Subgroups — C₂×#20241461 #26127858

Order 12032.

262 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×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241461	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15688876	—	2	1	no	—
11	#15688876	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#335	—	2	1	no	—
15	C₄₇×#335	C₂²	1	2	yes	—
16	C₂²×C₄₇⋊C₈	—	4	1	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	C₈	1	4	yes	complemented
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₄₇	—	4	1	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₄⋊C₈×C₄₇	—	4	1	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	8	2	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₄₇	D₈	1	1	yes	characteristic
47	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄₇⋊C₈	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	8	2	no	—
51	C₄₇⋊C₁₆	—	2	2	no	—
52	C₄₇⋊C₁₆	C₂×C₈	1	4	yes	complemented
53	C₂³×C₄₇	#107	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#108	1	2	yes	—
60	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
61	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	—	8	2	no	—
68	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
69	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
71	C₄₇⋊C₈	—	8	2	no	—
72	C₂×#1372	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	#335	1	2	yes	—
74	C₂²×C₄₇	#367	1	2	yes	—
75	C₂²×C₄₇	#368	1	2	yes	—
76	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	#335	1	2	yes	—
80	C₄×C₄₇	#367	1	2	yes	—
81	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
83	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
84	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
85	C₂×C₄×C₁₆	—	47	1	no	—
86	C₂×#335	—	47	1	no	—
87	C₂×#335	D₄₇	1	1	yes	characteristic
88	#1372	—	47	4	no	—
89	C₂×C₄₇	#1372	1	4	yes	complemented
90	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
91	C₂×C₄₇	C₂×#367	1	1	yes	characteristic
92	C₂×C₄₇	C₂×#368	1	1	yes	characteristic
93	C₂²×C₁₆	—	47	1	no	—
94	C₂²×C₁₆	—	47	1	no	—
95	C₂×C₄⋊C₈	—	2	1	no	—
96	C₂×C₄⋊C₈	—	94	1	no	—
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄×C₁₆	—	47	2	no	—
99	C₄×C₁₆	—	47	2	no	—
100	#335	—	2	1	no	—
101	#335	—	47	2	no	—
102	#335	—	94	1	no	—
103	#335	C₂×D₄₇	1	2	yes	—
104	C₄₇	C₂×#1372	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	—	188	1	no	—
107	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×C₁₆	—	47	4	no	—
111	C₂×C₁₆	—	47	2	no	—
112	C₂×C₁₆	—	47	2	no	—
113	C₂×C₁₆	—	94	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	94	2	no	—
117	C₄⋊C₈	—	188	1	no	—
118	C₄×C₈	Dic₉₄	1	1	yes	characteristic
119	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#10625334	1	1	yes	characteristic
124	C₂²×C₄	#10625337	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	376	2	no	—
129	C₂×C₈	—	188	2	no	—
130	C₂×C₈	C₈×D₄₇	1	2	yes	—
131	C₂×C₈	#10625324	1	2	yes	—
132	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
133	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
134	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
137	C₄²	#10625334	1	1	yes	characteristic
138	C₄²	#10625337	1	1	yes	characteristic
139	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₁₆	—	47	4	no	—
141	C₁₆	—	94	2	no	—
142	C₂³	#13093957	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
145	C₂×C₄	#13093957	1	1	yes	characteristic
146	C₂×C₄	#13093957	1	1	yes	characteristic
147	C₂×C₄	#13093957	1	1	yes	characteristic
148	C₂×C₄	#13093959	1	2	yes	—
149	C₂×C₄	#13093967	1	2	yes	—
150	C₂×C₄	#13093968	1	2	yes	—
151	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
153	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	8	2	no	—
157	C₈	—	376	2	no	—
158	C₈	Dic₄₇⋊C₈	1	2	yes	—
159	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
160	C₈	C₂×#10625324	1	1	yes	characteristic
161	C₂²	#15688875	1	2	yes	—
162	C₂²	#15688950	1	2	yes	—
163	C₂²	#15688952	1	2	yes	—
164	C₂²	C₂×#13093957	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	#15688875	1	2	yes	—
168	C₄	#15688950	1	2	yes	—
169	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
170	C₄	C₂×#13093959	1	1	yes	characteristic
171	C₄	C₂×#13093967	1	1	yes	characteristic
172	C₄	C₂×#13093968	1	1	yes	characteristic
173	C₂	#20241461	1	4	yes	direct product factor
174	C₂	C₂×#15688875	1	1	yes	characteristic
175	C₂	C₂×#15688950	1	1	yes	characteristic
176	C₂	C₂×#15688952	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	#107
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	C₂×#1372
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20241461	C₂	direct (N × H)
18. C₂×C₄₇⋊C₁₆	C₈	semidirect
52. C₄₇⋊C₁₆	C₂×C₈	semidirect
89. C₂×C₄₇	#1372	semidirect
104. C₄₇	C₂×#1372	semidirect
173. C₂	#20241461	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
C₂×#15688876	2
#20241461	2
C₂×C₄₇×#335	2
C₂×#1372	47

Subgroups — C2×#20241461 #26127858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241461 #26127858