Subgroups — #26131669

Order 12032.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688953	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243269	C₂	1	2	yes	maximal
4	#20241483	C₂	1	2	yes	maximal
5	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	#15689079	—	2	2	no	—
9	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688953	C₄	1	4	yes	complemented
11	C₁₆×Dic₄₇	C₄	1	4	yes	complemented
12	#15688882	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#427	C₂²	1	2	yes	—
15	C₄₇×#367	C₄	1	2	yes	complemented
16	C₄₇×#366	C₄	1	2	yes	complemented
17	C₂×#10625329	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	1	no	—
22	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	#13093960	—	2	2	no	—
24	#13093960	C₂×C₄	1	4	yes	—
25	#13093935	—	4	2	no	—
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₈×Dic₄₇	Q₈	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	#117×C₄₇	—	2	2	no	—
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
35	#107×C₄₇	C₂×C₄	1	4	yes	—
36	#106×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	1	no	—
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂²⋊C₄×C₄₇	—	4	2	no	—
47	C₄²×C₄₇	—	4	2	no	—
48	C₄×Dic₄₇	—	2	1	no	—
49	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
50	#10625329	—	2	4	no	—
51	#10625329	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
55	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
56	C₂×C₄₇⋊C₈	Q₁₆	1	2	yes	—
57	C₁₆×C₄₇	C₄²	1	2	yes	—
58	C₁₆×C₄₇	C₄²	1	2	yes	—
59	C₄₇⋊C₁₆	—	2	2	no	—
60	C₂³×C₄₇	#135	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	4	4	no	—
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₄₇	#117	1	1	yes	characteristic
72	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
74	C₄₇⋊C₈	—	4	4	no	—
75	C₄₇⋊C₈	#107	1	2	yes	—
76	C₄₇⋊C₈	#107	1	2	yes	—
77	#12670	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	#427	1	1	yes	characteristic
80	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
82	C₄×C₄₇	—	8	4	no	—
83	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
85	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
86	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
87	Dic₄₇	—	2	4	no	—
88	C₂×#368	—	47	1	no	—
89	#2158	D₄₇	1	1	yes	characteristic
90	C₂×C₄×C₁₆	—	47	1	no	—
91	#1772	—	47	2	no	—
92	#1382	—	47	2	no	—
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
96	C₂×C₄₇	#2178	1	2	yes	—
97	C₂²×C₁₆	—	47	1	no	—
98	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
99	C₂×#108	—	47	2	no	—
100	#427	C₂×D₄₇	1	2	yes	—
101	#405	—	94	2	no	—
102	C₂×C₄×C₈	—	47	1	no	—
103	#368	—	47	4	no	—
104	#367	Dic₄₇	1	2	yes	complemented
105	#366	Dic₄₇	1	2	yes	complemented
106	C₄×C₁₆	—	47	4	no	—
107	#339	—	94	2	no	—
108	C₄₇	#12670	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	94	2	no	—
110	C₂²×C₈	—	47	1	no	—
111	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
112	#117	—	2	2	no	—
113	C₂×C₄²	—	47	1	no	—
114	C₂×C₁₆	—	47	2	no	—
115	C₂×C₁₆	—	94	1	no	—
116	C₂×C₁₆	—	94	1	no	—
117	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
118	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
121	#108	—	47	4	no	—
122	#108	—	94	2	no	—
123	#107	C₂×Dic₄₇	1	4	yes	—
124	#106	—	2	2	no	—
125	#97	—	188	2	no	—
126	C₄×C₈	—	47	2	no	—
127	C₄×C₈	—	47	2	no	—
128	C₄×C₈	—	188	2	no	—
129	C₂²×C₄	—	94	1	no	—
130	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
131	M₁₆	—	94	4	no	—
132	M₁₆	—	188	2	no	—
133	C₂×C₈	—	47	2	no	—
134	C₂×C₈	—	47	1	no	—
135	C₂×C₈	—	47	1	no	—
136	C₂×C₈	—	47	2	no	—
137	C₂×C₈	—	188	2	no	—
138	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₈	#10625352	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
142	C₂×C₈	#10625354	1	1	yes	characteristic
143	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	47	2	no	—
148	C₄²	—	94	1	no	—
149	C₁₆	—	94	2	no	—
150	C₁₆	C₄×Dic₄₇	1	2	yes	—
151	C₁₆	C₄×Dic₄₇	1	2	yes	—
152	C₂³	#13094043	1	1	yes	characteristic
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₄	#13093986	1	1	yes	characteristic
158	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
159	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
160	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
161	C₂×C₄	#13094047	1	1	yes	characteristic
162	C₈	—	47	2	no	—
163	C₈	—	47	2	no	—
164	C₈	—	188	4	no	—
165	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
166	C₈	#13093982	1	1	yes	characteristic
167	C₈	#13093989	1	1	yes	characteristic
168	C₈	Q₈×Dic₄₇	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#15689160	1	1	yes	characteristic
171	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
172	C₂²	#15689217	1	1	yes	characteristic
173	C₄	—	8	4	no	—
174	C₄	—	94	4	no	—
175	C₄	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
176	C₄	#107×D₄₇	1	1	yes	characteristic
177	C₄	#15689205	1	1	yes	characteristic
178	C₄	Q₁₆×Dic₄₇	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#107×Dic₄₇	1	1	yes	characteristic
182	C₂	#20245269	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₄×D₈
prime radical	C₂³×C₄₇	#135
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#12670
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
10. #15688953	C₄	semidirect
11. C₁₆×Dic₄₇	C₄	semidirect
15. C₄₇×#367	C₄	semidirect
16. C₄₇×#366	C₄	semidirect
104. #367	Dic₄₇	semidirect
105. #366	Dic₄₇	semidirect
108. C₄₇	#12670	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688953	2
C₂×C₁₆×Dic₄₇	2
#20243269	2
#20241483	2
C₄₇×#2158	2
#12670	47