Subgroups — C₄×C₈×#3624 #17950261

Order 4352.

750 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#3624	C₂	1	2	yes	maximal
2	C₂×C₈×#3624	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#3624	C₄	1	8	yes	—
5	C₂×C₄×#3624	C₂²	1	4	yes	—
6	C₈×#3624	C₄	1	16	yes	direct product factor
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
10	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×#3624	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	C₂×C₄	1	4	yes	—
17	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
18	C₂²×#3622	C₂×C₄	1	2	yes	—
19	C₄×#3624	C₈	1	32	yes	direct product factor
20	C₄×#3624	C₂×C₄	1	16	yes	—
21	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
22	C₈×Dic₁₇	C₈	1	8	yes	complemented
23	C₈×Dic₁₇	C₂×C₄	1	8	yes	—
24	C₄×#3622	C₈	1	4	yes	complemented
25	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₁₇	C₄²	1	2	yes	—
28	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
29	C₂×#3624	C₄²	1	16	yes	—
30	C₂×#3624	C₂×C₈	1	32	yes	—
31	C₂×C₈×C₁₇	C₄²	1	4	yes	—
32	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
34	C₄²×C₁₇	C₂×C₈	1	2	yes	—
35	C₄²×C₁₇	C₂×C₈	1	2	yes	—
36	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
37	C₄×Dic₁₇	C₂×C₈	1	16	yes	—
38	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
39	C₂×#3622	C₄²	1	4	yes	—
40	C₂×#3622	C₂×C₈	1	4	yes	—
41	C₂×#3622	C₂×C₈	1	4	yes	—
42	C₄×C₈²	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄×C₈	1	8	yes	—
45	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
46	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
47	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
49	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₁₇	C₄×C₈	1	8	yes	—
51	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
52	C₂×Dic₁₇	C₂×C₄²	1	4	yes	—
53	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
54	#3624	C₄×C₈	1	32	yes	direct product factor
55	C₈×C₁₇	C₄×C₈	1	8	yes	complemented
56	#3622	C₄×C₈	1	8	yes	complemented
57	C₄²×C₈	—	17	2	no	—
58	C₄²×C₈	—	17	1	no	—
59	C₂×C₈²	—	17	4	no	—
60	C₂²×C₁₇	C₄³	1	1	yes	characteristic
61	C₂²×C₁₇	C₂×C₄×C₈	1	4	yes	—
62	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₁₇	C₈²	1	8	yes	complemented
65	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
66	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
67	Dic₁₇	C₈²	1	8	yes	complemented
68	Dic₁₇	C₂×C₄×C₈	1	8	yes	—
69	C₂×C₄×C₈	—	17	8	no	—
70	C₂×C₄×C₈	—	17	4	no	—
71	C₂×C₄×C₈	—	17	2	no	—
72	C₂×C₄×C₈	—	17	2	no	—
73	C₂×C₄×C₈	—	17	1	no	—
74	C₂×C₄×C₈	#595	1	1	yes	characteristic
75	C₄³	—	17	1	no	—
76	C₈²	—	17	16	no	—
77	C₂×C₁₇	C₂×C₈²	1	2	yes	—
78	C₂×C₁₇	C₂×C₈²	1	2	yes	—
79	C₂×C₁₇	C₄²×C₈	1	2	yes	—
80	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
81	C₂²×C₈	—	17	8	no	—
82	C₂²×C₈	—	17	2	no	—
83	C₂²×C₈	C₂×#595	1	2	yes	—
84	C₂×C₄²	—	17	4	no	—
85	C₂×C₄²	—	17	1	no	—
86	C₂×C₄²	—	17	1	no	—
87	C₂×C₄²	C₂×#595	1	1	yes	characteristic
88	C₄×C₈	—	17	16	no	—
89	C₄×C₈	—	17	32	no	—
90	C₄×C₈	—	17	8	no	—
91	C₄×C₈	—	17	8	no	—
92	C₄×C₈	—	17	4	no	—
93	C₄×C₈	#3624	1	4	yes	direct product factor
94	C₁₇	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	17	2	no	—
96	C₂²×C₄	—	17	2	no	—
97	C₂²×C₄	C₄×#595	1	2	yes	—
98	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
99	C₂×C₈	—	17	16	no	—
100	C₂×C₈	—	17	32	no	—
101	C₂×C₈	—	17	4	no	—
102	C₂×C₈	—	17	4	no	—
103	C₂×C₈	—	17	4	no	—
104	C₂×C₈	#63242	1	4	yes	—
105	C₂×C₈	C₄×#595	1	4	yes	—
106	C₂×C₈	C₂×#3624	1	4	yes	—
107	C₄²	—	17	4	no	—
108	C₄²	—	17	4	no	—
109	C₄²	—	17	16	no	—
110	C₄²	#63242	1	2	yes	—
111	C₄²	C₂×#3624	1	2	yes	—
112	C₂³	C₂×C₄×#595	1	1	yes	characteristic
113	C₂×C₄	—	17	4	no	—
114	C₂×C₄	—	17	8	no	—
115	C₂×C₄	—	17	8	no	—
116	C₂×C₄	—	17	4	no	—
117	C₂×C₄	C₈×#595	1	8	yes	—
118	C₂×C₄	C₄×#3624	1	4	yes	—
119	C₂×C₄	C₂×#63242	1	2	yes	—
120	C₂×C₄	C₂×#63242	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×#595	1	2	yes	—
122	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
123	C₈	—	17	32	no	—
124	C₈	—	17	8	no	—
125	C₈	C₄×#3624	1	8	yes	direct product factor
126	C₂²	C₄²×#595	1	1	yes	characteristic
127	C₂²	C₂×C₈×#595	1	4	yes	—
128	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
129	C₂²	C₄×#63242	1	1	yes	characteristic
130	C₄	—	17	8	no	—
131	C₄	—	17	8	no	—
132	C₄	C₈×#3624	1	8	yes	direct product factor
133	C₄	C₂×C₄×#3624	1	2	yes	—
134	C₄	C₄×#63242	1	2	yes	—
135	C₂	C₂×C₈×#3624	1	2	yes	—
136	C₂	C₈×#63242	1	2	yes	—
137	C₂	C₄×C₈×#595	1	2	yes	—
138	C₂	C₄²×#3624	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#595
commutator, hypocenter	C₁₇	C₄×C₈²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#3624	C₄	direct (N × H)
19. C₄×#3624	C₈	direct (N × H)
21. C₄×C₈×C₁₇	C₈	semidirect
22. C₈×Dic₁₇	C₈	semidirect
24. C₄×#3622	C₈	semidirect
54. #3624	C₄×C₈	direct (N × H)
55. C₈×C₁₇	C₄×C₈	semidirect
56. #3622	C₄×C₈	semidirect
64. C₄×C₁₇	C₈²	semidirect
67. Dic₁₇	C₈²	semidirect
93. C₄×C₈	#3624	direct (N × H)
94. C₁₇	C₄×C₈²	semidirect
125. C₈	C₄×#3624	direct (N × H)
132. C₄	C₈×#3624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×#3624	2
C₂×C₈×#3624	2
C₄×C₈×Dic₁₇	2
C₄×C₈²	17

Subgroups — C4×C8×#3624 #17950261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×#3624 #17950261