Subgroups — C₂×#20243133 #26213739

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243133	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688877	C₂	1	4	yes	maximal
5	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	C₄²×Dic₄₇	C₄	1	4	yes	—
13	C₄²×Dic₄₇	C₂²	1	4	yes	—
14	#15688877	C₂²	1	16	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₄₇	C₂²	1	4	yes	—
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	—	2	8	no	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₄₇	—	2	8	no	—
24	C₂×C₄×Dic₄₇	—	2	8	no	—
25	C₂×C₄×Dic₄₇	C₈	1	16	yes	complemented
26	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
30	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	2	8	no	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
42	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
44	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
46	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	#41	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	8	no	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	C₂×C₈	1	16	yes	—
57	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₄₇	—	2	16	no	—
59	C₂×C₈×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
61	C₄²×C₄₇	C₄²	1	8	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
64	C₄×Dic₄₇	—	2	8	no	—
65	C₄×Dic₄₇	—	2	16	no	—
66	C₄×Dic₄₇	—	2	8	no	—
67	C₄×Dic₄₇	C₂×C₈	1	32	yes	complemented
68	C₂×C₄₇⋊C₈	—	2	16	no	—
69	C₂×C₄₇⋊C₈	—	2	8	no	—
70	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
72	C₂³×C₄₇	#117	1	4	yes	—
73	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
74	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
75	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₄×C₈	1	16	yes	—
77	C₂×C₄×C₄₇	#97	1	16	yes	—
78	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₄₇	#117	1	8	yes	—
82	C₂×C₄×C₄₇	#117	1	4	yes	—
83	C₂×C₄×C₄₇	#117	1	8	yes	—
84	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
85	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
86	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
89	C₂×Dic₄₇	—	2	8	no	—
90	C₂×Dic₄₇	—	2	8	no	—
91	C₂×Dic₄₇	—	2	32	no	—
92	C₂×Dic₄₇	C₂²×C₈	1	16	yes	—
93	C₈×C₄₇	—	2	8	no	—
94	C₄₇⋊C₈	—	2	8	no	—
95	C₂×#1744	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	#376	1	4	yes	—
97	C₂²×C₄₇	C₂×C₄×C₈	1	4	yes	—
98	C₂²×C₄₇	C₂×#97	1	4	yes	—
99	C₂²×C₄₇	#431	1	16	yes	—
100	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
101	C₂²×C₄₇	C₂×#117	1	2	yes	—
102	C₂²×C₄₇	C₂³×C₈	1	2	yes	—
103	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
104	C₄×C₄₇	#376	1	4	yes	—
105	C₄×C₄₇	C₂×C₄×C₈	1	8	yes	—
106	C₄×C₄₇	C₂×#97	1	8	yes	—
107	C₄×C₄₇	C₂×#117	1	2	yes	—
108	C₄×C₄₇	C₂×#117	1	2	yes	—
109	Dic₄₇	—	2	16	no	—
110	C₂²×C₄×C₈	—	47	1	no	—
111	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
112	C₂×C₄³	—	47	1	no	—
113	#1744	—	47	8	no	—
114	C₂×#336	—	47	4	no	—
115	C₂×C₄₇	#1744	1	8	yes	complemented
116	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
117	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
119	C₂×C₄₇	C₂×#431	1	4	yes	—
120	C₂³×C₈	—	47	2	no	—
121	C₂³×C₈	C₂×D₄₇	1	2	yes	—
122	C₂²×C₄²	—	47	2	no	—
123	C₂²×C₄²	—	47	2	no	—
124	C₂²×C₄²	—	47	2	no	—
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	C₂×C₄×C₈	—	47	8	no	—
127	C₂×C₄×C₈	—	47	4	no	—
128	C₂×C₄×C₈	Dic₄₇	1	8	yes	complemented
129	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
130	C₄³	—	47	4	no	—
131	C₄³	—	47	4	no	—
132	#336	—	47	16	no	—
133	C₄₇	C₂×#1744	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	47	4	no	—
135	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
136	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂²×C₈	—	2	8	no	—
139	C₂²×C₈	—	47	8	no	—
140	C₂²×C₈	—	47	4	no	—
141	C₂²×C₈	—	94	8	no	—
142	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
143	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
144	C₂×C₄²	—	47	16	no	—
145	C₂×C₄²	—	47	8	no	—
146	C₂×C₄²	—	47	4	no	—
147	C₂×C₄²	—	47	8	no	—
148	C₂×C₄²	—	47	4	no	—
149	C₂×C₄²	—	94	8	no	—
150	C₂×C₄²	—	94	8	no	—
151	C₂×C₄²	C₄×D₄₇	1	2	yes	—
152	C₂×C₄²	C₄×D₄₇	1	4	yes	—
153	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
155	C₄×C₈	—	47	16	no	—
156	C₄×C₈	C₂×Dic₄₇	1	16	yes	complemented
157	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
158	C₂²×C₄	—	47	16	no	—
159	C₂²×C₄	—	47	8	no	—
160	C₂²×C₄	—	94	8	no	—
161	C₂²×C₄	—	94	8	no	—
162	C₂²×C₄	C₈×D₄₇	1	8	yes	—
163	C₂²×C₄	#10625324	1	8	yes	—
164	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
165	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
170	C₂²×C₄	#10625350	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	C₂×C₈	—	2	16	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	47	8	no	—
176	C₂×C₈	—	94	16	no	—
177	C₂×C₈	—	94	8	no	—
178	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
179	C₄²	—	47	32	no	—
180	C₄²	—	94	8	no	—
181	C₄²	—	94	16	no	—
182	C₄²	—	94	8	no	—
183	C₄²	C₄×Dic₄₇	1	8	yes	—
184	C₄²	C₂×C₄×D₄₇	1	4	yes	—
185	C₄²	C₂²×Dic₄₇	1	4	yes	—
186	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
187	C₂³	#13093981	1	4	yes	—
188	C₂³	C₂×C₈×D₄₇	1	4	yes	—
189	C₂³	C₂×#10625324	1	4	yes	—
190	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
191	C₂×C₄	—	47	16	no	—
192	C₂×C₄	—	94	8	no	—
193	C₂×C₄	—	94	8	no	—
194	C₂×C₄	—	94	32	no	—
195	C₂×C₄	C₈×Dic₄₇	1	16	yes	—
196	C₂×C₄	#13093934	1	16	yes	—
197	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
199	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
200	C₂×C₄	#13093981	1	4	yes	—
201	C₂×C₄	#13093981	1	8	yes	—
202	C₂×C₄	#13093983	1	8	yes	—
203	C₂×C₄	C₂×C₈×D₄₇	1	8	yes	—
204	C₂×C₄	C₂×#10625324	1	8	yes	—
205	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×#10625350	1	4	yes	—
208	C₈	—	2	8	no	—
209	C₈	—	94	8	no	—
210	C₂²	#15688969	1	4	yes	—
211	C₂²	C₂×C₈×Dic₄₇	1	4	yes	—
212	C₂²	C₂×#13093934	1	4	yes	—
213	C₂²	#15689174	1	16	yes	—
214	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
215	C₂²	C₂×#13093981	1	2	yes	—
216	C₂²	C₂²×C₈×D₄₇	1	2	yes	—
217	C₂²	C₂²×#10625324	1	2	yes	—
218	C₄	—	94	16	no	—
219	C₄	#15688969	1	4	yes	—
220	C₄	C₂×C₈×Dic₄₇	1	8	yes	—
221	C₄	C₂×#13093934	1	8	yes	—
222	C₄	C₂×#13093981	1	2	yes	—
223	C₄	C₂×#13093983	1	2	yes	—
224	C₂	#20243133	1	8	yes	direct product factor
225	C₂	C₂×#15688969	1	1	yes	characteristic
226	C₂	C₂²×C₈×Dic₄₇	1	1	yes	characteristic
227	C₂	C₂²×#13093934	1	1	yes	characteristic
228	C₂	C₂×#15689174	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#1744
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20243133	C₂	direct (N × H)
10. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
17. C₂×C₄×C₈×C₄₇	C₄	semidirect
25. C₂×C₄×Dic₄₇	C₈	semidirect
30. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
41. C₄×C₈×C₄₇	C₂×C₄	semidirect
67. C₄×Dic₄₇	C₂×C₈	semidirect
115. C₂×C₄₇	#1744	semidirect
128. C₂×C₄×C₈	Dic₄₇	semidirect
133. C₄₇	C₂×#1744	semidirect
156. C₄×C₈	C₂×Dic₄₇	semidirect
224. C₂	#20243133	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×C₄×C₄₇⋊C₈	2
C₂×C₄²×Dic₄₇	2
#20243133	2
C₂×#15688877	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1744	47

Subgroups — C2×#20243133 #26213739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243133 #26213739