Subgroups — C₂×#22558281 #30767409

Order 17152.

262 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863149	C₂	1	2	yes	maximal
2	#22558281	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
5	#16863149	—	2	2	no	—
6	#16863149	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₆₇⋊C₈	—	4	2	no	—
13	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
14	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	—	2	2	no	—
19	C₂×C₁₆×C₆₇	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₆₇⋊C₈	—	4	4	no	—
27	C₂×C₆₇⋊C₈	—	8	4	no	—
28	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₆₇	D₈	1	1	yes	characteristic
40	C₄²×C₆₇	Q₁₆	1	1	yes	characteristic
41	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₆₇	—	2	2	no	—
43	C₁₆×C₆₇	C₂×C₈	1	4	yes	complemented
44	C₂³×C₆₇	#107	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#108	1	2	yes	—
51	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
52	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₆₇	—	2	1	no	—
57	C₈×C₆₇	—	2	1	no	—
58	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
59	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
61	C₆₇⋊C₈	—	8	4	no	—
62	C₂²×C₆₇	#335	1	2	yes	—
63	C₂²×C₆₇	#366	1	2	yes	—
64	C₂²×C₆₇	#368	1	2	yes	—
65	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	—	2	1	no	—
68	C₄×C₆₇	#335	1	2	yes	—
69	C₄×C₆₇	#366	1	2	yes	—
70	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
72	C₄×C₆₇	C₂×D₁₆	1	1	yes	characteristic
73	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
74	C₂×#1371	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	#1371	1	4	yes	complemented
76	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
77	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
78	C₂×C₆₇	C₂×#368	1	1	yes	characteristic
79	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
80	C₂×#335	—	67	2	no	—
81	#1371	—	67	4	no	—
82	C₆₇	C₂×#1371	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
84	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	134	2	no	—
86	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄×C₁₆	Dic₆₇	1	2	yes	—
88	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
89	#335	—	67	4	no	—
90	#335	—	134	2	no	—
91	C₂²×C₈	—	268	2	no	—
92	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₆₇⋊C₈	1	4	yes	complemented
97	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
98	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
99	C₄⋊C₈	—	268	2	no	—
100	C₄⋊C₈	—	134	4	no	—
101	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
102	C₄×C₈	D₂₆₈	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
104	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
107	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	268	4	no	—
110	C₂×C₈	—	536	4	no	—
111	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
113	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
114	C₂×C₈	#11761612	1	2	yes	—
115	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
117	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
118	C₄²	D₅₃₆	1	1	yes	characteristic
119	C₄²	Dic₂₆₈	1	1	yes	characteristic
120	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	C₂×C₆₇⋊C₈	1	4	yes	complemented
123	C₂³	#14523977	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
126	C₂×C₄	#14523977	1	1	yes	characteristic
127	C₂×C₄	#14523977	1	1	yes	characteristic
128	C₂×C₄	#14523977	1	1	yes	characteristic
129	C₂×C₄	#14523979	1	2	yes	—
130	C₂×C₄	D₁₀₇₂	1	2	yes	—
131	C₂×C₄	Dic₅₃₆	1	2	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	536	4	no	—
138	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
139	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
140	C₈	C₂×#11761612	1	1	yes	characteristic
141	C₂²	#16863149	1	2	yes	—
142	C₂²	#16863222	1	2	yes	—
143	C₂²	#16863226	1	2	yes	—
144	C₂²	C₂×#14523977	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	#16863149	1	2	yes	—
148	C₄	#16863222	1	2	yes	—
149	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
150	C₄	C₂×#14523979	1	1	yes	characteristic
151	C₄	C₂×D₁₀₇₂	1	1	yes	characteristic
152	C₄	C₂×Dic₅₃₆	1	1	yes	characteristic
153	C₂	#22558281	1	4	yes	direct product factor
154	C₂	C₂×#16863149	1	1	yes	characteristic
155	C₂	C₂×#16863222	1	1	yes	characteristic
156	C₂	C₂×#16863226	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₃₆
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#107
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1371
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ 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. #22558281	C₂	direct (N × H)
19. C₂×C₁₆×C₆₇	C₈	semidirect
43. C₁₆×C₆₇	C₂×C₈	semidirect
75. C₂×C₆₇	#1371	semidirect
82. C₆₇	C₂×#1371	semidirect
96. C₂×C₁₆	C₆₇⋊C₈	semidirect
122. C₁₆	C₂×C₆₇⋊C₈	semidirect
153. C₂	#22558281	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×#16863149	2
#22558281	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#1371	67

Subgroups — C2×#22558281 #30767409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558281 #30767409