Subgroups — C₂²×C₈×#3624 #17963966

Order 4352.

1342 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#3624	C₂	1	2	yes	maximal
2	C₂×C₈×#3624	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3624	C₄	1	4	yes	—
5	C₂×C₄×#3624	C₄	1	12	yes	—
6	C₂×C₄×#3624	C₂²	1	12	yes	—
7	C₈×#3624	C₂²	1	16	yes	direct product factor
8	C₂³×#3622	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	6	yes	—
11	C₂×C₈×Dic₁₇	C₂²	1	6	yes	—
12	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×Dic₁₇	C₂×C₄	1	2	yes	—
15	C₂²×#3624	C₈	1	32	yes	direct product factor
16	C₂²×#3624	C₂×C₄	1	24	yes	—
17	C₂²×C₈×C₁₇	C₈	1	8	yes	complemented
18	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
19	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₁₇	C₂³	1	3	yes	—
22	C₂²×#3622	C₈	1	8	yes	complemented
23	C₂²×#3622	C₂×C₄	1	6	yes	—
24	C₄×#3624	C₂×C₄	1	24	yes	—
25	C₄×#3624	C₂³	1	8	yes	—
26	C₈×Dic₁₇	C₂×C₄	1	12	yes	—
27	C₈×Dic₁₇	C₂³	1	4	yes	—
28	C₂⁴×C₁₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₄²	1	3	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
33	C₂²×Dic₁₇	C₄²	1	6	yes	—
34	C₂²×Dic₁₇	C₂×C₈	1	16	yes	—
35	C₂²×Dic₁₇	C₂²×C₄	1	6	yes	—
36	C₂×#3624	C₂×C₈	1	96	yes	direct product factor
37	C₂×#3624	C₂²×C₄	1	16	yes	—
38	C₂×C₈×C₁₇	C₂×C₈	1	24	yes	complemented
39	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
40	C₄×Dic₁₇	C₄²	1	6	yes	—
41	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
42	C₄×Dic₁₇	C₂²×C₄	1	6	yes	—
43	C₄×Dic₁₇	C₂⁴	1	1	yes	characteristic
44	C₂×#3622	C₂×C₈	1	24	yes	complemented
45	C₂×#3622	C₂²×C₄	1	4	yes	—
46	C₂²×C₈²	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₄×C₈	1	8	yes	—
48	C₂³×C₁₇	C₄×C₈	1	4	yes	—
49	C₂³×C₁₇	C₂×C₄²	1	3	yes	—
50	C₂×C₄×C₁₇	C₄×C₈	1	12	yes	—
51	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
52	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
53	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
54	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	C₄×C₈	1	24	yes	—
56	C₂×Dic₁₇	C₂×C₄²	1	6	yes	—
57	C₂×Dic₁₇	C₂²×C₈	1	24	yes	—
58	C₂×Dic₁₇	C₂³×C₄	1	2	yes	—
59	#3624	C₂²×C₈	1	32	yes	direct product factor
60	C₈×C₁₇	C₂²×C₈	1	8	yes	complemented
61	#3622	C₂²×C₈	1	8	yes	complemented
62	C₂²×C₄×C₈	—	17	2	no	—
63	C₂²×C₄×C₈	—	17	1	no	—
64	C₂×C₈²	—	17	12	no	—
65	C₂²×C₁₇	C₈²	1	16	yes	complemented
66	C₂²×C₁₇	C₂×C₄×C₈	1	12	yes	—
67	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
68	C₂²×C₁₇	C₂²×C₄²	1	1	yes	characteristic
69	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
70	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
72	Dic₁₇	C₂×C₄×C₈	1	12	yes	—
73	Dic₁₇	C₂³×C₈	1	4	yes	—
74	C₂³×C₈	—	17	4	no	—
75	C₂³×C₈	—	17	1	no	—
76	C₂³×C₈	#595	1	1	yes	characteristic
77	C₂²×C₄²	—	17	1	no	—
78	C₂×C₄×C₈	—	17	12	no	—
79	C₂×C₄×C₈	—	17	12	no	—
80	C₂×C₄×C₈	—	17	6	no	—
81	C₂×C₄×C₈	—	17	6	no	—
82	C₈²	—	17	16	no	—
83	C₂×C₁₇	C₂×C₈²	1	12	yes	complemented
84	C₂×C₁₇	C₂²×C₄×C₈	1	2	yes	—
85	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
86	C₂³×C₄	—	17	2	no	—
87	C₂³×C₄	C₂×#595	1	1	yes	characteristic
88	C₂²×C₈	—	17	32	no	—
89	C₂²×C₈	—	17	24	no	—
90	C₂²×C₈	—	17	8	no	—
91	C₂²×C₈	—	17	6	no	—
92	C₂²×C₈	#3624	1	8	yes	direct product factor
93	C₂²×C₈	C₂×#595	1	6	yes	—
94	C₂×C₄²	—	17	6	no	—
95	C₂×C₄²	—	17	3	no	—
96	C₂×C₄²	—	17	3	no	—
97	C₄×C₈	—	17	8	no	—
98	C₄×C₈	—	17	24	no	—
99	C₄×C₈	—	17	4	no	—
100	C₄×C₈	—	17	12	no	—
101	C₁₇	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
102	C₂⁴	C₄×#595	1	1	yes	characteristic
103	C₂²×C₄	—	17	6	no	—
104	C₂²×C₄	—	17	6	no	—
105	C₂²×C₄	—	17	16	no	—
106	C₂²×C₄	#63242	1	4	yes	—
107	C₂²×C₄	C₄×#595	1	3	yes	—
108	C₂²×C₄	C₂×#3624	1	4	yes	—
109	C₂²×C₄	C₂²×#595	1	3	yes	—
110	C₂×C₈	—	17	96	no	—
111	C₂×C₈	—	17	16	no	—
112	C₂×C₈	—	17	24	no	—
113	C₂×C₈	—	17	4	no	—
114	C₂×C₈	C₂×#3624	1	24	yes	direct product factor
115	C₂×C₈	C₂²×#595	1	4	yes	—
116	C₄²	—	17	6	no	—
117	C₄²	—	17	6	no	—
118	C₄²	—	17	1	no	—
119	C₄²	—	17	3	no	—
120	C₂³	C₈×#595	1	8	yes	—
121	C₂³	C₄×#3624	1	4	yes	—
122	C₂³	C₂×C₄×#595	1	3	yes	—
123	C₂×C₄	—	17	24	no	—
124	C₂×C₄	—	17	24	no	—
125	C₂×C₄	—	17	2	no	—
126	C₂×C₄	—	17	6	no	—
127	C₂×C₄	C₄×#3624	1	12	yes	—
128	C₂×C₄	C₂×#63242	1	6	yes	—
129	C₂×C₄	C₂×C₄×#595	1	3	yes	—
130	C₂×C₄	C₂²×#3624	1	6	yes	—
131	C₂×C₄	C₂³×#595	1	1	yes	characteristic
132	C₈	—	17	32	no	—
133	C₈	—	17	8	no	—
134	C₈	C₂²×#3624	1	8	yes	direct product factor
135	C₂²	C₈×#3624	1	16	yes	direct product factor
136	C₂²	C₂×C₈×#595	1	12	yes	—
137	C₂²	C₂×C₄×#3624	1	6	yes	—
138	C₂²	C₂²×C₄×#595	1	1	yes	characteristic
139	C₄	—	17	4	no	—
140	C₄	—	17	12	no	—
141	C₄	C₂×C₄×#3624	1	6	yes	—
142	C₄	C₂³×#3624	1	1	yes	characteristic
143	C₄	C₂²×#63242	1	1	yes	characteristic
144	C₂	C₂×C₈×#3624	1	12	yes	direct product factor
145	C₂	C₂²×C₈×#595	1	2	yes	—
146	C₂	C₂²×C₄×#3624	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#595
commutator, hypocenter	C₁₇	C₂²×C₈²
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂³×C₈×C₁₇	C₄
socle, prime radical	C₂⁴×C₁₇	C₄²
cosocle	C₄×Dic₁₇	C₂⁴
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×#3624	C₂	direct (N × H)
7. C₈×#3624	C₂²	direct (N × H)
15. C₂²×#3624	C₈	direct (N × H)
17. C₂²×C₈×C₁₇	C₈	semidirect
22. C₂²×#3622	C₈	semidirect
36. C₂×#3624	C₂×C₈	direct (N × H)
38. C₂×C₈×C₁₇	C₂×C₈	semidirect
44. C₂×#3622	C₂×C₈	semidirect
59. #3624	C₂²×C₈	direct (N × H)
60. C₈×C₁₇	C₂²×C₈	semidirect
61. #3622	C₂²×C₈	semidirect
65. C₂²×C₁₇	C₈²	semidirect
83. C₂×C₁₇	C₂×C₈²	semidirect
92. C₂²×C₈	#3624	direct (N × H)
101. C₁₇	C₂²×C₈²	semidirect
114. C₂×C₈	C₂×#3624	direct (N × H)
134. C₈	C₂²×#3624	direct (N × H)
135. C₂²	C₈×#3624	direct (N × H)
144. C₂	C₂×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
C₂²×C₄×#3624	2
C₂×C₈×#3624	2
C₂²×C₈×Dic₁₇	2
C₂²×C₈²	17

Subgroups — C22×C8×#3624 #17963966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×#3624 #17963966