Subgroups — C₂×#20251770 #26422296

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251770	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689029	C₂	1	3	yes	maximal
3	C₂×#15689021	C₂	1	3	yes	maximal
4	C₂×#15689007	C₂	1	6	yes	maximal
5	C₂×#15688991	C₂	1	2	yes	maximal
6	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	—
9	C₂²×Dic₄₇⋊C₄	C₂²	1	6	yes	—
10	#15689029	C₂²	1	24	yes	—
11	#15689021	C₂²	1	24	yes	—
12	#15689007	C₂²	1	48	yes	—
13	#15688991	C₂²	1	16	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	8	yes	—
21	C₂²×Dic₉₄	—	2	12	no	—
22	C₂²×Dic₉₄	C₂³	1	4	yes	—
23	C₂³×Dic₄₇	C₂³	1	2	yes	—
24	C₂³×Dic₄₇	C₂³	1	6	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	12	no	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
27	C₂×Dic₄₇⋊C₄	—	2	24	no	—
28	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
29	C₄₇⋊#95	C₂³	1	24	yes	—
30	C₄₇⋊#95	C₂³	1	24	yes	—
31	C₄₇⋊#95	C₂³	1	24	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	—	2	12	no	—
36	C₂×C₄²×C₄₇	—	2	12	no	—
37	C₂×C₄²×C₄₇	Q₈	1	8	yes	complemented
38	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
41	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	6	no	—
43	C₂²×C₄×C₄₇	—	2	6	no	—
44	C₂²×C₄×C₄₇	—	2	6	no	—
45	C₂²×C₄×C₄₇	—	2	6	no	—
46	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
47	C₂²×C₄×C₄₇	#41	1	12	yes	—
48	C₂²×C₄×C₄₇	#41	1	12	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
50	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
52	C₂²×Dic₄₇	—	2	24	no	—
53	C₂²×Dic₄₇	—	2	12	no	—
54	C₂²×Dic₄₇	—	2	12	no	—
55	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
56	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
57	C₂×Dic₉₄	—	2	24	no	—
58	C₂×Dic₉₄	—	4	16	no	—
59	C₄²×C₄₇	—	2	24	no	—
60	C₄²×C₄₇	—	2	24	no	—
61	C₄²×C₄₇	C₂×Q₈	1	16	yes	complemented
62	C₄⋊Dic₄₇	—	2	24	no	—
63	Dic₄₇⋊C₄	—	2	48	no	—
64	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
65	C₂³×C₄₇	C₂×#41	1	6	yes	—
66	C₂³×C₄₇	C₂×#41	1	6	yes	—
67	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	24	no	—
69	C₂×C₄×C₄₇	—	2	24	no	—
70	C₂×C₄×C₄₇	—	2	6	no	—
71	C₂×C₄×C₄₇	—	2	6	no	—
72	C₂×C₄×C₄₇	—	2	6	no	—
73	C₂×C₄×C₄₇	—	2	6	no	—
74	C₂×C₄×C₄₇	#126	1	16	yes	—
75	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
76	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
77	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
78	C₂×Dic₄₇	—	2	24	no	—
79	C₂×Dic₄₇	—	2	12	no	—
80	C₂×Dic₄₇	—	2	12	no	—
81	C₂×Dic₄₇	—	4	8	no	—
82	C₂×Dic₄₇	—	4	24	no	—
83	Dic₉₄	—	4	16	no	—
84	C₂×#2863	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	C₂×#126	1	4	yes	—
86	C₂²×C₄₇	#529	1	12	yes	—
87	C₂²×C₄₇	#533	1	12	yes	—
88	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
89	C₂²×C₄₇	C₂²×#41	1	3	yes	—
90	C₂²×C₄₇	C₂²×#41	1	3	yes	—
91	C₄×C₄₇	—	2	12	no	—
92	C₄×C₄₇	—	2	12	no	—
93	C₄×C₄₇	C₂×#126	1	8	yes	—
94	Dic₄₇	—	4	4	no	—
95	Dic₄₇	—	4	12	no	—
96	#2863	—	47	16	no	—
97	C₂×#391	—	47	3	no	—
98	C₂×#389	—	47	3	no	—
99	C₂×#387	—	47	6	no	—
100	C₂×#383	—	47	2	no	—
101	C₂×C₄³	D₄₇	1	1	yes	characteristic
102	C₂×C₄₇	#2863	1	8	yes	complemented
103	C₂×C₄₇	C₂²×#126	1	1	yes	characteristic
104	C₂×C₄₇	C₂×#529	1	3	yes	—
105	C₂×C₄₇	C₂×#533	1	3	yes	—
106	C₂³×Q₈	—	47	1	no	—
107	C₂²×C₄⋊C₄	—	47	6	no	—
108	C₂²×C₄⋊C₄	—	47	3	no	—
109	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
110	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
111	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#391	—	47	24	no	—
113	#389	—	47	24	no	—
114	#387	—	47	48	no	—
115	#383	—	47	16	no	—
116	C₂×#95	—	47	6	no	—
117	C₂×#95	—	47	6	no	—
118	C₂×#95	—	47	6	no	—
119	C₄³	C₂×D₄₇	1	8	yes	—
120	C₄₇	C₂×#2863	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	47	4	no	—
122	C₂²×Q₈	—	94	12	no	—
123	C₂³×C₄	—	47	6	no	—
124	C₂³×C₄	—	47	2	no	—
125	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
126	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
127	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	47	24	no	—
129	C₂×C₄⋊C₄	—	47	12	no	—
130	C₂×C₄⋊C₄	—	94	24	no	—
131	C₂×C₄⋊C₄	—	94	12	no	—
132	C₂×C₄²	—	2	12	no	—
133	C₂×C₄²	—	2	12	no	—
134	C₂×C₄²	Dic₉₄	1	8	yes	complemented
135	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
136	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
137	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
138	#95	—	47	24	no	—
139	#95	—	47	24	no	—
140	#95	—	47	24	no	—
141	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
142	C₂×Q₈	—	94	24	no	—
143	C₂×Q₈	—	188	16	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	6	no	—
148	C₂²×C₄	—	47	4	no	—
149	C₂²×C₄	—	47	12	no	—
150	C₂²×C₄	—	94	12	no	—
151	C₂²×C₄	—	94	24	no	—
152	C₂²×C₄	—	94	12	no	—
153	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
154	C₂²×C₄	#10625350	1	12	yes	—
155	C₂²×C₄	#10625350	1	12	yes	—
156	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
157	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
158	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
159	C₄⋊C₄	—	94	48	no	—
160	C₄⋊C₄	—	94	24	no	—
161	C₄²	—	2	24	no	—
162	C₄²	—	2	24	no	—
163	C₄²	C₂×Dic₉₄	1	16	yes	complemented
164	C₂³	C₂²×Dic₉₄	1	2	yes	—
165	C₂³	C₂×#10625350	1	6	yes	—
166	C₂³	C₂×#10625350	1	6	yes	—
167	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
168	Q₈	—	188	16	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	24	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	94	12	no	—
176	C₂×C₄	—	94	24	no	—
177	C₂×C₄	—	94	12	no	—
178	C₂×C₄	—	188	8	no	—
179	C₂×C₄	—	188	24	no	—
180	C₂×C₄	#13094018	1	16	yes	—
181	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
182	C₂×C₄	C₂×#10625350	1	12	yes	—
183	C₂×C₄	C₂×#10625350	1	12	yes	—
184	C₂²	C₂×#13094018	1	4	yes	—
185	C₂²	#15689657	1	12	yes	—
186	C₂²	#15689680	1	12	yes	—
187	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
188	C₂²	C₂²×#10625350	1	3	yes	—
189	C₂²	C₂²×#10625350	1	3	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	2	12	no	—
192	C₄	—	188	4	no	—
193	C₄	—	188	12	no	—
194	C₄	C₂×#13094018	1	8	yes	—
195	C₂	#20251770	1	8	yes	direct product factor
196	C₂	C₂²×#13094018	1	1	yes	characteristic
197	C₂	C₂×#15689657	1	3	yes	—
198	C₂	C₂×#15689680	1	3	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×#2863
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
1. #20251770	C₂	direct (N × H)
37. C₂×C₄²×C₄₇	Q₈	semidirect
61. C₄²×C₄₇	C₂×Q₈	semidirect
102. C₂×C₄₇	#2863	semidirect
120. C₄₇	C₂×#2863	semidirect
134. C₂×C₄²	Dic₉₄	semidirect
163. C₄²	C₂×Dic₉₄	semidirect
195. C₂	#20251770	direct (N × H)

Maximal subgroups

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

subgroup	index
#20251770	2
C₂×#15689029	2
C₂×#15689021	2
C₂×#15689007	2
C₂×#15688991	2
C₂×C₄³×C₄₇	2
C₂×#2863	47

Subgroups — C2×#20251770 #26422296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251770 #26422296