Subgroups — C₂×#14531059 #17189801

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	2	yes	maximal
2	#14531059	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762635	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	—	2	2	no	—
6	#11763214	—	2	4	no	—
7	#11763214	C₂²	1	8	yes	—
8	C₂×#10587489	—	2	2	no	—
9	C₂×#10587489	C₂²	1	4	yes	—
10	#11762635	C₂²	1	16	yes	—
11	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₇	C₂²	1	4	yes	—
14	C₂²×Dic₃₄	—	4	2	no	—
15	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	—	2	4	no	—
19	C₂×C₄⋊Dic₁₇	—	4	2	no	—
20	#10587555	—	2	8	no	—
21	#10587555	—	4	4	no	—
22	C₂×C₁₆×C₁₇	Q₈	1	8	yes	complemented
23	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
24	C₄×C₈×C₁₇	D₄	1	2	yes	—
25	C₄×C₈×C₁₇	C₂³	1	2	yes	—
26	#10587489	—	2	4	no	—
27	#10587489	—	2	8	no	—
28	#10587489	C₂³	1	8	yes	—
29	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	—	4	4	no	—
33	C₂×Dic₃₄	—	4	8	no	—
34	C₂×Dic₃₄	—	8	8	no	—
35	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
36	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
39	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
40	C₄²×C₁₇	D₈	1	2	yes	—
41	C₄²×C₁₇	C₂×D₄	1	2	yes	—
42	C₄⋊Dic₁₇	—	2	8	no	—
43	C₄⋊Dic₁₇	—	4	8	no	—
44	C₄⋊Dic₁₇	—	4	4	no	—
45	C₁₆×C₁₇	C₂×Q₈	1	8	yes	complemented
46	C₂×#2252	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	D₁₆	1	4	yes	—
49	C₂×C₄×C₁₇	Q₃₂	1	4	yes	—
50	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
51	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₁₇	—	4	8	no	—
57	C₂×Dic₁₇	—	8	8	no	—
58	Dic₃₄	—	8	16	no	—
59	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
60	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
61	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
62	C₂×#500	—	17	2	no	—
63	#2252	—	17	8	no	—
64	C₂×#366	—	17	4	no	—
65	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
66	C₂²×C₁₇	#500	1	2	yes	—
67	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
68	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
69	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₁₇	#500	1	2	yes	—
71	C₄×C₁₇	C₂×D₁₆	1	4	yes	—
72	C₄×C₁₇	C₂×Q₃₂	1	4	yes	—
73	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
75	Dic₁₇	—	8	8	no	—
76	C₂×C₄⋊Q₈	—	34	2	no	—
77	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
78	#500	—	17	8	no	—
79	#500	—	34	4	no	—
80	C₂×#107	—	17	4	no	—
81	C₂×#107	—	34	2	no	—
82	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
83	#366	—	17	16	no	—
84	C₄×C₁₆	C₂×D₁₇	1	4	yes	—
85	C₂×C₁₇	#2252	1	4	yes	complemented
86	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
87	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
88	C₂×C₁₇	C₂²×Q₃₂	1	1	yes	characteristic
89	C₂²×Q₈	—	68	2	no	—
90	C₂²×C₈	D₆₈	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	34	8	no	—
93	C₄⋊Q₈	—	68	4	no	—
94	C₂×C₄⋊C₄	—	34	4	no	—
95	C₂×C₄⋊C₄	—	68	2	no	—
96	C₂×C₄²	D₆₈	1	1	yes	characteristic
97	C₂×C₁₆	Dic₃₄	1	8	yes	complemented
98	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
99	#107	—	17	8	no	—
100	#107	—	34	4	no	—
101	#107	—	34	8	no	—
102	C₄×C₈	D₆₈	1	2	yes	—
103	C₄×C₈	C₂²×D₁₇	1	2	yes	—
104	C₁₇	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₈	—	68	8	no	—
106	C₂×Q₈	—	136	8	no	—
107	C₂²×C₄	—	68	4	no	—
108	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
109	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
111	C₂×C₈	Dic₆₈	1	4	yes	—
112	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
113	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
114	C₂×C₈	C₂×D₆₈	1	2	yes	—
115	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
116	C₄⋊C₄	—	34	8	no	—
117	C₄⋊C₄	—	68	4	no	—
118	C₄⋊C₄	—	68	8	no	—
119	C₄²	D₁₃₆	1	2	yes	—
120	C₄²	C₂×D₆₈	1	2	yes	—
121	C₁₆	C₂×Dic₃₄	1	8	yes	complemented
122	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
123	Q₈	—	136	16	no	—
124	C₂×C₄	—	68	8	no	—
125	C₂×C₄	—	136	8	no	—
126	C₂×C₄	D₂₇₂	1	4	yes	—
127	C₂×C₄	Dic₁₃₆	1	4	yes	—
128	C₂×C₄	#10587555	1	2	yes	—
129	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
130	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
131	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
132	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
133	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
134	C₈	#10587555	1	2	yes	—
135	C₈	C₂×Dic₆₈	1	4	yes	—
136	C₈	C₂²×Dic₃₄	1	2	yes	—
137	C₂²	#11763214	1	2	yes	—
138	C₂²	C₂×D₂₇₂	1	2	yes	—
139	C₂²	C₂×Dic₁₃₆	1	2	yes	—
140	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
141	C₄	—	136	8	no	—
142	C₄	#11763214	1	2	yes	—
143	C₄	C₂×D₂₇₂	1	4	yes	—
144	C₄	C₂×Dic₁₃₆	1	4	yes	—
145	C₄	C₂×#10587555	1	1	yes	characteristic
146	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
147	C₂	#14531059	1	4	yes	direct product factor
148	C₂	C₂×#11763214	1	1	yes	characteristic
149	C₂	C₂²×D₂₇₂	1	1	yes	characteristic
150	C₂	C₂²×Dic₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	C₂×#2252
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14531059	C₂	direct (N × H)
22. C₂×C₁₆×C₁₇	Q₈	semidirect
45. C₁₆×C₁₇	C₂×Q₈	semidirect
85. C₂×C₁₇	#2252	semidirect
97. C₂×C₁₆	Dic₃₄	semidirect
104. C₁₇	C₂×#2252	semidirect
121. C₁₆	C₂×Dic₃₄	semidirect
147. C₂	#14531059	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14531059	2
C₂×#11762635	2
C₂×C₄×C₁₆×C₁₇	2
C₂×#2252	17

Subgroups — C2×#14531059 #17189801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531059 #17189801