Subgroups — C₂×#22562483 #31011307

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863800	C₂	1	2	yes	maximal
2	#22562483	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863221	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524042	—	2	2	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#16863800	—	2	4	no	—
10	#16863800	C₂²	1	8	yes	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	C₂×#14523976	—	2	2	no	—
13	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863222	C₂²	1	4	yes	—
15	#16863221	C₂²	1	8	yes	—
16	C₁₆×Dic₆₇	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	#366×C₆₇	C₂²	1	4	yes	—
20	C₂²×Dic₁₃₄	—	4	2	no	—
21	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
22	#14524042	—	2	8	no	—
23	#14524042	—	4	4	no	—
24	C₂×C₄⋊Dic₆₇	—	2	2	no	—
25	C₂×Dic₆₇⋊C₄	—	4	2	no	—
26	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
28	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
29	#14523977	—	2	2	no	—
30	#14523977	C₂³	1	4	yes	—
31	#14523976	—	2	8	no	—
32	C₈×Dic₆₇	D₄	1	2	yes	—
33	C₈×Dic₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
38	#107×C₆₇	—	2	2	no	—
39	#107×C₆₇	C₂³	1	4	yes	—
40	C₂×Dic₁₃₄	—	4	8	no	—
41	C₂×Dic₁₃₄	—	8	8	no	—
42	C₂²×Dic₆₇	—	4	2	no	—
43	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
44	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
45	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
46	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
47	C₄⋊Dic₆₇	—	2	4	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	Dic₆₇⋊C₄	—	4	8	no	—
50	C₄×Dic₆₇	D₈	1	2	yes	—
51	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
52	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₄×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
62	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	4	4	no	—
64	C₂×C₄×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₆₇	—	4	4	no	—
70	C₂×Dic₆₇	—	8	4	no	—
71	C₂×Dic₆₇	D₁₆	1	4	yes	—
72	C₂×Dic₆₇	Q₃₂	1	4	yes	—
73	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
74	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
75	Dic₁₃₄	—	8	16	no	—
76	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
77	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
80	C₂²×C₆₇	#500	1	2	yes	—
81	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
82	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
83	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
84	C₄×C₆₇	—	8	4	no	—
85	C₄×C₆₇	#500	1	2	yes	—
86	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
88	Dic₆₇	—	8	4	no	—
89	Dic₆₇	C₂×D₁₆	1	4	yes	—
90	Dic₆₇	C₂×Q₃₂	1	4	yes	—
91	C₂×#2252	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#2252	1	4	yes	complemented
93	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
94	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
96	C₂×#500	—	67	2	no	—
97	#2252	—	67	8	no	—
98	C₂×#366	—	67	1	no	—
99	C₂×#366	—	67	2	no	—
100	C₂×#366	D₆₇	1	1	yes	characteristic
101	C₂×C₄×C₁₆	—	67	1	no	—
102	C₆₇	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	134	2	no	—
104	C₂²×C₁₆	—	67	1	no	—
105	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
106	#500	—	67	8	no	—
107	#500	—	134	4	no	—
108	C₂×#107	—	67	2	no	—
109	C₂×#107	—	134	2	no	—
110	C₂×#107	C₂×D₆₇	1	2	yes	—
111	C₂×C₄×C₈	—	67	1	no	—
112	#366	—	67	4	no	—
113	#366	—	67	8	no	—
114	#366	C₂×D₆₇	1	4	yes	—
115	C₄×C₁₆	—	67	4	no	—
116	C₂²×Q₈	—	268	2	no	—
117	C₂²×C₈	—	67	1	no	—
118	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	C₄⋊Q₈	—	268	4	no	—
120	C₄⋊Q₈	—	134	8	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	268	2	no	—
123	C₂×C₄⋊C₄	—	134	2	no	—
124	C₂×C₄²	—	67	1	no	—
125	C₂×C₁₆	—	67	4	no	—
126	C₂×C₁₆	—	67	2	no	—
127	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
129	#107	—	2	2	no	—
130	#107	—	67	4	no	—
131	#107	—	134	8	no	—
132	#107	—	134	2	no	—
133	#107	C₂²×D₆₇	1	4	yes	—
134	C₄×C₈	—	67	2	no	—
135	C₄×C₈	—	67	2	no	—
136	C₂×Q₈	—	268	8	no	—
137	C₂×Q₈	—	536	8	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	268	2	no	—
140	C₂²×C₄	—	67	1	no	—
141	C₂²×C₄	—	67	1	no	—
142	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₈	—	67	4	no	—
144	C₂×C₈	—	67	2	no	—
145	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₆₇	1	2	yes	—
147	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
148	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	268	8	no	—
152	C₄⋊C₄	—	268	2	no	—
153	C₄⋊C₄	—	134	4	no	—
154	C₄²	—	67	2	no	—
155	C₄²	—	67	2	no	—
156	C₁₆	—	67	4	no	—
157	C₁₆	C₂×Dic₁₃₄	1	4	yes	complemented
158	C₂³	D₈×D₆₇	1	1	yes	characteristic
159	Q₈	—	536	16	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	268	4	no	—
163	C₂×C₄	—	536	4	no	—
164	C₂×C₄	—	67	4	no	—
165	C₂×C₄	—	67	4	no	—
166	C₂×C₄	—	67	2	no	—
167	C₂×C₄	—	67	2	no	—
168	C₂×C₄	#14524042	1	2	yes	—
169	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
170	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
171	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
172	C₈	—	67	4	no	—
173	C₈	#14524042	1	2	yes	—
174	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
176	C₂²	#16863800	1	2	yes	—
177	C₂²	D₁₆×D₆₇	1	2	yes	—
178	C₂²	Q₃₂×D₆₇	1	2	yes	—
179	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
180	C₄	—	8	4	no	—
181	C₄	—	536	4	no	—
182	C₄	—	67	4	no	—
183	C₄	—	67	4	no	—
184	C₄	#16863800	1	2	yes	—
185	C₄	C₂×#14524042	1	1	yes	characteristic
186	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
187	C₂	#22562483	1	4	yes	direct product factor
188	C₂	C₂×#16863800	1	1	yes	characteristic
189	C₂	C₂×D₁₆×D₆₇	1	1	yes	characteristic
190	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	C₂×#2252
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #22562483	C₂	direct (N × H)
27. C₂×C₆₇⋊C₁₆	Q₈	semidirect
36. C₂×C₁₆×C₆₇	Q₈	semidirect
52. C₆₇⋊C₁₆	C₂×Q₈	semidirect
61. C₁₆×C₆₇	C₂×Q₈	semidirect
92. C₂×C₆₇	#2252	semidirect
102. C₆₇	C₂×#2252	semidirect
127. C₂×C₁₆	Dic₁₃₄	semidirect
157. C₁₆	C₂×Dic₁₃₄	semidirect
187. C₂	#22562483	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×#16863800	2
#22562483	2
C₂×#16863222	2
C₂×#16863221	2
C₂×C₁₆×Dic₆₇	2
C₂×#366×C₆₇	2
C₂×#2252	67

Subgroups — C2×#22562483 #31011307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562483 #31011307