Subgroups — C₂×#20243516 #26216462

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688993	C₂	1	2	yes	maximal
2	#20243516	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688886	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
6	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	#15688993	—	2	4	no	—
9	#15688993	C₂²	1	8	yes	—
10	#15688886	C₂²	1	16	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	8	no	—
19	C₂×C₄⋊Dic₄₇	—	2	8	no	—
20	C₂×C₄⋊Dic₄₇	—	4	8	no	—
21	C₂×C₄⋊Dic₄₇	—	4	8	no	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
23	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	—	2	8	no	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
33	C₄×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
34	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
37	C₂²×C₄×C₄₇	D₈	1	4	yes	—
38	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	#41	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂²×Dic₄₇	—	2	8	no	—
48	C₂²×Dic₄₇	—	4	8	no	—
49	C₂²×Dic₄₇	—	4	8	no	—
50	C₂²×Dic₄₇	—	4	8	no	—
51	C₂×C₈×C₄₇	—	2	16	no	—
52	C₂×C₈×C₄₇	—	2	8	no	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
54	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
56	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₄₇	—	2	16	no	—
58	C₄⋊Dic₄₇	—	4	8	no	—
59	C₄⋊Dic₄₇	—	4	16	no	—
60	C₄⋊Dic₄₇	—	4	16	no	—
61	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₄₇	#124	1	4	yes	—
63	C₂³×C₄₇	C₂×D₈	1	2	yes	—
64	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
65	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
66	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#106	1	16	yes	—
68	C₂×C₄×C₄₇	#107	1	16	yes	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₄₇	#117	1	8	yes	—
73	C₂×C₄×C₄₇	#124	1	4	yes	—
74	C₂×C₄×C₄₇	#125	1	8	yes	—
75	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
78	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
81	C₂×Dic₄₇	—	4	8	no	—
82	C₂×Dic₄₇	—	4	8	no	—
83	C₂×Dic₄₇	—	4	8	no	—
84	C₂×Dic₄₇	—	8	16	no	—
85	C₈×C₄₇	—	2	8	no	—
86	C₂×#1834	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	#382	1	4	yes	—
88	C₂²×C₄₇	C₂×#106	1	4	yes	—
89	C₂²×C₄₇	C₂×#107	1	4	yes	—
90	C₂²×C₄₇	#486	1	8	yes	—
91	C₂²×C₄₇	#487	1	8	yes	—
92	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#124	1	2	yes	—
94	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
95	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
97	C₄×C₄₇	#382	1	4	yes	—
98	C₄×C₄₇	C₂×#106	1	8	yes	—
99	C₄×C₄₇	C₂×#107	1	8	yes	—
100	C₄×C₄₇	C₂×#117	1	2	yes	—
101	C₄×C₄₇	C₂×#125	1	2	yes	—
102	Dic₄₇	—	8	8	no	—
103	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
104	C₂×#384	—	47	2	no	—
105	#1834	—	47	8	no	—
106	C₂×#340	—	47	4	no	—
107	C₂×C₄₇	#1834	1	8	yes	complemented
108	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
109	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
110	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#486	1	2	yes	—
112	C₂×C₄₇	C₂×#487	1	2	yes	—
113	C₂³×C₈	C₂×D₄₇	1	2	yes	—
114	C₂²×C₄⋊C₄	—	47	4	no	—
115	C₂²×C₄⋊C₄	—	94	2	no	—
116	C₂²×C₄⋊C₄	—	94	2	no	—
117	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	C₂×C₄×C₈	Dic₄₇	1	8	yes	complemented
119	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
120	#384	—	47	8	no	—
121	#384	—	94	4	no	—
122	#340	—	47	16	no	—
123	C₄₇	C₂×#1834	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	—	94	4	no	—
125	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
126	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
127	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₂²×C₈	—	2	8	no	—
129	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
130	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
131	C₂×C₄⋊C₄	—	47	8	no	—
132	C₂×C₄⋊C₄	—	94	16	no	—
133	C₂×C₄⋊C₄	—	94	4	no	—
134	C₂×C₄⋊C₄	—	94	8	no	—
135	C₂×C₄⋊C₄	—	94	8	no	—
136	C₂×C₄⋊C₄	—	188	8	no	—
137	C₂×C₄⋊C₄	—	188	8	no	—
138	C₂×C₄²	Dic₉₄	1	4	yes	—
139	C₂×C₄²	D₁₈₈	1	2	yes	—
140	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
141	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
142	C₄×C₈	C₂×Dic₄₇	1	16	yes	complemented
143	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
144	C₂²×C₄	—	94	8	no	—
145	C₂²×C₄	—	188	8	no	—
146	C₂²×C₄	—	188	8	no	—
147	C₂²×C₄	—	188	8	no	—
148	C₂²×C₄	#10625325	1	8	yes	—
149	C₂²×C₄	D₃₇₆	1	4	yes	—
150	C₂²×C₄	Dic₁₈₈	1	4	yes	—
151	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
153	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
154	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
156	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
157	C₂²×C₄	#10625350	1	8	yes	—
158	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
159	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
163	C₄⋊C₄	—	94	16	no	—
164	C₄⋊C₄	—	188	8	no	—
165	C₄⋊C₄	—	188	16	no	—
166	C₄⋊C₄	—	188	16	no	—
167	C₄²	C₄⋊Dic₄₇	1	8	yes	—
168	C₄²	C₂×Dic₉₄	1	4	yes	—
169	C₄²	C₂²×Dic₄₇	1	4	yes	—
170	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
171	C₂³	#13094012	1	4	yes	—
172	C₂³	C₂×D₃₇₆	1	2	yes	—
173	C₂³	C₂×#10625325	1	4	yes	—
174	C₂³	C₂×Dic₁₈₈	1	2	yes	—
175	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
176	C₂×C₄	—	376	16	no	—
177	C₂×C₄	—	188	8	no	—
178	C₂×C₄	—	188	8	no	—
179	C₂×C₄	—	188	8	no	—
180	C₂×C₄	#13093956	1	16	yes	—
181	C₂×C₄	#13093958	1	16	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
185	C₂×C₄	#13093983	1	8	yes	—
186	C₂×C₄	#13094012	1	4	yes	—
187	C₂×C₄	#13094016	1	8	yes	—
188	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
189	C₂×C₄	C₂×#10625325	1	8	yes	—
190	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
191	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
193	C₂×C₄	C₂×#10625350	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	#15688989	1	4	yes	—
196	C₂²	C₂×#13093956	1	4	yes	—
197	C₂²	C₂×#13093958	1	4	yes	—
198	C₂²	#15689470	1	8	yes	—
199	C₂²	#15689474	1	8	yes	—
200	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
201	C₂²	C₂×#13094012	1	2	yes	—
202	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
203	C₂²	C₂²×#10625325	1	2	yes	—
204	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
205	C₄	—	376	8	no	—
206	C₄	#15688989	1	4	yes	—
207	C₄	C₂×#13093956	1	8	yes	—
208	C₄	C₂×#13093958	1	8	yes	—
209	C₄	C₂×#13093983	1	2	yes	—
210	C₄	C₂×#13094016	1	2	yes	—
211	C₂	#20243516	1	8	yes	direct product factor
212	C₂	C₂×#15688989	1	1	yes	characteristic
213	C₂	C₂²×#13093956	1	1	yes	characteristic
214	C₂	C₂²×#13093958	1	1	yes	characteristic
215	C₂	C₂×#15689470	1	2	yes	—
216	C₂	C₂×#15689474	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×#1834
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
2. #20243516	C₂	direct (N × H)
13. C₂×C₄×C₈×C₄₇	C₄	semidirect
33. C₄×C₈×C₄₇	C₂×C₄	semidirect
107. C₂×C₄₇	#1834	semidirect
118. C₂×C₄×C₈	Dic₄₇	semidirect
123. C₄₇	C₂×#1834	semidirect
142. C₄×C₈	C₂×Dic₄₇	semidirect
211. C₂	#20243516	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×#15688993	2
#20243516	2
C₂×#15688886	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1834	47

Subgroups — C2×#20243516 #26216462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243516 #26216462