Subgroups — C₂²×C₄⋊#63216 #17978393

Order 4352.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 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	—	2	8	no	—
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	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₈	1	6	yes	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
17	C₂²×#3622	—	2	4	no	—
18	C₂²×#3622	C₂×C₄	1	6	yes	—
19	C₂²×#3622	C₂×C₄	1	3	yes	—
20	C₂²×#3622	D₄	1	4	yes	—
21	C₂²×#3622	Q₈	1	4	yes	—
22	C₂²×#3622	C₂³	1	3	yes	—
23	C₂×#63216	—	2	24	no	—
24	C₂×#63216	C₂³	1	8	yes	—
25	C₄×#3622	C₂×C₄	1	12	yes	—
26	C₄×#3622	C₂³	1	4	yes	—
27	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	—	2	4	no	—
29	C₂²×C₄×C₁₇	C₁₆	1	8	yes	complemented
30	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
31	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
34	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
35	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
36	C₄²×C₁₇	C₂×C₈	1	12	yes	—
37	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
38	C₂×#3622	—	2	12	no	—
39	C₂×#3622	C₄⋊C₄	1	12	yes	—
40	C₂×#3622	C₂²×C₄	1	4	yes	—
41	C₂×#3622	C₂²×C₄	1	3	yes	—
42	C₂×#3622	C₂×D₄	1	6	yes	—
43	C₂×#3622	C₂×Q₈	1	6	yes	—
44	C₂×#3622	C₂⁴	1	1	yes	characteristic
45	#63216	—	2	8	no	—
46	C₂²×C₄⋊C₁₆	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₄⋊C₈	1	4	yes	—
48	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
49	C₂³×C₁₇	M₃₂	1	4	yes	—
50	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
51	C₂×C₄×C₁₇	—	2	12	no	—
52	C₂×C₄×C₁₇	C₄⋊C₈	1	12	yes	—
53	C₂×C₄×C₁₇	C₂×C₁₆	1	24	yes	complemented
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
55	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
56	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
57	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
58	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
59	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
60	#3622	—	2	4	no	—
61	#3622	C₂×C₄⋊C₄	1	6	yes	—
62	#3622	C₂²×D₄	1	1	yes	characteristic
63	#3622	C₂²×Q₈	1	1	yes	characteristic
64	C₂³×C₁₆	—	17	2	no	—
65	C₂²×C₄×C₈	—	17	1	no	—
66	C₂×C₄⋊C₁₆	—	17	12	no	—
67	C₂²×C₁₇	C₄⋊C₁₆	1	16	yes	complemented
68	C₂²×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
69	C₂²×C₁₇	C₂²×C₁₆	1	6	yes	—
70	C₂²×C₁₇	C₂×M₃₂	1	6	yes	—
71	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
72	C₄×C₁₇	—	2	4	no	—
73	C₄×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
74	C₄×C₁₇	C₂²×C₁₆	1	8	yes	complemented
75	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
77	C₂³×C₈	—	17	1	no	—
78	C₂³×C₈	—	17	1	no	—
79	C₂²×C₄²	#595	1	1	yes	characteristic
80	C₂²×C₁₆	—	17	12	no	—
81	C₂²×C₁₆	—	34	8	no	—
82	C₂×C₄×C₈	—	17	6	no	—
83	C₂×C₄×C₈	—	17	6	no	—
84	C₄⋊C₁₆	—	17	16	no	—
85	C₂×C₁₇	C₂×C₄⋊C₁₆	1	12	yes	complemented
86	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
88	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
89	C₂³×C₄	#3624	1	1	yes	characteristic
90	C₂³×C₄	#3624	1	1	yes	characteristic
91	C₂³×C₄	C₂×#595	1	1	yes	characteristic
92	C₂²×C₈	—	17	4	no	—
93	C₂²×C₈	—	17	4	no	—
94	C₂²×C₈	—	17	3	no	—
95	C₂²×C₈	—	17	3	no	—
96	C₂²×C₈	—	17	6	no	—
97	C₂²×C₈	—	34	4	no	—
98	C₂×C₄²	#3624	1	6	yes	—
99	C₂×C₄²	C₂×#595	1	6	yes	—
100	C₂×C₁₆	—	17	8	no	—
101	C₂×C₁₆	—	34	24	no	—
102	C₄×C₈	—	17	12	no	—
103	C₄×C₈	—	17	4	no	—
104	C₁₇	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂⁴	C₂×#3624	1	1	yes	characteristic
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	#63216	1	8	yes	complemented
108	C₂²×C₄	#63243	1	4	yes	—
109	C₂²×C₄	C₄⋊#595	1	4	yes	—
110	C₂²×C₄	C₂×#3624	1	6	yes	—
111	C₂²×C₄	C₂×#3624	1	6	yes	—
112	C₂²×C₄	C₂×#3624	1	3	yes	—
113	C₂²×C₄	C₂²×#595	1	3	yes	—
114	C₂×C₈	—	17	6	no	—
115	C₂×C₈	—	17	12	no	—
116	C₂×C₈	—	17	6	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	17	3	no	—
119	C₂×C₈	—	17	4	no	—
120	C₂×C₈	—	34	12	no	—
121	C₄²	C₂×#3624	1	12	yes	—
122	C₄²	C₂²×#595	1	4	yes	—
123	C₁₆	—	34	8	no	—
124	C₂³	C₂×#63216	1	4	yes	—
125	C₂³	#10587538	1	4	yes	—
126	C₂³	C₄⋊#3624	1	4	yes	—
127	C₂³	C₂²×#3624	1	3	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₂×#63216	1	24	yes	complemented
130	C₂×C₄	C₄⋊#3624	1	12	yes	—
131	C₂×C₄	C₂×#63243	1	6	yes	—
132	C₂×C₄	C₂×C₄⋊#595	1	6	yes	—
133	C₂×C₄	C₂²×#3624	1	4	yes	—
134	C₂×C₄	C₂²×#3624	1	3	yes	—
135	C₂×C₄	C₂²×#3624	1	4	yes	—
136	C₂×C₄	C₂³×#595	1	1	yes	characteristic
137	C₈	—	17	1	no	—
138	C₈	—	17	6	no	—
139	C₈	—	17	1	no	—
140	C₈	—	34	4	no	—
141	C₂²	C₄⋊#63216	1	16	yes	direct product factor
142	C₂²	C₂×C₄⋊#3624	1	6	yes	—
143	C₂²	C₂²×#63216	1	6	yes	—
144	C₂²	C₂×#10587538	1	6	yes	—
145	C₂²	C₂³×#3624	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	C₂×C₄⋊#3624	1	6	yes	—
148	C₄	C₂²×#63216	1	8	yes	complemented
149	C₄	C₂²×C₄⋊#595	1	1	yes	characteristic
150	C₄	C₂²×#63243	1	1	yes	characteristic
151	C₂	C₂×C₄⋊#63216	1	12	yes	direct product factor
152	C₂	C₂²×C₄⋊#3624	1	1	yes	characteristic
153	C₂	C₂³×#63216	1	1	yes	characteristic
154	C₂	C₂²×#10587538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#595
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂²×C₄²×C₁₇	C₄
socle, prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂²×C₄²	#595
hypocenter	C₁₇	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ 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)
6. C₄⋊#63216	C₂²	direct (N × H)
29. C₂²×C₄×C₁₇	C₁₆	semidirect
53. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
67. C₂²×C₁₇	C₄⋊C₁₆	semidirect
74. C₄×C₁₇	C₂²×C₁₆	semidirect
85. C₂×C₁₇	C₂×C₄⋊C₁₆	semidirect
104. C₁₇	C₂²×C₄⋊C₁₆	semidirect
107. C₂²×C₄	#63216	semidirect
129. C₂×C₄	C₂×#63216	semidirect
141. C₂²	C₄⋊#63216	direct (N × H)
148. C₄	C₂²×#63216	semidirect
151. 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 #17978393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊#63216 #17978393