Subgroups — C₂²×#15688953 #26369330

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	2	yes	maximal
2	C₂×#15688953	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	C₂×#13093960	—	2	6	no	—
6	C₂×#13093960	C₂²	1	12	yes	—
7	#15688953	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₄₇	C₄	1	6	yes	—
10	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
11	C₂²×C₁₆×C₄₇	C₂²	1	6	yes	—
12	C₂×#10625329	—	2	12	no	—
13	C₂×#10625329	—	4	6	no	—
14	C₂²×C₄₇⋊C₈	—	4	2	no	—
15	#13093960	—	2	12	no	—
16	#13093960	C₂³	1	8	yes	—
17	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	D₄	1	3	yes	—
21	C₂²×C₈×C₄₇	Q₈	1	3	yes	—
22	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
24	C₂×C₁₆×C₄₇	C₂×C₄	1	12	yes	—
25	C₂×C₁₆×C₄₇	C₂×C₄	1	12	yes	—
26	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
27	C₂⁴×C₄₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₄⋊C₄	1	3	yes	—
29	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	D₈	1	3	yes	—
31	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	Q₁₆	1	3	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	6	yes	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	3	yes	—
36	C₂×C₈×C₄₇	C₄⋊C₄	1	3	yes	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
38	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
39	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
40	C₂×C₈×C₄₇	C₂×Q₈	1	3	yes	—
41	C₂×C₈×C₄₇	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
43	#10625329	—	2	8	no	—
44	#10625329	—	4	12	no	—
45	C₂×C₄₇⋊C₈	—	4	12	no	—
46	C₁₆×C₄₇	C₂²×C₄	1	8	yes	—
47	C₂³×C₄₇	—	2	4	no	—
48	C₂³×C₄₇	#107	1	3	yes	—
49	C₂³×C₄₇	#107	1	1	yes	characteristic
50	C₂³×C₄₇	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄×C₄₇	#107	1	6	yes	—
52	C₂×C₄×C₄₇	#107	1	3	yes	—
53	C₂×C₄×C₄₇	#107	1	3	yes	—
54	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₄₇	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
60	C₈×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
61	C₈×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
64	C₄₇⋊C₈	—	4	8	no	—
65	C₂²×#368	—	47	1	no	maximal, Sylow
66	C₂²×C₄₇	—	2	6	no	—
67	C₂²×C₄₇	—	2	6	no	—
68	C₂²×C₄₇	#368	1	4	yes	complemented
69	C₂²×C₄₇	C₂×#107	1	3	yes	—
70	C₂²×C₄₇	C₂×#107	1	3	yes	—
71	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
72	C₄×C₄₇	C₂×#107	1	3	yes	—
73	C₄×C₄₇	C₂×#107	1	3	yes	—
74	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
76	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
77	C₂²×#108	—	47	2	no	—
78	C₂×#368	—	47	12	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	—	2	3	no	—
81	C₂×C₄₇	C₂×#368	1	6	yes	complemented
82	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
83	C₂²×M₁₆	—	94	2	no	—
84	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
85	C₂²×C₁₆	Dic₄₇	1	2	yes	—
86	C₂²×C₁₆	Dic₄₇	1	6	yes	—
87	C₂²×C₁₆	C₂×D₄₇	1	6	yes	—
88	C₂×#108	—	47	12	no	—
89	C₂×#108	—	94	6	no	—
90	#368	—	47	16	no	—
91	C₄₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
93	C₂×M₁₆	—	94	12	no	—
94	C₂×M₁₆	—	188	6	no	—
95	C₂²×C₈	—	188	2	no	—
96	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
97	C₂²×C₈	Dic₉₄	1	3	yes	—
98	C₂²×C₈	D₁₈₈	1	3	yes	—
99	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
100	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
101	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
102	C₂×C₁₆	C₂×Dic₄₇	1	12	yes	—
103	C₂×C₁₆	C₂×Dic₄₇	1	12	yes	—
104	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
105	#108	—	47	8	no	—
106	#108	—	94	12	no	—
107	C₂⁴	Dic₁₈₈	1	1	yes	characteristic
108	C₂²×C₄	D₃₇₆	1	3	yes	—
109	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
110	C₂²×C₄	Dic₁₈₈	1	3	yes	—
111	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₄₇	1	3	yes	—
113	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
114	M₁₆	—	94	8	no	—
115	M₁₆	—	188	12	no	—
116	C₂×C₈	—	188	12	no	—
117	C₂×C₈	C₄⋊Dic₄₇	1	6	yes	—
118	C₂×C₈	C₄⋊Dic₄₇	1	3	yes	—
119	C₂×C₈	C₄⋊Dic₄₇	1	3	yes	—
120	C₂×C₈	C₂×Dic₉₄	1	3	yes	—
121	C₂×C₈	C₂×Dic₉₄	1	3	yes	—
122	C₂×C₈	C₂×D₁₈₈	1	3	yes	—
123	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
124	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
125	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₄₇	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#13093958	1	3	yes	—
129	C₂³	#13093958	1	1	yes	characteristic
130	C₂³	C₂×Dic₁₈₈	1	3	yes	—
131	C₂×C₄	#13093958	1	6	yes	—
132	C₂×C₄	#13093958	1	3	yes	—
133	C₂×C₄	#13093958	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
136	C₂×C₄	C₂×D₃₇₆	1	3	yes	—
137	C₂×C₄	C₂×D₃₇₆	1	3	yes	—
138	C₂×C₄	C₂×Dic₁₈₈	1	3	yes	—
139	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
140	C₈	—	188	8	no	—
141	C₈	C₂×C₄⋊Dic₄₇	1	3	yes	—
142	C₈	C₂×C₄⋊Dic₄₇	1	3	yes	—
143	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
144	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#15688953	1	4	yes	direct product factor
148	C₂²	C₂×#13093958	1	3	yes	—
149	C₂²	C₂×#13093958	1	3	yes	—
150	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
151	C₄	C₂×#13093958	1	3	yes	—
152	C₄	C₂×#13093958	1	3	yes	—
153	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
154	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	3	no	—
157	C₂	C₂×#15688953	1	6	yes	direct product factor
158	C₂	C₂²×#13093958	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₇₆
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle	C₂³×C₄₇	#107
prime radical	C₂⁴×C₄₇	Q₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	C₂²×#368
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15688953	C₂	direct (N × H)
7. #15688953	C₂²	direct (N × H)
68. C₂²×C₄₇	#368	semidirect
81. C₂×C₄₇	C₂×#368	semidirect
91. C₄₇	C₂²×#368	semidirect
147. C₂²	#15688953	direct (N × H)
157. C₂	C₂×#15688953	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×#13093960	2
C₂×#15688953	2
C₂³×C₁₆×C₄₇	2
C₂²×#368	47

Subgroups — C22×#15688953 #26369330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688953 #26369330