Subgroups — C₂³×C₄⋊#3622 #17897932

Order 4352.

3870 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#3622	C₂	1	2	yes	maximal
2	C₂²×C₄⋊#3622	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3622	—	2	16	no	—
5	C₂³×#3622	C₂²	1	28	yes	—
6	C₂×C₄⋊#3622	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₄×C₁₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₇	C₄	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₄	1	14	yes	—
11	C₂²×C₄²×C₁₇	C₂²	1	14	yes	—
12	C₂⁵×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	—	2	8	no	—
14	C₂³×C₄×C₁₇	C₈	1	16	yes	complemented
15	C₂³×C₄×C₁₇	C₂×C₄	1	14	yes	—
16	C₂³×C₄×C₁₇	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₁₇	C₂×C₄	1	14	yes	—
18	C₂³×C₄×C₁₇	D₄	1	8	yes	—
19	C₂³×C₄×C₁₇	Q₈	1	8	yes	—
20	C₂³×C₄×C₁₇	C₂³	1	7	yes	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	84	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	28	yes	—
23	C₂²×#3622	—	2	112	no	—
24	C₂²×#3622	C₂³	1	56	yes	—
25	C₄⋊#3622	C₂³	1	64	yes	direct product factor
26	C₂⁴×C₁₇	C₄⋊C₄	1	8	yes	—
27	C₂⁴×C₁₇	C₂×C₈	1	8	yes	—
28	C₂⁴×C₁₇	M₁₆	1	8	yes	—
29	C₂⁴×C₁₇	C₂²×C₄	1	7	yes	—
30	C₂²×C₄×C₁₇	—	2	56	no	—
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	56	yes	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	112	yes	complemented
33	C₂²×C₄×C₁₇	C₂²×C₄	1	28	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	21	yes	—
35	C₂²×C₄×C₁₇	C₂²×C₄	1	28	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	28	yes	—
37	C₂²×C₄×C₁₇	C₂×Q₈	1	28	yes	—
38	C₂²×C₄×C₁₇	C₂⁴	1	7	yes	—
39	C₄²×C₁₇	C₂²×C₄	1	56	yes	—
40	C₄²×C₁₇	C₂⁴	1	8	yes	—
41	C₂×#3622	—	2	112	no	—
42	C₂×#3622	C₂⁴	1	16	yes	—
43	C₂³×C₄⋊C₈	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₄⋊C₈	1	64	yes	complemented
45	C₂³×C₁₇	C₂×C₄⋊C₄	1	28	yes	—
46	C₂³×C₁₇	C₂²×C₈	1	28	yes	—
47	C₂³×C₁₇	C₂×M₁₆	1	28	yes	—
48	C₂³×C₁₇	C₂³×C₄	1	7	yes	—
49	C₂×C₄×C₁₇	—	2	56	no	—
50	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	84	yes	—
51	C₂×C₄×C₁₇	C₂²×C₈	1	112	yes	complemented
52	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
53	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
54	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
55	C₂×C₄×C₁₇	C₂²×D₄	1	14	yes	—
56	C₂×C₄×C₁₇	C₂²×Q₈	1	14	yes	—
57	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
58	#3622	—	2	16	no	—
59	C₂⁴×C₈	—	17	2	no	—
60	C₂³×C₄²	D₁₇	1	1	yes	characteristic
61	C₂²×C₄⋊C₈	—	17	28	no	—
62	C₂²×C₁₇	C₂×C₄⋊C₈	1	112	yes	complemented
63	C₂²×C₁₇	C₂²×C₄⋊C₄	1	14	yes	—
64	C₂²×C₁₇	C₂³×C₈	1	14	yes	—
65	C₂²×C₁₇	C₂²×M₁₆	1	14	yes	—
66	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
67	C₄×C₁₇	—	2	8	no	—
68	C₄×C₁₇	C₂²×C₄⋊C₄	1	14	yes	—
69	C₄×C₁₇	C₂³×C₈	1	16	yes	complemented
70	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₁₇	C₂³×Q₈	1	1	yes	characteristic
72	C₂⁴×C₄	Dic₁₇	1	1	yes	characteristic
73	C₂⁴×C₄	Dic₁₇	1	1	yes	characteristic
74	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
75	C₂³×C₈	—	17	28	no	—
76	C₂³×C₈	—	34	16	no	—
77	C₂²×C₄²	Dic₁₇	1	14	yes	—
78	C₂²×C₄²	C₂×D₁₇	1	14	yes	—
79	C₂×C₄⋊C₈	—	17	112	no	—
80	C₂×C₁₇	C₂²×C₄⋊C₈	1	28	yes	complemented
81	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
83	C₂×C₁₇	C₂³×M₁₆	1	1	yes	characteristic
84	C₂⁵	C₂×Dic₁₇	1	1	yes	characteristic
85	C₂³×C₄	—	2	8	no	—
86	C₂³×C₄	#3622	1	16	yes	complemented
87	C₂³×C₄	Dic₃₄	1	8	yes	—
88	C₂³×C₄	D₆₈	1	8	yes	—
89	C₂³×C₄	C₂×Dic₁₇	1	14	yes	—
90	C₂³×C₄	C₂×Dic₁₇	1	14	yes	—
91	C₂³×C₄	C₂×Dic₁₇	1	7	yes	—
92	C₂³×C₄	C₂²×D₁₇	1	7	yes	—
93	C₂²×C₈	—	17	56	no	—
94	C₂²×C₈	—	34	112	no	—
95	C₂×C₄²	C₂×Dic₁₇	1	84	yes	—
96	C₂×C₄²	C₂²×D₁₇	1	28	yes	—
97	C₄⋊C₈	—	17	64	no	—
98	C₁₇	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
99	C₂⁴	C₂×#3622	1	8	yes	—
100	C₂⁴	#63223	1	8	yes	—
101	C₂⁴	C₄⋊Dic₁₇	1	8	yes	—
102	C₂⁴	C₂²×Dic₁₇	1	7	yes	—
103	C₂²×C₄	—	2	56	no	—
104	C₂²×C₄	C₂×#3622	1	112	yes	complemented
105	C₂²×C₄	C₄⋊Dic₁₇	1	56	yes	—
106	C₂²×C₄	C₂×Dic₃₄	1	28	yes	—
107	C₂²×C₄	C₂×D₆₈	1	28	yes	—
108	C₂²×C₄	C₂²×Dic₁₇	1	28	yes	—
109	C₂²×C₄	C₂²×Dic₁₇	1	28	yes	—
110	C₂²×C₄	C₂²×Dic₁₇	1	21	yes	—
111	C₂²×C₄	C₂³×D₁₇	1	7	yes	—
112	C₂×C₈	—	17	16	no	—
113	C₂×C₈	—	34	112	no	—
114	C₄²	C₂²×Dic₁₇	1	56	yes	—
115	C₄²	C₂³×D₁₇	1	8	yes	—
116	C₂³	C₄⋊#3622	1	64	yes	direct product factor
117	C₂³	C₂²×#3622	1	28	yes	—
118	C₂³	C₂×#63223	1	28	yes	—
119	C₂³	C₂×C₄⋊Dic₁₇	1	28	yes	—
120	C₂³	C₂³×Dic₁₇	1	7	yes	—
121	C₂×C₄	—	2	56	no	—
122	C₂×C₄	C₂²×#3622	1	112	yes	complemented
123	C₂×C₄	C₂×C₄⋊Dic₁₇	1	84	yes	—
124	C₂×C₄	C₂²×Dic₃₄	1	14	yes	—
125	C₂×C₄	C₂²×D₆₈	1	14	yes	—
126	C₂×C₄	C₂³×Dic₁₇	1	8	yes	—
127	C₂×C₄	C₂³×Dic₁₇	1	7	yes	—
128	C₂×C₄	C₂³×Dic₁₇	1	8	yes	—
129	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
130	C₈	—	34	16	no	—
131	C₂²	C₂×C₄⋊#3622	1	112	yes	direct product factor
132	C₂²	C₂²×C₄⋊Dic₁₇	1	14	yes	—
133	C₂²	C₂³×#3622	1	14	yes	—
134	C₂²	C₂²×#63223	1	14	yes	—
135	C₂²	C₂⁴×Dic₁₇	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	C₂²×C₄⋊Dic₁₇	1	14	yes	—
138	C₄	C₂³×#3622	1	16	yes	complemented
139	C₄	C₂³×D₆₈	1	1	yes	characteristic
140	C₄	C₂³×Dic₃₄	1	1	yes	characteristic
141	C₂	C₂²×C₄⋊#3622	1	28	yes	direct product factor
142	C₂	C₂³×C₄⋊Dic₁₇	1	1	yes	characteristic
143	C₂	C₂⁴×#3622	1	1	yes	characteristic
144	C₂	C₂³×#63223	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂³×C₄²×C₁₇	C₂
socle, prime radical	C₂⁵×C₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂³×C₄²	D₁₇
hypocenter	C₁₇	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊#3622	C₂	direct (N × H)
6. C₂×C₄⋊#3622	C₂²	direct (N × H)
14. C₂³×C₄×C₁₇	C₈	semidirect
25. C₄⋊#3622	C₂³	direct (N × H)
32. C₂²×C₄×C₁₇	C₂×C₈	semidirect
44. C₂³×C₁₇	C₄⋊C₈	semidirect
51. C₂×C₄×C₁₇	C₂²×C₈	semidirect
62. C₂²×C₁₇	C₂×C₄⋊C₈	semidirect
69. C₄×C₁₇	C₂³×C₈	semidirect
80. C₂×C₁₇	C₂²×C₄⋊C₈	semidirect
86. C₂³×C₄	#3622	semidirect
98. C₁₇	C₂³×C₄⋊C₈	semidirect
104. C₂²×C₄	C₂×#3622	semidirect
116. C₂³	C₄⋊#3622	direct (N × H)
122. C₂×C₄	C₂²×#3622	semidirect
131. C₂²	C₂×C₄⋊#3622	direct (N × H)
138. C₄	C₂³×#3622	semidirect
141. C₂	C₂²×C₄⋊#3622	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#3622	2
C₂²×C₄⋊#3622	2
C₂³×C₄²×C₁₇	2
C₂³×C₄⋊C₈	17

Subgroups — C23×C4⋊#3622 #17897932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊#3622 #17897932