Subgroups — C₂²×C₄×#63216 #17978114

Order 4352.

1022 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63216	C₂	1	2	yes	maximal
2	C₂×C₄×#63216	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	C₄	1	16	yes	direct product factor
5	C₂²×#63216	C₂²	1	12	yes	—
6	C₄×#63216	C₂²	1	16	yes	direct product factor
7	C₂³×#3622	C₄	1	1	yes	characteristic
8	C₂³×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₄	1	6	yes	—
10	C₂×C₄×#3622	C₂²	1	6	yes	—
11	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
12	C₂³×C₄×C₁₇	C₈	1	2	yes	—
13	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	C₈	1	6	yes	—
15	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
16	C₂²×#3622	C₈	1	8	yes	—
17	C₂²×#3622	C₂×C₄	1	8	yes	—
18	C₂²×#3622	C₂×C₄	1	3	yes	—
19	C₂²×#3622	C₂×C₄	1	6	yes	—
20	C₂²×#3622	C₂³	1	3	yes	—
21	C₂×#63216	C₂×C₄	1	48	yes	direct product factor
22	C₂×#63216	C₂³	1	8	yes	—
23	C₄×#3622	C₂×C₄	1	12	yes	—
24	C₄×#3622	C₂³	1	4	yes	—
25	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₁₆	1	16	yes	complemented
27	C₂²×C₄×C₁₇	C₄²	1	4	yes	—
28	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
29	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	12	yes	—
31	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
32	C₄²×C₁₇	C₂×C₈	1	12	yes	—
33	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
34	C₂×#3622	C₄²	1	12	yes	—
35	C₂×#3622	C₂×C₈	1	24	yes	—
36	C₂×#3622	C₂²×C₄	1	3	yes	—
37	C₂×#3622	C₂²×C₄	1	12	yes	—
38	C₂×#3622	C₂²×C₄	1	4	yes	—
39	C₂×#3622	C₂⁴	1	1	yes	characteristic
40	#63216	C₂²×C₄	1	16	yes	direct product factor
41	C₂²×C₄×C₁₆	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	C₄×C₈	1	4	yes	—
43	C₂³×C₁₇	C₂×C₁₆	1	8	yes	—
44	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
45	C₂×C₄×C₁₇	C₄×C₈	1	12	yes	—
46	C₂×C₄×C₁₇	C₂×C₁₆	1	48	yes	complemented
47	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
48	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
49	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
51	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
52	#3622	C₂×C₄²	1	6	yes	—
53	#3622	C₂²×C₈	1	8	yes	—
54	#3622	C₂³×C₄	1	2	yes	—
55	C₂³×C₁₆	—	17	2	no	—
56	C₂²×C₄×C₈	—	17	1	no	—
57	C₂×C₄×C₁₆	—	17	12	no	—
58	C₂²×C₁₇	C₄×C₁₆	1	16	yes	complemented
59	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
60	C₂²×C₁₇	C₂²×C₁₆	1	12	yes	—
61	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
62	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
63	C₄×C₁₇	C₂²×C₁₆	1	16	yes	complemented
64	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
65	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
66	C₂³×C₈	—	17	1	no	—
67	C₂³×C₈	—	17	1	no	—
68	C₂²×C₄²	#595	1	1	yes	characteristic
69	C₂²×C₁₆	—	17	12	no	—
70	C₂²×C₁₆	—	17	16	no	—
71	C₂×C₄×C₈	—	17	6	no	—
72	C₂×C₄×C₈	—	17	6	no	—
73	C₄×C₁₆	—	17	16	no	—
74	C₂×C₁₇	C₂×C₄×C₁₆	1	12	yes	complemented
75	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
76	C₂×C₁₇	C₂³×C₁₆	1	2	yes	—
77	C₂³×C₄	#3624	1	2	yes	—
78	C₂³×C₄	C₂×#595	1	1	yes	characteristic
79	C₂²×C₈	—	17	8	no	—
80	C₂²×C₈	—	17	8	no	—
81	C₂²×C₈	—	17	6	no	—
82	C₂²×C₈	—	17	3	no	—
83	C₂²×C₈	—	17	3	no	—
84	C₂×C₄²	#3624	1	6	yes	—
85	C₂×C₄²	C₂×#595	1	6	yes	—
86	C₂×C₁₆	—	17	8	no	—
87	C₂×C₁₆	—	17	48	no	—
88	C₄×C₈	—	17	12	no	—
89	C₄×C₈	—	17	4	no	—
90	C₁₇	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	C₂×#3624	1	1	yes	characteristic
92	C₂²×C₄	#63216	1	16	yes	direct product factor
93	C₂²×C₄	#63242	1	4	yes	—
94	C₂²×C₄	C₄×#595	1	4	yes	—
95	C₂²×C₄	C₂×#3624	1	12	yes	—
96	C₂²×C₄	C₂×#3624	1	3	yes	—
97	C₂²×C₄	C₂²×#595	1	3	yes	—
98	C₂×C₈	—	17	24	no	—
99	C₂×C₈	—	17	12	no	—
100	C₂×C₈	—	17	12	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	17	3	no	—
103	C₂×C₈	—	17	4	no	—
104	C₄²	C₂×#3624	1	12	yes	—
105	C₄²	C₂²×#595	1	4	yes	—
106	C₁₆	—	17	16	no	—
107	C₂³	C₂×#63216	1	8	yes	—
108	C₂³	C₄×#3624	1	4	yes	—
109	C₂³	C₂²×#3624	1	3	yes	—
110	C₂×C₄	C₂×#63216	1	48	yes	direct product factor
111	C₂×C₄	C₄×#3624	1	12	yes	—
112	C₂×C₄	C₂×#63242	1	6	yes	—
113	C₂×C₄	C₂×C₄×#595	1	6	yes	—
114	C₂×C₄	C₂²×#3624	1	3	yes	—
115	C₂×C₄	C₂²×#3624	1	8	yes	—
116	C₂×C₄	C₂³×#595	1	1	yes	characteristic
117	C₈	—	17	2	no	—
118	C₈	—	17	6	no	—
119	C₈	—	17	8	no	—
120	C₂²	C₄×#63216	1	16	yes	direct product factor
121	C₂²	C₂×C₄×#3624	1	6	yes	—
122	C₂²	C₂²×#63216	1	12	yes	—
123	C₂²	C₂³×#3624	1	1	yes	characteristic
124	C₄	C₂×C₄×#3624	1	6	yes	—
125	C₄	C₂²×#63216	1	16	yes	direct product factor
126	C₄	C₂²×C₄×#595	1	1	yes	characteristic
127	C₄	C₂²×#63242	1	1	yes	characteristic
128	C₂	C₂×C₄×#63216	1	12	yes	direct product factor
129	C₂	C₂²×C₄×#3624	1	1	yes	characteristic
130	C₂	C₂³×#63216	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#595
commutator, hypocenter	C₁₇	C₂²×C₄×C₁₆
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂²×C₄²×C₁₇	C₄
socle, prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×#3622	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. C₂×C₄×#63216	C₂	direct (N × H)
4. C₂²×#63216	C₄	direct (N × H)
6. C₄×#63216	C₂²	direct (N × H)
21. C₂×#63216	C₂×C₄	direct (N × H)
26. C₂²×C₄×C₁₇	C₁₆	semidirect
40. #63216	C₂²×C₄	direct (N × H)
46. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
58. C₂²×C₁₇	C₄×C₁₆	semidirect
63. C₄×C₁₇	C₂²×C₁₆	semidirect
74. C₂×C₁₇	C₂×C₄×C₁₆	semidirect
90. C₁₇	C₂²×C₄×C₁₆	semidirect
92. C₂²×C₄	#63216	direct (N × H)
110. C₂×C₄	C₂×#63216	direct (N × H)
120. C₂²	C₄×#63216	direct (N × H)
125. C₄	C₂²×#63216	direct (N × H)
128. C₂	C₂×C₄×#63216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63216	2
C₂×C₄×#63216	2
C₂²×C₄×#3622	2
C₂²×C₄×C₁₆	17

Subgroups — C22×C4×#63216 #17978114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#63216 #17978114