Subgroups — C₂×#22558344 #30776015

Order 17152.

222 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	C₂×#16863230	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863171	C₂	1	1	yes	maximal, characteristic
3	#22558344	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
7	#16863230	C₄	1	2	yes	complemented
8	#16863230	C₂²	1	2	yes	—
9	C₂×C₆₇⋊C₃₂	C₄	1	2	yes	complemented
10	#16863171	C₄	1	2	yes	—
11	#16863171	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	M₆₄×C₆₇	C₄	1	2	yes	complemented
14	M₆₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₃₂×C₆₇	C₄	1	2	yes	complemented
16	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
18	C₂×C₆₇⋊C₁₆	C₈	1	2	yes	—
19	C₂×C₆₇⋊C₁₆	C₈	1	2	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₈×Dic₆₇	C₈	1	2	yes	—
22	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₆₇⋊C₃₂	C₂×C₄	1	4	yes	complemented
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
31	C₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
32	C₂²×Dic₆₇	—	2	1	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
36	C₄×Dic₆₇	—	2	1	no	—
37	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
38	C₆₇⋊C₁₆	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₆₇	C₄²	1	1	yes	characteristic
47	C₁₆×C₆₇	C₄²	1	1	yes	characteristic
48	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₆₇	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	4	2	no	—
59	C₈×C₆₇	C₄×C₈	1	1	yes	characteristic
60	C₈×C₆₇	C₄×C₈	1	1	yes	characteristic
61	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
62	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₆₇⋊C₈	—	2	2	no	—
64	C₂²×C₆₇	—	2	2	no	—
65	C₂²×C₆₇	#346	1	1	yes	characteristic
66	C₂²×C₆₇	#346	1	1	yes	characteristic
67	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₆₇	#346	1	1	yes	characteristic
69	C₄×C₆₇	#346	1	1	yes	characteristic
70	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
72	Dic₆₇	—	4	2	no	—
73	C₂×#1398	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	#1398	1	2	yes	complemented
77	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
78	C₂×M₆₄	—	67	1	no	—
79	C₂×M₆₄	D₆₇	1	1	yes	characteristic
80	C₂×#346	—	67	1	no	—
81	#1398	—	67	4	no	—
82	C₆₇	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	67	1	no	—
84	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
85	C₂×C₄×C₈	—	67	1	no	—
86	M₆₄	—	67	2	no	—
87	M₆₄	—	67	2	no	—
88	M₆₄	Dic₆₇	1	2	yes	complemented
89	M₆₄	C₂×D₆₇	1	2	yes	—
90	C₂×C₃₂	—	67	2	no	—
91	C₂×C₃₂	Dic₆₇	1	2	yes	complemented
92	#346	—	67	2	no	—
93	#346	—	67	2	no	—
94	C₂²×C₈	—	67	1	no	—
95	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
96	C₂×C₄²	—	67	1	no	—
97	C₂×C₁₆	—	67	2	no	—
98	C₂×C₁₆	—	67	2	no	—
99	C₂×C₁₆	—	67	2	no	—
100	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
106	C₄×C₈	—	67	2	no	—
107	C₄×C₈	—	67	2	no	—
108	C₃₂	—	67	4	no	—
109	C₃₂	C₂×Dic₆₇	1	4	yes	complemented
110	C₂²×C₄	—	134	1	no	—
111	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	—	134	1	no	—
114	C₂×C₈	—	134	1	no	—
115	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
116	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
117	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
118	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
119	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
121	C₄²	—	67	2	no	—
122	C₄²	—	134	1	no	—
123	C₁₆	—	67	4	no	—
124	C₁₆	C₄×Dic₆₇	1	1	yes	characteristic
125	C₁₆	C₄×Dic₆₇	1	1	yes	characteristic
126	C₁₆	C₂×C₄×D₆₇	1	1	yes	characteristic
127	C₁₆	C₂²×Dic₆₇	1	1	yes	characteristic
128	C₂³	#14523982	1	1	yes	characteristic
129	C₂×C₄	—	268	2	no	—
130	C₂×C₄	—	134	2	no	—
131	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
132	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
133	C₂×C₄	#14523982	1	1	yes	characteristic
134	C₂×C₄	#14523982	1	1	yes	characteristic
135	C₂×C₄	#14523982	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
137	C₈	—	134	2	no	—
138	C₈	C₈×Dic₆₇	1	1	yes	characteristic
139	C₈	C₈×Dic₆₇	1	1	yes	characteristic
140	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
141	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	#16863171	1	1	yes	characteristic
144	C₂²	#16863171	1	1	yes	characteristic
145	C₂²	C₂×#14523982	1	1	yes	characteristic
146	C₄	—	268	2	no	—
147	C₄	#16863171	1	1	yes	characteristic
148	C₄	#16863171	1	1	yes	characteristic
149	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
150	C₄	C₂×#14523982	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	2	1	no	—
153	C₂	#22558344	1	2	yes	direct product factor
154	C₂	C₂×#16863171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₂×M₆₄×C₆₇	C₂
socle	C₂²×C₆₇	#346
prime radical	C₂³×C₆₇	M₃₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₂×M₆₄	D₆₇
hypocenter	C₆₇	C₂×#1398
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #22558344	C₂	direct (N × H)
7. #16863230	C₄	semidirect
9. C₂×C₆₇⋊C₃₂	C₄	semidirect
13. M₆₄×C₆₇	C₄	semidirect
15. C₂×C₃₂×C₆₇	C₄	semidirect
23. C₆₇⋊C₃₂	C₂×C₄	semidirect
31. C₃₂×C₆₇	C₂×C₄	semidirect
76. C₂×C₆₇	#1398	semidirect
82. C₆₇	C₂×#1398	semidirect
88. M₆₄	Dic₆₇	semidirect
91. C₂×C₃₂	Dic₆₇	semidirect
109. C₃₂	C₂×Dic₆₇	semidirect
153. C₂	#22558344	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×#16863230	2
C₂×#16863171	2
#22558344	2
C₂×M₆₄×C₆₇	2
C₂×#1398	67

Subgroups — C2×#22558344 #30776015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558344 #30776015