Subgroups — C₂×#20243545 #26216676

Order 12032.

1358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093956	C₂	1	4	yes	maximal
2	C₂×#15688993	C₂	1	2	yes	maximal
3	#20243545	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	4	no	—
6	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
7	C₂×#13093956	—	2	16	no	—
8	C₂×#13093956	C₂²	1	16	yes	—
9	#15688993	—	2	4	no	—
10	#15688993	C₂²	1	8	yes	—
11	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₇	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₄₇	C₂²	1	4	yes	—
15	C₂³×Dic₄₇	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	16	no	—
17	C₂×C₄⋊Dic₄₇	—	2	4	no	—
18	C₂×C₄⋊Dic₄₇	—	2	16	no	—
19	C₂×C₄⋊Dic₄₇	—	4	16	no	—
20	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
21	#13093956	—	2	32	no	—
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₄₇	D₄	1	8	yes	—
26	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
27	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
32	C₄×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
33	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₄₇	SD₁₆	1	16	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
40	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	8	no	—
44	C₂²×Dic₄₇	—	4	8	no	—
45	C₂²×Dic₄₇	—	4	16	no	—
46	C₂×C₈×C₄₇	C₄⋊C₄	1	32	yes	complemented
47	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
48	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
49	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
50	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
52	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
53	C₄⋊Dic₄₇	—	2	16	no	—
54	C₄⋊Dic₄₇	—	4	8	no	—
55	C₄⋊Dic₄₇	—	4	32	no	—
56	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₄₇	#127	1	2	yes	—
58	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
59	C₂³×C₄₇	C₂×SD₁₆	1	8	yes	—
60	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#106	1	32	yes	complemented
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
65	C₂×C₄×C₄₇	#127	1	2	yes	—
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×SD₁₆	1	16	yes	—
69	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×Dic₄₇	—	4	8	no	—
74	C₂×Dic₄₇	—	4	16	no	—
75	C₂×Dic₄₇	—	8	16	no	—
76	C₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	complemented
77	C₂×#1844	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	#384	1	4	yes	—
79	C₂²×C₄₇	C₂×#106	1	8	yes	—
80	C₂²×C₄₇	#492	1	8	yes	—
81	C₂²×C₄₇	#498	1	8	yes	—
82	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
84	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
85	C₂²×C₄₇	C₂²×SD₁₆	1	4	yes	—
86	C₄×C₄₇	#384	1	4	yes	—
87	C₄×C₄₇	C₂×#106	1	16	yes	complemented
88	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
89	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
90	Dic₄₇	—	8	8	no	—
91	C₂²×#106	—	47	4	no	—
92	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
93	C₂×#384	—	47	2	no	—
94	#1844	—	47	8	no	—
95	C₂×C₄₇	#1844	1	8	yes	complemented
96	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
97	C₂×C₄₇	C₂²×#106	1	2	yes	—
98	C₂×C₄₇	C₂×#492	1	2	yes	—
99	C₂×C₄₇	C₂×#498	1	2	yes	—
100	C₂³×C₈	C₂×D₄₇	1	2	yes	—
101	C₂²×C₄⋊C₄	—	47	4	no	—
102	C₂²×C₄⋊C₄	—	94	4	no	—
103	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
104	C₂×#106	—	47	16	no	—
105	C₂×#106	—	94	16	no	—
106	C₂×C₄×C₈	Dic₄₇	1	8	yes	complemented
107	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
108	#384	—	47	8	no	—
109	#384	—	94	4	no	—
110	C₄₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₄	—	94	4	no	—
112	C₂³×C₄	D₁₈₈	1	2	yes	—
113	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂²×C₈	Dic₉₄	1	8	yes	—
115	C₂²×C₈	D₁₈₈	1	8	yes	—
116	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
117	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
118	C₂×C₄⋊C₄	—	47	8	no	—
119	C₂×C₄⋊C₄	—	94	16	no	—
120	C₂×C₄⋊C₄	—	94	4	no	—
121	C₂×C₄⋊C₄	—	94	16	no	—
122	C₂×C₄⋊C₄	—	188	16	no	—
123	C₂×C₄²	Dic₉₄	1	4	yes	—
124	C₂×C₄²	D₁₈₈	1	2	yes	—
125	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
127	#106	—	94	32	no	—
128	C₄×C₈	C₂×Dic₄₇	1	16	yes	complemented
129	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂²×C₄	—	94	8	no	—
131	C₂²×C₄	—	188	8	no	—
132	C₂²×C₄	—	188	16	no	—
133	C₂²×C₄	#10625325	1	16	yes	—
134	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
135	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
136	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
137	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
139	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
140	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
141	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₄₇	1	32	yes	complemented
143	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
144	C₂×C₈	C₂×D₁₈₈	1	8	yes	—
145	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
146	C₄⋊C₄	—	94	16	no	—
147	C₄⋊C₄	—	188	8	no	—
148	C₄⋊C₄	—	188	32	no	—
149	C₄²	C₄⋊Dic₄₇	1	8	yes	—
150	C₄²	C₂×Dic₉₄	1	4	yes	—
151	C₄²	C₂²×Dic₄₇	1	4	yes	—
152	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
153	C₂³	#13094021	1	2	yes	—
154	C₂³	#13094024	1	2	yes	—
155	C₂³	C₂×#10625325	1	8	yes	—
156	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
157	C₂×C₄	—	376	16	no	—
158	C₂×C₄	—	188	16	no	—
159	C₂×C₄	—	188	8	no	—
160	C₂×C₄	#13093956	1	32	yes	complemented
161	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
164	C₂×C₄	#13094021	1	2	yes	—
165	C₂×C₄	#13094024	1	2	yes	—
166	C₂×C₄	#13094024	1	8	yes	—
167	C₂×C₄	C₂×#10625325	1	16	yes	—
168	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
170	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
171	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₄₇	1	16	yes	complemented
173	C₂²	#15688993	1	4	yes	—
174	C₂²	C₂×#13093956	1	8	yes	—
175	C₂²	#15689496	1	8	yes	—
176	C₂²	#15689521	1	8	yes	—
177	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
178	C₂²	C₂×#13094021	1	1	yes	characteristic
179	C₂²	C₂×#13094024	1	1	yes	characteristic
180	C₂²	C₂²×#10625325	1	4	yes	—
181	C₄	—	376	8	no	—
182	C₄	#15688993	1	4	yes	—
183	C₄	C₂×#13093956	1	16	yes	complemented
184	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
185	C₄	C₂×#13094024	1	2	yes	—
186	C₂	#20243545	1	8	yes	direct product factor
187	C₂	C₂×#15688993	1	1	yes	characteristic
188	C₂	C₂²×#13093956	1	2	yes	—
189	C₂	C₂×#15689496	1	2	yes	—
190	C₂	C₂×#15689521	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20243545	C₂	direct (N × H)
13. C₂×C₄×C₈×C₄₇	C₄	semidirect
32. C₄×C₈×C₄₇	C₂×C₄	semidirect
46. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
61. C₂×C₄×C₄₇	#106	semidirect
76. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
87. C₄×C₄₇	C₂×#106	semidirect
95. C₂×C₄₇	#1844	semidirect
106. C₂×C₄×C₈	Dic₄₇	semidirect
110. C₄₇	C₂×#1844	semidirect
128. C₄×C₈	C₂×Dic₄₇	semidirect
142. C₂×C₈	C₄⋊Dic₄₇	semidirect
160. C₂×C₄	#13093956	semidirect
172. C₈	C₂×C₄⋊Dic₄₇	semidirect
183. C₄	C₂×#13093956	semidirect
186. C₂	#20243545	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093956	2
C₂×#15688993	2
#20243545	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1844	47

Subgroups — C2×#20243545 #26216676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243545 #26216676