Subgroups — C₂×#20245297 #26369766

Order 12032.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245297	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688899	C₂	1	2	yes	maximal
5	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	8	yes	—
11	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
12	#15688899	C₂²	1	8	yes	—
13	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
15	C₈×Q₈×C₄₇	C₄	1	4	yes	—
16	C₈×Q₈×C₄₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	4	no	—
23	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
24	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
25	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
26	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
27	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	C₈	1	4	yes	—
32	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	C₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
38	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₄₇	C₂³	1	2	yes	—
42	C₄×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂×Q₈×C₄₇	—	2	4	no	—
44	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₄₇	#41	1	2	yes	—
57	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
60	C₄²×C₄₇	C₂×C₈	1	4	yes	—
61	C₄²×C₄₇	C₂×C₈	1	2	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
64	C₄₇⋊C₁₆	—	2	2	no	—
65	C₄₇⋊C₁₆	—	2	4	no	—
66	C₄₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
67	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
73	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₄₇	#131	1	2	yes	—
80	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₄₇	—	2	2	no	—
82	C₈×C₄₇	—	2	4	no	—
83	C₈×C₄₇	C₄×Q₈	1	2	yes	—
84	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
86	C₂×#2183	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
88	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
89	C₂²×C₄₇	#504	1	2	yes	—
90	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	2	4	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	C₈×Q₈	1	2	yes	—
95	C₄×C₄₇	C₂×M₃₂	1	4	yes	—
96	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
98	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
99	#2183	—	47	8	no	—
100	C₂×C₄⋊C₁₆	—	47	1	no	—
101	C₂×C₄⋊C₁₆	—	47	2	no	—
102	C₂×#346	—	47	2	no	—
103	C₂×C₄×C₁₆	—	47	1	no	—
104	C₂×C₄₇	#2183	1	4	yes	complemented
105	C₂×C₄₇	C₂×C₈×Q₈	1	1	yes	characteristic
106	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
107	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
108	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
109	C₂²×C₁₆	—	47	1	no	—
110	C₂²×C₁₆	—	47	1	no	—
111	C₂²×C₁₆	—	47	2	no	—
112	C₈×Q₈	Dic₄₇	1	4	yes	—
113	C₈×Q₈	C₂×D₄₇	1	4	yes	—
114	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
115	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
118	C₄⋊C₁₆	—	47	4	no	—
119	C₄⋊C₁₆	—	47	8	no	—
120	#346	—	47	8	no	—
121	C₄×C₁₆	—	47	4	no	—
122	C₄₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₄×Q₈	C₄₇⋊C₈	1	4	yes	—
128	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄⋊C₄	C₄₇⋊C₈	1	2	yes	—
130	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
132	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
133	C₂×C₁₆	—	47	4	no	—
134	C₂×C₁₆	—	47	2	no	—
135	C₂×C₁₆	—	47	2	no	—
136	C₂×C₁₆	—	47	4	no	—
137	C₂×C₁₆	—	94	2	no	—
138	C₂×C₁₆	—	94	4	no	—
139	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
140	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
141	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
142	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
143	C₄×C₈	C₂²×D₄₇	1	2	yes	—
144	C₄×C₈	C₂²×D₄₇	1	4	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
147	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
148	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
150	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	Q₈×D₄₇	1	2	yes	—
155	C₂×C₈	#10625354	1	2	yes	—
156	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
157	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
158	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
161	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
162	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
163	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
164	C₄²	C₂²×Dic₄₇	1	4	yes	—
165	C₄²	C₂²×Dic₄₇	1	2	yes	—
166	C₁₆	—	47	4	no	—
167	C₁₆	—	94	2	no	—
168	C₁₆	—	94	4	no	—
169	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#13093964	1	4	yes	—
175	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
176	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
177	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
179	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
181	C₂×C₄	#13094031	1	2	yes	—
182	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	4	no	—
185	C₈	Q₈×Dic₄₇	1	2	yes	—
186	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
187	C₈	C₂×#10625354	1	1	yes	characteristic
188	C₂²	Q₈×C₄₇⋊C₈	1	2	yes	—
189	C₂²	C₂×#13093964	1	2	yes	—
190	C₂²	#15689540	1	2	yes	—
191	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	Q₈×C₄₇⋊C₈	1	2	yes	—
196	C₄	C₂×#13093964	1	4	yes	—
197	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
198	C₄	C₂×#13094031	1	1	yes	characteristic
199	C₂	#20245297	1	4	yes	direct product factor
200	C₂	C₂×Q₈×C₄₇⋊C₈	1	1	yes	characteristic
201	C₂	C₂²×#13093964	1	1	yes	characteristic
202	C₂	C₂×#15689540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	C₂×#2183
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #20245297	C₂	direct (N × H)
23. C₂×C₄₇⋊C₁₆	Q₈	semidirect
66. C₄₇⋊C₁₆	C₂×Q₈	semidirect
104. C₂×C₄₇	#2183	semidirect
122. C₄₇	C₂×#2183	semidirect
199. C₂	#20245297	direct (N × H)

Maximal subgroups

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

subgroup	index
#20245297	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688899	2
C₂×C₄×C₄₇⋊C₁₆	2
C₂×C₈×Q₈×C₄₇	2
C₂×#2183	47

Subgroups — C2×#20245297 #26369766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245297 #26369766