₇	1	2	yes	—
109	C₄₇	#13162	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
111	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	#117	C₂×Dic₄₇	1	2	yes	—
114	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
115	M₃₂	—	94	2	no	—
116	M₃₂	—	94	2	no	—
117	C₂×C₁₆	—	47	2	no	—
118	C₂×C₁₆	—	47	2	no	—
119	C₂×C₁₆	—	94	1	no	—
120	C₂×C₁₆	—	94	1	no	—
121	C₂×C₁₆	—	94	1	no	—
122	C₂×C₁₆	—	94	1	no	—
123	C₂×C₁₆	—	94	2	no	—
124	#108	—	2	2	no	—
125	#107	C₄₇⋊C₈	1	2	yes	—
126	#106	C₄₇⋊C₈	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	Dic₉₄	1	2	yes	—
130	C₄×C₈	D₁₈₈	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
138	C₂×C₈	#10625329	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
140	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
141	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
142	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
143	C₂×C₈	#10625352	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
145	C₂×C₈	#10625354	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊Dic₄₇	1	2	yes	—
151	C₁₆	—	94	2	no	—
152	C₁₆	—	94	2	no	—
153	C₁₆	—	188	2	no	—
154	C₂³	#13094031	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
159	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
160	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
163	C₂×C₄	#13094031	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
166	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
167	C₈	#13093999	1	1	yes	characteristic
168	C₈	#13094004	1	1	yes	characteristic
169	C₈	#13094015	1	1	yes	characteristic
170	C₈	#13094023	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#15689150	1	1	yes	characteristic
173	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
174	C₂²	#15689254	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	#15689022	1	1	yes	characteristic
178	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
179	C₄	#15689201	1	1	yes	characteristic
180	C₄	Q₈×C₄₇⋊C₈	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#20244107	1	1	yes	characteristic
184	C₂	#20245230	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1772	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1772	D₄₇
hypocenter	C₄₇	#13162
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
109. C₄₇	#13162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688944	2
#20245147	2
C₂×#15688899	2
#20241477	2
C₄₇×#1772	2
#13162	47