Subgroups — #368×Dic₄₇ #26131692

Order 12032.

314 conjugacy classes of subgroups.

The subgroup lattice has 178 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	#108×Dic₄₇	C₂	1	2	yes	maximal
4	#20241481	C₂	1	2	yes	maximal
5	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	M₁₆×Dic₄₇	—	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	—
12	#15688887	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₄₇×#368	C₄	1	4	yes	complemented
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	#13093960	—	2	2	no	—
23	#13093960	C₂×C₄	1	4	yes	—
24	#13093934	—	4	2	no	—
25	C₈×Dic₄₇	—	4	2	no	—
26	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₈×Dic₄₇	Q₈	1	2	yes	—
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	#108×C₄₇	—	2	2	no	—
34	#108×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
37	M₁₆×C₄₇	—	2	4	no	—
38	M₁₆×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
46	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₄₇	D₈	1	1	yes	characteristic
48	C₄×Dic₄₇	D₈	1	1	yes	characteristic
49	#10625329	—	2	4	no	—
50	#10625329	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₁₆×C₄₇	C₄²	1	4	yes	—
56	C₄₇⋊C₁₆	—	2	2	no	—
57	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	#107	1	2	yes	—
66	C₂×Dic₄₇	#107	1	2	yes	—
67	C₈×C₄₇	—	4	4	no	—
68	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₄₇	#117	1	1	yes	characteristic
70	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
72	C₄₇⋊C₈	—	2	2	no	—
73	C₄₇⋊C₈	—	4	4	no	—
74	C₄×#368	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₄₇	#427	1	1	yes	characteristic
81	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
82	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
83	Dic₄₇	—	2	2	no	—
84	Dic₄₇	#368	1	4	yes	direct product factor
85	C₂×#368	—	47	1	no	—
86	C₂×#368	D₄₇	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	47	1	no	—
88	C₄×#108	—	47	2	no	—
89	#1381	—	47	2	no	—
90	C₂×C₄₇	—	2	1	no	—
91	C₂×C₄₇	—	2	1	no	—
92	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
93	C₂×C₄₇	C₂×#368	1	2	yes	—
94	C₂²×C₁₆	—	47	1	no	—
95	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
96	C₂×#108	—	47	2	no	—
97	C₂×#108	C₂×D₄₇	1	2	yes	—
98	C₄×M₁₆	—	94	2	no	—
99	C₂×C₄×C₈	—	47	1	no	—
100	#368	—	47	4	no	—
101	#368	Dic₄₇	1	4	yes	direct product factor
102	C₄×C₁₆	—	47	4	no	—
103	#341	—	94	2	no	—
104	C₄₇	C₄×#368	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	94	2	no	—
107	C₂²×C₈	—	47	1	no	—
108	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₂×C₄²	—	47	1	no	—
110	C₂×C₁₆	—	47	2	no	—
111	C₂×C₁₆	—	94	1	no	—
112	C₂×C₁₆	—	94	1	no	—
113	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
114	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	47	4	no	—
118	#108	—	94	2	no	—
119	#108	C₂×Dic₄₇	1	4	yes	—
120	#97	—	188	2	no	—
121	C₄×C₈	—	47	2	no	—
122	C₄×C₈	—	47	2	no	—
123	C₄×C₈	—	188	2	no	—
124	C₂²×C₄	—	47	2	no	—
125	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	94	4	no	—
129	M₁₆	—	188	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	47	2	no	—
132	C₂×C₈	—	94	1	no	—
133	C₂×C₈	—	94	1	no	—
134	C₂×C₈	—	188	2	no	—
135	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₈	#10625352	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
139	C₂×C₈	#10625354	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
141	C₄²	—	47	2	no	—
142	C₄²	—	47	1	no	—
143	C₄²	—	47	1	no	—
144	C₁₆	—	94	2	no	—
145	C₁₆	C₄×Dic₄₇	1	4	yes	—
146	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
147	C₂×C₄	—	47	2	no	—
148	C₂×C₄	—	47	2	no	—
149	C₂×C₄	—	94	4	no	—
150	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₄	#13093986	1	1	yes	characteristic
152	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
154	C₂×C₄	#13094043	1	1	yes	characteristic
155	C₂×C₄	#13094047	1	1	yes	characteristic
156	C₈	—	4	4	no	—
157	C₈	—	94	2	no	—
158	C₈	—	188	4	no	—
159	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
160	C₈	#13093982	1	1	yes	characteristic
161	C₈	#13093989	1	1	yes	characteristic
162	C₈	Q₈×Dic₄₇	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#107×D₄₇	1	1	yes	characteristic
165	C₂²	#15689217	1	1	yes	characteristic
166	C₂²	Q₁₆×Dic₄₇	1	1	yes	characteristic
167	C₄	—	47	4	no	—
168	C₄	—	94	2	no	—
169	C₄	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
170	C₄	#15689160	1	1	yes	characteristic
171	C₄	#15689205	1	1	yes	characteristic
172	C₄	D₈×Dic₄₇	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#107×Dic₄₇	1	1	yes	characteristic
176	C₂	#368×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#368	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#368	D₄₇
hypocenter	C₄₇	C₄×#368
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #15688953	C₄	semidirect
15. C₄₇×#368	C₄	semidirect
84. Dic₄₇	#368	direct (N × H)
101. #368	Dic₄₇	direct (N × H)
104. C₄₇	C₄×#368	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
#108×Dic₄₇	2
#20241481	2
C₂×C₄₇×#368	2
C₄×#368	47

Subgroups — #368×Dic47 #26131692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #368×Dic₄₇ #26131692