Subgroups — C₂×#20251751 #26422258

Order 12032.

2558 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094016	C₂	1	2	yes	maximal
2	#20251751	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689021	C₂	1	8	yes	maximal
4	C₂×#15688993	C₂	1	2	yes	maximal
5	C₂×#15688989	C₂	1	2	yes	maximal
6	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094016	—	2	16	no	—
8	C₂×#13094016	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	8	yes	—
11	C₂²×Dic₄₇⋊C₄	C₂²	1	8	yes	—
12	#15689021	C₂²	1	64	yes	—
13	#15688993	C₂²	1	16	yes	—
14	#15688989	C₂²	1	16	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	8	yes	—
21	C₂³×Dic₄₇	C₂³	1	8	yes	—
22	#13094016	—	2	64	no	—
23	C₂×C₄⋊Dic₄₇	—	2	16	no	—
24	C₂×C₄⋊Dic₄₇	—	2	32	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	—
27	C₂×Dic₄₇⋊C₄	—	2	32	no	—
28	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
29	C₄₇⋊#95	C₂³	1	32	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂×C₄²×C₄₇	—	2	16	no	—
35	C₂×C₄²×C₄₇	D₄	1	8	yes	—
36	C₂×C₄²×C₄₇	Q₈	1	8	yes	—
37	C₂×C₄²×C₄₇	Q₈	1	8	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
42	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	8	no	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
49	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₄₇	#41	1	16	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
56	C₂²×Dic₄₇	—	2	16	no	—
57	C₂²×Dic₄₇	—	2	32	no	—
58	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
59	C₄²×C₄₇	—	2	32	no	—
60	C₄²×C₄₇	C₂×D₄	1	16	yes	—
61	C₄²×C₄₇	C₂×Q₈	1	16	yes	—
62	C₄²×C₄₇	C₂×Q₈	1	16	yes	—
63	C₄⋊Dic₄₇	—	2	32	no	—
64	C₄⋊Dic₄₇	—	2	64	no	—
65	Dic₄₇⋊C₄	—	2	64	no	—
66	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
67	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
68	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
69	C₂³×C₄₇	C₂×#41	1	8	yes	—
70	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	16	no	—
72	C₂×C₄×C₄₇	—	2	16	no	—
73	C₂×C₄×C₄₇	—	2	8	no	—
74	C₂×C₄×C₄₇	—	2	8	no	—
75	C₂×C₄×C₄₇	#125	1	32	yes	complemented
76	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	—
77	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
78	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
79	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
80	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
81	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
82	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
84	C₂×Dic₄₇	—	2	16	no	—
85	C₂×Dic₄₇	—	2	32	no	—
86	C₂×Dic₄₇	—	4	32	no	—
87	C₂×#2859	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	C₂×#125	1	8	yes	—
89	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
90	C₂²×C₄₇	#532	1	8	yes	—
91	C₂²×C₄₇	#533	1	8	yes	—
92	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
93	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
94	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
95	C₂²×C₄₇	C₂²×#41	1	4	yes	—
96	C₄×C₄₇	—	2	8	no	—
97	C₄×C₄₇	—	2	8	no	—
98	C₄×C₄₇	C₂×#125	1	16	yes	complemented
99	C₄×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
100	Dic₄₇	—	4	16	no	—
101	C₂²×#125	—	47	2	no	—
102	#2859	—	47	16	no	—
103	C₂×#389	—	47	8	no	—
104	C₂×#384	—	47	2	no	—
105	C₂×#382	—	47	2	no	—
106	C₂×C₄³	D₄₇	1	1	yes	characteristic
107	C₂×C₄₇	#2859	1	8	yes	complemented
108	C₂×C₄₇	C₂²×#125	1	2	yes	—
109	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄₇	C₂×#532	1	2	yes	—
111	C₂×C₄₇	C₂×#533	1	2	yes	—
112	C₂×#125	—	47	16	no	—
113	C₂×#125	—	94	16	no	—
114	C₂²×C₄⋊C₄	—	47	8	no	—
115	C₂²×C₄⋊C₄	—	47	4	no	—
116	C₂²×C₄⋊C₄	—	47	8	no	—
117	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
118	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
121	#389	—	47	64	no	—
122	#384	—	47	16	no	—
123	#382	—	47	16	no	—
124	C₂×#95	—	47	8	no	—
125	C₄³	C₂×D₄₇	1	8	yes	—
126	C₄₇	C₂×#2859	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	—	47	8	no	—
128	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
129	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
130	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
132	#125	—	94	64	no	—
133	C₂×C₄⋊C₄	—	47	32	no	—
134	C₂×C₄⋊C₄	—	47	16	no	—
135	C₂×C₄⋊C₄	—	47	32	no	—
136	C₂×C₄⋊C₄	—	94	32	no	—
137	C₂×C₄⋊C₄	—	94	16	no	—
138	C₂×C₄⋊C₄	—	94	32	no	—
139	C₂×C₄²	—	2	16	no	—
140	C₂×C₄²	Dic₉₄	1	8	yes	—
141	C₂×C₄²	Dic₉₄	1	8	yes	—
142	C₂×C₄²	D₁₈₈	1	8	yes	—
143	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
144	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
145	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
147	#95	—	47	32	no	—
148	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	47	16	no	—
152	C₂²×C₄	—	94	16	no	—
153	C₂²×C₄	—	94	32	no	—
154	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
155	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
156	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
157	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
158	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
159	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
160	C₂²×C₄	#10625350	1	16	yes	—
161	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
162	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
163	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
164	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
165	C₄⋊C₄	—	94	64	no	—
166	C₄⋊C₄	—	94	32	no	—
167	C₄⋊C₄	—	94	64	no	—
168	C₄²	—	2	32	no	—
169	C₄²	C₂×Dic₉₄	1	16	yes	—
170	C₄²	C₂×Dic₉₄	1	16	yes	—
171	C₄²	C₂×D₁₈₈	1	16	yes	—
172	C₂³	C₂²×D₁₈₈	1	2	yes	—
173	C₂³	C₂²×Dic₉₄	1	2	yes	—
174	C₂³	C₂²×Dic₉₄	1	2	yes	—
175	C₂³	C₂×#10625350	1	8	yes	—
176	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	94	16	no	—
182	C₂×C₄	—	94	32	no	—
183	C₂×C₄	—	188	32	no	—
184	C₂×C₄	#13094016	1	32	yes	complemented
185	C₂×C₄	#13094024	1	16	yes	—
186	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
187	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
188	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
189	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
190	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
191	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
192	C₂×C₄	C₂×#10625350	1	16	yes	—
193	C₂²	C₂×#13094016	1	8	yes	—
194	C₂²	C₂×#13094024	1	4	yes	—
195	C₂²	#15689673	1	8	yes	—
196	C₂²	#15689680	1	8	yes	—
197	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
198	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
199	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
200	C₂²	C₂²×#10625350	1	4	yes	—
201	C₄	—	2	8	no	—
202	C₄	—	2	8	no	—
203	C₄	—	188	16	no	—
204	C₄	C₂×#13094016	1	16	yes	complemented
205	C₄	C₂×#13094024	1	8	yes	—
206	C₂	#20251751	1	8	yes	direct product factor
207	C₂	C₂²×#13094016	1	2	yes	—
208	C₂	C₂²×#13094024	1	1	yes	characteristic
209	C₂	C₂×#15689673	1	2	yes	—
210	C₂	C₂×#15689680	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	C₂×#2859
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20251751	C₂	direct (N × H)
75. C₂×C₄×C₄₇	#125	semidirect
98. C₄×C₄₇	C₂×#125	semidirect
107. C₂×C₄₇	#2859	semidirect
126. C₄₇	C₂×#2859	semidirect
184. C₂×C₄	#13094016	semidirect
204. C₄	C₂×#13094016	semidirect
206. C₂	#20251751	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×#13094016	2
#20251751	2
C₂×#15689021	2
C₂×#15688993	2
C₂×#15688989	2
C₂×C₄³×C₄₇	2
C₂×#2859	47

Subgroups — C2×#20251751 #26422258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251751 #26422258