Subgroups — C₂×C₄²×#3624 #17968121

Order 4352.

2046 conjugacy classes of subgroups.

The subgroup lattice has 144 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	6	yes	maximal
2	C₄²×#3624	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#3624	C₂²	1	4	yes	—
5	C₂×C₄×#3624	C₄	1	48	yes	direct product factor
6	C₂×C₄×#3624	C₂²	1	24	yes	—
7	C₂²×C₄×Dic₁₇	C₄	1	3	yes	—
8	C₂²×C₄×Dic₁₇	C₂²	1	3	yes	—
9	C₄²×Dic₁₇	C₄	1	4	yes	—
10	C₄²×Dic₁₇	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
12	C₂³×C₄×C₁₇	C₂×C₄	1	3	yes	—
13	C₂³×Dic₁₇	C₂×C₄	1	3	yes	—
14	C₂³×Dic₁₇	C₂³	1	1	yes	characteristic
15	C₂²×#3624	C₂×C₄	1	48	yes	—
16	C₂²×#3624	C₂³	1	8	yes	—
17	C₂×C₄²×C₁₇	C₈	1	8	yes	complemented
18	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₁₇	C₈	1	24	yes	complemented
20	C₂×C₄×Dic₁₇	C₂×C₄	1	24	yes	—
21	C₂×C₄×Dic₁₇	C₂×C₄	1	6	yes	—
22	C₂×C₄×Dic₁₇	C₂×C₄	1	12	yes	—
23	C₂×C₄×Dic₁₇	C₂³	1	6	yes	—
24	C₄×#3624	C₂×C₄	1	96	yes	direct product factor
25	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₄²	1	12	yes	—
27	C₂²×C₄×C₁₇	C₂×C₈	1	12	yes	—
28	C₂²×C₄×C₁₇	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
30	C₂²×Dic₁₇	C₄²	1	12	yes	—
31	C₂²×Dic₁₇	C₂×C₈	1	24	yes	—
32	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₇	C₂²×C₄	1	12	yes	—
34	C₂²×Dic₁₇	C₂²×C₄	1	6	yes	—
35	C₂²×Dic₁₇	C₂⁴	1	1	yes	characteristic
36	C₂×#3624	C₄²	1	64	yes	direct product factor
37	C₂×#3624	C₂²×C₄	1	48	yes	—
38	C₄²×C₁₇	C₂×C₈	1	16	yes	complemented
39	C₄×Dic₁₇	C₄²	1	24	yes	—
40	C₄×Dic₁₇	C₂×C₈	1	48	yes	complemented
41	C₄×Dic₁₇	C₂²×C₄	1	24	yes	—
42	C₂×C₄²×C₈	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₂×C₄²	1	6	yes	—
44	C₂³×C₁₇	C₂²×C₈	1	6	yes	—
45	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
46	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₄×C₈	1	48	yes	complemented
48	C₂×C₄×C₁₇	C₂×C₄²	1	12	yes	—
49	C₂×C₄×C₁₇	C₂²×C₈	1	12	yes	—
50	C₂×C₄×C₁₇	C₂²×C₈	1	12	yes	—
51	C₂×Dic₁₇	C₄×C₈	1	48	yes	complemented
52	C₂×Dic₁₇	C₂×C₄²	1	6	yes	—
53	C₂×Dic₁₇	C₂×C₄²	1	16	yes	—
54	C₂×Dic₁₇	C₂×C₄²	1	12	yes	—
55	C₂×Dic₁₇	C₂²×C₈	1	24	yes	—
56	C₂×Dic₁₇	C₂³×C₄	1	6	yes	—
57	#3624	C₂×C₄²	1	32	yes	direct product factor
58	C₂²×C₄×C₈	—	17	6	no	—
59	C₂×C₄³	—	17	1	no	—
60	C₄²×C₈	—	17	8	no	—
61	C₂²×C₁₇	C₄³	1	4	yes	—
62	C₂²×C₁₇	C₂×C₄×C₈	1	12	yes	—
63	C₂²×C₁₇	C₂×C₄×C₈	1	12	yes	—
64	C₂²×C₁₇	C₂²×C₄²	1	3	yes	—
65	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
66	C₂²×C₁₇	C₂³×C₈	1	3	yes	—
67	C₄×C₁₇	C₂×C₄×C₈	1	24	yes	complemented
68	Dic₁₇	C₄³	1	4	yes	—
69	Dic₁₇	C₂×C₄×C₈	1	24	yes	complemented
70	Dic₁₇	C₂²×C₄²	1	4	yes	—
71	C₂³×C₈	—	17	4	no	—
72	C₂²×C₄²	—	17	3	no	—
73	C₂²×C₄²	—	17	3	no	—
74	C₂²×C₄²	#595	1	1	yes	characteristic
75	C₂×C₄×C₈	—	17	48	no	—
76	C₂×C₄×C₈	—	17	24	no	—
77	C₄³	—	17	4	no	—
78	C₄³	—	17	4	no	—
79	C₂×C₁₇	C₄²×C₈	1	8	yes	complemented
80	C₂×C₁₇	C₂×C₄³	1	1	yes	characteristic
81	C₂×C₁₇	C₂²×C₄×C₈	1	3	yes	—
82	C₂×C₁₇	C₂²×C₄×C₈	1	3	yes	—
83	C₂³×C₄	—	17	3	no	—
84	C₂³×C₄	—	17	1	no	—
85	C₂³×C₄	C₂×#595	1	3	yes	—
86	C₂²×C₈	—	17	48	no	—
87	C₂²×C₈	—	17	8	no	—
88	C₂×C₄²	—	17	24	no	—
89	C₂×C₄²	—	17	24	no	—
90	C₂×C₄²	—	17	6	no	—
91	C₂×C₄²	—	17	12	no	—
92	C₂×C₄²	—	17	6	no	—
93	C₂×C₄²	#3624	1	8	yes	direct product factor
94	C₂×C₄²	C₂×#595	1	4	yes	—
95	C₄×C₈	—	17	96	no	—
96	C₁₇	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	C₂²×#595	1	1	yes	characteristic
98	C₂²×C₄	—	17	24	no	—
99	C₂²×C₄	—	17	12	no	—
100	C₂²×C₄	—	17	12	no	—
101	C₂²×C₄	—	17	1	no	—
102	C₂²×C₄	—	17	6	no	—
103	C₂²×C₄	—	17	1	no	—
104	C₂²×C₄	#63242	1	12	yes	—
105	C₂²×C₄	C₄×#595	1	12	yes	—
106	C₂²×C₄	C₂×#3624	1	12	yes	—
107	C₂²×C₄	C₂²×#595	1	6	yes	—
108	C₂×C₈	—	17	64	no	—
109	C₂×C₈	—	17	48	no	—
110	C₄²	—	17	48	no	—
111	C₄²	—	17	24	no	—
112	C₄²	—	17	24	no	—
113	C₄²	C₂×#3624	1	16	yes	direct product factor
114	C₂³	C₂×#63242	1	6	yes	—
115	C₂³	C₂×C₄×#595	1	6	yes	—
116	C₂³	C₂²×#3624	1	2	yes	—
117	C₂³	C₂³×#595	1	1	yes	characteristic
118	C₂×C₄	—	17	16	no	—
119	C₂×C₄	—	17	48	no	—
120	C₂×C₄	—	17	6	no	—
121	C₂×C₄	—	17	6	no	—
122	C₂×C₄	—	17	12	no	—
123	C₂×C₄	—	17	24	no	—
124	C₂×C₄	C₄×#3624	1	48	yes	direct product factor
125	C₂×C₄	C₂×#63242	1	12	yes	—
126	C₂×C₄	C₂×C₄×#595	1	12	yes	—
127	C₂×C₄	C₂²×#3624	1	12	yes	—
128	C₈	—	17	32	no	—
129	C₂²	C₄²×#595	1	4	yes	—
130	C₂²	C₂×C₄×#3624	1	12	yes	—
131	C₂²	C₄×#63242	1	12	yes	—
132	C₂²	C₂²×C₄×#595	1	3	yes	—
133	C₂²	C₂³×#3624	1	1	yes	characteristic
134	C₂²	C₂²×#63242	1	3	yes	—
135	C₄	—	17	4	no	—
136	C₄	—	17	4	no	—
137	C₄	—	17	24	no	—
138	C₄	C₂×C₄×#3624	1	24	yes	direct product factor
139	C₂	C₄²×#3624	1	8	yes	direct product factor
140	C₂	C₂×C₄²×#595	1	1	yes	characteristic
141	C₂	C₂²×C₄×#3624	1	3	yes	—
142	C₂	C₂×C₄×#63242	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄²×#3624	C₂	direct (N × H)
5. C₂×C₄×#3624	C₄	direct (N × H)
17. C₂×C₄²×C₁₇	C₈	semidirect
19. C₂×C₄×Dic₁₇	C₈	semidirect
24. C₄×#3624	C₂×C₄	direct (N × H)
36. C₂×#3624	C₄²	direct (N × H)
38. C₄²×C₁₇	C₂×C₈	semidirect
40. C₄×Dic₁₇	C₂×C₈	semidirect
47. C₂×C₄×C₁₇	C₄×C₈	semidirect
51. C₂×Dic₁₇	C₄×C₈	semidirect
57. #3624	C₂×C₄²	direct (N × H)
67. C₄×C₁₇	C₂×C₄×C₈	semidirect
69. Dic₁₇	C₂×C₄×C₈	semidirect
79. C₂×C₁₇	C₄²×C₈	semidirect
93. C₂×C₄²	#3624	direct (N × H)
96. C₁₇	C₂×C₄²×C₈	semidirect
113. C₄²	C₂×#3624	direct (N × H)
124. C₂×C₄	C₄×#3624	direct (N × H)
138. C₄	C₂×C₄×#3624	direct (N × H)
139. C₂	C₄²×#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
C₄²×#3624	2
C₂×C₄²×Dic₁₇	2
C₂×C₄²×C₈	17

Subgroups — C2×C42×#3624 #17968121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×#3624 #17968121