Subgroups — C₂×#20251766 #26422291

Order 12032.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	2	yes	maximal
2	#20251766	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689029	C₂	1	8	yes	maximal
4	C₂×#15688989	C₂	1	4	yes	maximal
5	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₉₄	C₂²	1	4	yes	—
7	C₂×#13094024	—	2	16	no	—
8	C₂×#13094024	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	#15689029	C₂²	1	64	yes	—
12	#15688989	C₂²	1	32	yes	—
13	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₄³×C₄₇	C₂²	1	8	yes	—
18	C₂²×Dic₉₄	—	2	16	no	—
19	C₂²×Dic₉₄	—	2	32	no	—
20	C₂²×Dic₉₄	C₂³	1	16	yes	—
21	C₂³×Dic₄₇	C₂³	1	8	yes	—
22	#13094024	—	2	64	no	—
23	C₂×C₄⋊Dic₄₇	—	2	16	no	—
24	C₂×C₄⋊Dic₄₇	—	2	16	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
27	C₄₇⋊#95	C₂³	1	64	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₄₇	—	2	16	no	—
32	C₂×C₄²×C₄₇	D₄	1	8	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	16	yes	complemented
34	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
37	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	16	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
42	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
43	C₂²×C₄×C₄₇	#41	1	16	yes	—
44	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
47	C₂²×Dic₄₇	—	2	16	no	—
48	C₂²×Dic₄₇	—	2	32	no	—
49	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
50	C₂×Dic₉₄	—	2	32	no	—
51	C₂×Dic₉₄	—	2	64	no	—
52	C₂×Dic₉₄	—	4	64	no	—
53	C₄²×C₄₇	—	2	32	no	—
54	C₄²×C₄₇	C₂×D₄	1	16	yes	—
55	C₄²×C₄₇	C₂×Q₈	1	32	yes	complemented
56	C₄⋊Dic₄₇	—	2	32	no	—
57	C₄⋊Dic₄₇	—	2	32	no	—
58	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
59	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
60	C₂³×C₄₇	C₂×#41	1	8	yes	—
61	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	32	no	—
63	C₂×C₄×C₄₇	—	2	16	no	—
64	C₂×C₄×C₄₇	#124	1	32	yes	—
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	—
66	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
67	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
68	C₂×C₄×C₄₇	C₂²×Q₈	1	16	yes	—
69	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
70	C₂×C₄×C₄₇	C₂×#41	1	16	yes	—
71	C₂×Dic₄₇	—	2	16	no	—
72	C₂×Dic₄₇	—	2	32	no	—
73	C₂×Dic₄₇	—	4	32	no	—
74	Dic₉₄	—	4	64	no	—
75	C₂×#2862	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	C₂×#124	1	8	yes	—
77	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
78	C₂²×C₄₇	#533	1	16	yes	—
79	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
80	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
81	C₂²×C₄₇	C₂²×#41	1	4	yes	—
82	C₄×C₄₇	—	2	16	no	—
83	C₄×C₄₇	C₂×#124	1	16	yes	—
84	C₄×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
85	Dic₄₇	—	4	16	no	—
86	C₂²×C₄⋊Q₈	—	47	2	no	—
87	#2862	—	47	16	no	—
88	C₂×#391	—	47	8	no	—
89	C₂×#382	—	47	4	no	—
90	C₂×C₄³	D₄₇	1	1	yes	characteristic
91	C₂×C₄₇	#2862	1	8	yes	complemented
92	C₂×C₄₇	C₂²×#124	1	2	yes	—
93	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄₇	C₂×#533	1	4	yes	—
95	C₂³×Q₈	—	47	4	no	—
96	C₂×C₄⋊Q₈	—	47	16	no	—
97	C₂×C₄⋊Q₈	—	94	16	no	—
98	C₂²×C₄⋊C₄	—	47	4	no	—
99	C₂²×C₄⋊C₄	—	47	4	no	—
100	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
101	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
102	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	#391	—	47	64	no	—
104	#382	—	47	32	no	—
105	C₂×#95	—	47	16	no	—
106	C₄³	C₂×D₄₇	1	8	yes	—
107	C₄₇	C₂×#2862	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	47	16	no	—
109	C₂²×Q₈	—	94	16	no	—
110	C₂²×Q₈	—	94	32	no	—
111	C₂³×C₄	—	47	8	no	—
112	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
113	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
115	C₄⋊Q₈	—	94	64	no	—
116	C₂×C₄⋊C₄	—	47	16	no	—
117	C₂×C₄⋊C₄	—	47	16	no	—
118	C₂×C₄⋊C₄	—	94	16	no	—
119	C₂×C₄⋊C₄	—	94	16	no	—
120	C₂×C₄²	—	2	16	no	—
121	C₂×C₄²	Dic₉₄	1	16	yes	complemented
122	C₂×C₄²	D₁₈₈	1	8	yes	—
123	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
124	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
125	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
126	#95	—	47	64	no	—
127	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
128	C₂×Q₈	—	94	32	no	—
129	C₂×Q₈	—	94	64	no	—
130	C₂×Q₈	—	188	64	no	—
131	C₂²×C₄	—	2	16	no	—
132	C₂²×C₄	—	47	16	no	—
133	C₂²×C₄	—	94	16	no	—
134	C₂²×C₄	—	94	32	no	—
135	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
136	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
137	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
138	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
139	C₂²×C₄	#10625350	1	16	yes	—
140	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
141	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
142	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
143	C₄⋊C₄	—	94	32	no	—
144	C₄⋊C₄	—	94	32	no	—
145	C₄²	—	2	32	no	—
146	C₄²	C₂×Dic₉₄	1	32	yes	complemented
147	C₄²	C₂×D₁₈₈	1	16	yes	—
148	C₂³	C₂²×D₁₈₈	1	2	yes	—
149	C₂³	C₂²×Dic₉₄	1	4	yes	—
150	C₂³	C₂×#10625350	1	8	yes	—
151	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
152	Q₈	—	188	64	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	94	16	no	—
156	C₂×C₄	—	94	32	no	—
157	C₂×C₄	—	188	32	no	—
158	C₂×C₄	#13094012	1	32	yes	—
159	C₂×C₄	#13094024	1	16	yes	—
160	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
161	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
162	C₂×C₄	C₂²×Dic₉₄	1	16	yes	—
163	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
164	C₂×C₄	C₂×#10625350	1	16	yes	—
165	C₂²	C₂×#13094012	1	8	yes	—
166	C₂²	C₂×#13094024	1	4	yes	—
167	C₂²	#15689680	1	16	yes	—
168	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
169	C₂²	C₂³×Dic₉₄	1	2	yes	—
170	C₂²	C₂²×#10625350	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	188	16	no	—
173	C₄	C₂×#13094012	1	16	yes	—
174	C₄	C₂×#13094024	1	8	yes	—
175	C₂	#20251766	1	8	yes	direct product factor
176	C₂	C₂²×#13094012	1	2	yes	—
177	C₂	C₂²×#13094024	1	1	yes	characteristic
178	C₂	C₂×#15689680	1	4	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×#2862
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. #20251766	C₂	direct (N × H)
33. C₂×C₄²×C₄₇	Q₈	semidirect
55. C₄²×C₄₇	C₂×Q₈	semidirect
91. C₂×C₄₇	#2862	semidirect
107. C₄₇	C₂×#2862	semidirect
121. C₂×C₄²	Dic₉₄	semidirect
146. C₄²	C₂×Dic₉₄	semidirect
175. C₂	#20251766	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×#13094024	2
#20251766	2
C₂×#15689029	2
C₂×#15688989	2
C₂×C₄³×C₄₇	2
C₂×#2862	47

Subgroups — C2×#20251766 #26422291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251766 #26422291