Subgroups — C₂×C₂²⋊#88034 #15783110

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88034	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#88034	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×C₁₆	—	2	4	no	—
6	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₄	1	2	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#88034	—	2	4	no	—
11	C₂×#88034	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	—	2	2	no	—
14	C₂²×C₈×C₁₃	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₈	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	—	2	2	no	—
20	C₂×C₁₃×C₁₆	—	2	4	no	—
21	C₂×C₁₃×C₁₆	—	2	4	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
25	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
26	#88034	—	2	4	no	—
27	C₂×C₂²⋊C₃₂	—	13	1	no	maximal, Sylow
28	C₂⁴×C₁₃	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	C₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₁₆	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	—	2	4	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×C₁₆	C₂²⋊C₄	1	2	yes	—
46	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
47	C₂³×C₁₆	D₁₃	1	1	yes	characteristic
48	C₂²×C₃₂	—	13	2	no	—
49	C₂²⋊C₃₂	—	13	4	no	—
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	C₃₂	1	4	yes	complemented
53	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
54	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
63	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
66	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
69	C₂²×C₁₆	—	2	4	no	—
70	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
71	C₂²×C₁₆	Dic₁₃	1	2	yes	—
72	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
73	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
74	C₂×C₃₂	—	13	4	no	—
75	C₂×C₃₂	—	26	4	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	—	2	8	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
80	C₂²×C₁₃	C₂×C₃₂	1	2	yes	—
81	C₂²×C₁₃	C₂×C₃₂	1	4	yes	complemented
82	C₂²×C₁₃	M₆₄	1	2	yes	—
83	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
86	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
88	C₂³×C₄	#1055	1	1	yes	characteristic
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	#1055	1	1	yes	characteristic
92	C₂²×C₈	#1055	1	2	yes	—
93	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
94	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	4	no	—
97	C₂×C₁₆	—	2	4	no	—
98	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	—
101	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
102	C₃₂	—	26	4	no	—
103	C₂×C₁₃	—	2	4	no	—
104	C₂×C₁₃	C₂²⋊C₃₂	1	4	yes	complemented
105	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
106	C₂×C₁₃	C₂²×C₃₂	1	1	yes	characteristic
107	C₂×C₁₃	C₂×M₆₄	1	1	yes	characteristic
108	C₂⁴	#6090	1	1	yes	characteristic
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	#6090	1	1	yes	characteristic
112	C₂²×C₄	#6090	1	2	yes	—
113	C₂²×C₄	C₂×#1055	1	2	yes	—
114	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₂×#1055	1	2	yes	—
120	C₂×C₈	C₂×#1055	1	1	yes	characteristic
121	C₂×C₈	#6099	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
123	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	C₂²⋊Dic₁₃	1	2	yes	—
126	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
127	C₁₃	C₂×C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	#88034	1	4	yes	complemented
131	C₂³	C₂×#6090	1	2	yes	—
132	C₂³	C₂×#6090	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	C₂×#6090	1	2	yes	—
138	C₂×C₄	C₂×#6090	1	1	yes	characteristic
139	C₂×C₄	#88052	1	2	yes	—
140	C₂×C₄	C₂²⋊#1055	1	2	yes	—
141	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	C₂²⋊#1055	1	2	yes	—
144	C₈	C₂×#6099	1	1	yes	characteristic
145	C₈	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	8	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂×#88034	1	2	yes	—
150	C₂²	C₂×#88034	1	4	yes	complemented
151	C₂²	#11718091	1	2	yes	—
152	C₂²	C₂²⋊#6090	1	2	yes	—
153	C₂²	C₂²×#6090	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	C₂²⋊#6090	1	2	yes	—
156	C₄	C₂×#88052	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
158	C₂	—	2	4	no	—
159	C₂	C₂²⋊#88034	1	4	yes	direct product factor
160	C₂	C₂×C₂²⋊#6090	1	1	yes	characteristic
161	C₂	C₂²×#88034	1	1	yes	characteristic
162	C₂	C₂×#11718091	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₁₃
commutator	C₂×C₁₃	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₂³×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂×C₁₆
prime radical	C₂⁴×C₁₃	C₁₆
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂³×C₁₆	D₁₃
hypocenter	C₁₃	C₂×C₂²⋊C₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88034	C₂	semidirect
2. C₂²⋊#88034	C₂	direct (N × H)
11. C₂×#88034	C₂²	semidirect
52. C₂³×C₁₃	C₃₂	semidirect
81. C₂²×C₁₃	C₂×C₃₂	semidirect
104. C₂×C₁₃	C₂²⋊C₃₂	semidirect
127. C₁₃	C₂×C₂²⋊C₃₂	semidirect
130. C₂³	#88034	semidirect
150. C₂²	C₂×#88034	semidirect
159. C₂	C₂²⋊#88034	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×#88034	2
C₂²⋊#88034	2
C₂³×C₁₃×C₁₆	2
C₂×C₂²⋊C₃₂	13

Subgroups — C2×C22⋊#88034 #15783110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#88034 #15783110