Subgroups — C₂²×#15689525 #27085872

Order 12032.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 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	C₂×#15689525	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093958	C₂	1	2	yes	maximal
4	C₂²×#13093956	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	—	2	12	no	—
7	C₂×#13094016	C₂²	1	24	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂²×Dic₄₇⋊C₄	—	2	4	no	—
10	#15689525	C₂²	1	64	yes	direct product factor
11	C₂×#13093958	C₂²	1	24	yes	—
12	C₂×#13093956	C₂²	1	24	yes	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	12	yes	—
17	C₂³×Dic₄₇	—	2	4	no	—
18	#13094016	—	2	48	no	—
19	#13094016	C₂³	1	32	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	12	no	—
21	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
22	C₂×Dic₄₇⋊C₄	—	2	48	no	—
23	#13093958	C₂³	1	32	yes	—
24	#13093956	C₂³	1	32	yes	—
25	C₂³×C₄×C₄₇	D₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
28	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
29	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
31	C₂×C₄²×C₄₇	D₄	1	6	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
33	C₄×C₈×C₄₇	C₂³	1	16	yes	—
34	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
41	C₂²×Dic₄₇	—	2	24	no	—
42	C₂²×Dic₄₇	—	4	16	no	—
43	C₂×C₈×C₄₇	C₂×Q₈	1	24	yes	—
44	C₂×C₈×C₄₇	C₂×Q₈	1	24	yes	—
45	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
46	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
47	C₄²×C₄₇	C₂×D₄	1	12	yes	—
48	C₄²×C₄₇	C₂⁴	1	4	yes	—
49	C₄⋊Dic₄₇	—	2	24	no	—
50	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
51	Dic₄₇⋊C₄	—	2	64	no	—
52	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
53	C₂³×C₄₇	#135	1	8	yes	—
54	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
55	C₂×C₄×C₄₇	—	2	24	no	—
56	C₂×C₄×C₄₇	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
58	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
59	C₂×C₄×C₄₇	C₂²×Q₈	1	6	yes	—
60	C₂×C₄×C₄₇	C₂²×Q₈	1	6	yes	—
61	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
62	C₂×Dic₄₇	—	2	16	no	—
63	C₂×Dic₄₇	—	4	48	no	—
64	C₈×C₄₇	C₂²×Q₈	1	8	yes	—
65	C₈×C₄₇	C₂²×Q₈	1	8	yes	—
66	C₂²×#499	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	#499	1	16	yes	complemented
68	C₂²×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
69	C₂²×C₄₇	C₂×#135	1	12	yes	—
70	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₄₇	—	2	8	no	—
72	C₄×C₄₇	C₂×C₄⋊Q₈	1	6	yes	—
73	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
74	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
75	Dic₄₇	—	4	16	no	—
76	C₂²×#125	—	47	2	no	—
77	C₂×#499	—	47	24	no	—
78	C₂²×#107	—	47	2	no	—
79	C₂²×#106	—	47	2	no	—
80	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
81	C₂×C₄₇	C₂×#499	1	12	yes	complemented
82	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄₇	C₂²×#135	1	2	yes	—
84	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
85	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
86	C₂×#125	—	47	24	no	—
87	C₂×#125	—	94	12	no	—
88	C₂²×C₄⋊C₄	—	47	4	no	—
89	C₂²×C₄⋊C₄	—	94	4	no	—
90	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
91	#499	—	47	64	no	—
92	C₂×#107	—	47	24	no	—
93	C₂×#106	—	47	24	no	—
94	C₂×C₄×C₈	C₂×D₄₇	1	12	yes	—
95	C₄₇	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₄	—	94	4	no	—
97	C₂³×C₄	D₁₈₈	1	2	yes	—
98	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂²×C₈	Dic₉₄	1	8	yes	—
100	C₂²×C₈	Dic₉₄	1	8	yes	—
101	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
102	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
103	#125	—	47	32	no	—
104	#125	—	94	48	no	—
105	C₂×C₄⋊C₄	—	47	24	no	—
106	C₂×C₄⋊C₄	—	94	12	no	—
107	C₂×C₄⋊C₄	—	94	48	no	—
108	C₂×C₄²	D₁₈₈	1	6	yes	—
109	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
110	#107	—	47	32	no	—
111	#106	—	47	32	no	—
112	C₄×C₈	C₂²×D₄₇	1	16	yes	—
113	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	94	24	no	—
116	C₂²×C₄	—	188	16	no	—
117	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
118	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
119	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
120	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
121	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
122	C₂×C₈	C₂×Dic₉₄	1	24	yes	—
123	C₂×C₈	C₂×Dic₉₄	1	24	yes	—
124	C₂×C₈	C₂³×D₄₇	1	4	yes	—
125	C₂×C₈	C₂³×D₄₇	1	4	yes	—
126	C₄⋊C₄	—	47	16	no	—
127	C₄⋊C₄	—	94	64	no	—
128	C₄⋊C₄	—	94	24	no	—
129	C₄²	C₂×D₁₈₈	1	12	yes	—
130	C₄²	C₂³×D₄₇	1	4	yes	—
131	C₂³	#13094024	1	4	yes	—
132	C₂³	#13094046	1	8	yes	—
133	C₂³	C₂²×D₁₈₈	1	3	yes	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	94	16	no	—
136	C₂×C₄	—	188	48	no	—
137	C₂×C₄	#13094024	1	12	yes	—
138	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
139	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
140	C₂×C₄	C₂²×Dic₉₄	1	6	yes	—
141	C₂×C₄	C₂²×Dic₉₄	1	6	yes	—
142	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
143	C₈	C₂²×Dic₉₄	1	8	yes	—
144	C₈	C₂²×Dic₉₄	1	8	yes	—
145	C₂²	#15689525	1	16	yes	direct product factor
146	C₂²	C₂×#13094024	1	6	yes	—
147	C₂²	C₂×#13094046	1	12	yes	—
148	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	188	16	no	—
151	C₄	C₂×#13094024	1	6	yes	—
152	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
153	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
154	C₂	C₂×#15689525	1	12	yes	direct product factor
155	C₂	C₂²×#13094024	1	1	yes	characteristic
156	C₂	C₂²×#13094046	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂²×#499
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. C₂×#15689525	C₂	direct (N × H)
10. #15689525	C₂²	direct (N × H)
67. C₂²×C₄₇	#499	semidirect
81. C₂×C₄₇	C₂×#499	semidirect
95. C₄₇	C₂²×#499	semidirect
145. C₂²	#15689525	direct (N × H)
154. C₂	C₂×#15689525	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
C₂×#15689525	2
C₂²×#13093958	2
C₂²×#13093956	2
C₂²×C₄×C₈×C₄₇	2
C₂²×#499	47

Subgroups — C22×#15689525 #27085872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689525 #27085872