Subgroups — C₂×#22562488 #31011313

Order 17152.

494 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	C₂×#16863798	C₂	1	2	yes	maximal
2	#22562488	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863224	C₂	1	2	yes	maximal
4	C₂×#16863222	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524035	—	2	2	no	—
7	#16863798	—	2	4	no	—
8	#16863798	C₂²	1	8	yes	—
9	C₂×#14523977	C₂²	1	4	yes	—
10	C₂×#14523975	—	2	2	no	—
11	#16863224	C₂²	1	8	yes	—
12	#16863222	C₂²	1	8	yes	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
17	#14524035	—	2	8	no	—
18	#14524035	—	4	4	no	—
19	C₂×C₄⋊Dic₆₇	—	2	4	no	—
20	C₂×Dic₆₇⋊C₄	—	4	4	no	—
21	#14523977	—	2	4	no	—
22	#14523977	C₂³	1	8	yes	—
23	#14523975	—	2	8	no	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	Q₈	1	4	yes	—
28	C₂×C₁₆×C₆₇	Q₈	1	4	yes	—
29	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
30	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₄×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×Dic₆₇	—	4	4	no	—
34	C₄⋊Dic₆₇	—	2	8	no	—
35	C₄⋊Dic₆₇	—	4	4	no	—
36	Dic₆₇⋊C₄	—	4	16	no	—
37	C₂²×C₄×C₆₇	D₈	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
45	C₄²×C₆₇	D₈	1	2	yes	—
46	C₄²×C₆₇	C₂×D₄	1	2	yes	—
47	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
48	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	4	no	—
51	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₆₇	—	4	8	no	—
57	C₂×Dic₆₇	—	8	8	no	—
58	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
60	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
61	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₆₇	#500	1	2	yes	—
63	C₂²×C₆₇	#508	1	4	yes	—
64	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₆₇	—	2	4	no	—
66	C₄×C₆₇	#500	1	2	yes	—
67	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₆₇	—	8	8	no	—
70	C₂×#2253	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	#2253	1	4	yes	complemented
72	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
73	C₂×C₆₇	C₂×#508	1	2	yes	—
74	C₂×#499	—	67	2	no	—
75	#2253	—	67	8	no	—
76	C₂×#367	—	67	2	no	—
77	C₂×#366	—	67	2	no	—
78	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
79	C₆₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
80	C₂×#125	—	134	2	no	—
81	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
83	#499	—	67	8	no	—
84	#499	—	134	4	no	—
85	C₂×#107	—	67	4	no	—
86	C₂×#106	—	134	2	no	—
87	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
88	#367	—	67	8	no	—
89	#366	—	67	8	no	—
90	C₄×C₁₆	C₂×D₆₇	1	4	yes	—
91	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
93	#125	—	268	4	no	—
94	#125	—	134	8	no	—
95	C₂×C₄⋊C₄	—	268	4	no	—
96	C₂×C₄⋊C₄	—	134	4	no	—
97	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₃₄	1	4	yes	—
99	C₂×C₁₆	Dic₁₃₄	1	4	yes	—
100	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
101	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
102	#107	—	67	8	no	—
103	#107	—	134	4	no	—
104	#106	—	134	8	no	—
105	C₄×C₈	D₂₆₈	1	2	yes	—
106	C₄×C₈	C₂²×D₆₇	1	2	yes	—
107	C₂²×C₄	—	268	4	no	—
108	C₂²×C₄	D₅₃₆	1	2	yes	—
109	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
110	C₂×C₈	Dic₂₆₈	1	4	yes	—
111	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
112	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
113	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
114	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
115	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
116	C₄⋊C₄	—	268	16	no	—
117	C₄⋊C₄	—	268	4	no	—
118	C₄⋊C₄	—	134	8	no	—
119	C₄²	D₅₃₆	1	2	yes	—
120	C₄²	C₂×D₂₆₈	1	2	yes	—
121	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
122	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
123	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	268	8	no	—
126	C₂×C₄	—	536	8	no	—
127	C₂×C₄	#14524043	1	2	yes	—
128	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
129	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
130	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
131	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
132	C₈	#14524043	1	2	yes	—
133	C₈	C₂×Dic₂₆₈	1	4	yes	—
134	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
135	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
136	C₂²	#16863801	1	2	yes	—
137	C₂²	#16863828	1	4	yes	—
138	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	536	8	no	—
141	C₄	#16863801	1	2	yes	—
142	C₄	C₂×#14524043	1	1	yes	characteristic
143	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
144	C₂	#22562488	1	4	yes	direct product factor
145	C₂	C₂×#16863801	1	1	yes	characteristic
146	C₂	C₂×#16863828	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2253
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
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. #22562488	C₂	direct (N × H)
71. C₂×C₆₇	#2253	semidirect
79. C₆₇	C₂×#2253	semidirect
144. C₂	#22562488	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×#16863798	2
#22562488	2
C₂×#16863224	2
C₂×#16863222	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2253	67

Subgroups — C2×#22562488 #31011313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562488 #31011313