Subgroups — C₂×#22558283 #30767412

Order 17152.

262 conjugacy classes of subgroups.

The subgroup lattice has 154 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	#22558283	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	SD₃₂	1	4	yes	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₈×C₆₇	—	2	1	no	—
56	C₈×C₆₇	—	2	1	no	—
57	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
58	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
60	C₆₇⋊C₈	—	8	4	no	—
61	C₂²×C₆₇	#335	1	2	yes	—
62	C₂²×C₆₇	#367	1	2	yes	—
63	C₂²×C₆₇	#368	1	2	yes	—
64	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
65	C₄×C₆₇	—	2	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	#335	1	2	yes	—
68	C₄×C₆₇	#367	1	2	yes	—
69	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
71	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
72	C₂×#1372	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	#1372	1	4	yes	complemented
74	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
75	C₂×C₆₇	C₂×#367	1	1	yes	characteristic
76	C₂×C₆₇	C₂×#368	1	1	yes	characteristic
77	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
78	C₂×#335	—	67	2	no	—
79	#1372	—	67	4	no	—
80	C₆₇	C₂×#1372	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
82	C₂²×C₁₆	Dic₆₇	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	134	2	no	—
84	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
85	C₄×C₁₆	Dic₆₇	1	2	yes	—
86	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
87	#335	—	67	4	no	—
88	#335	—	134	2	no	—
89	C₂²×C₈	—	268	2	no	—
90	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
92	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₆₇⋊C₈	1	4	yes	complemented
95	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
97	C₄⋊C₈	—	268	2	no	—
98	C₄⋊C₈	—	134	4	no	—
99	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
100	C₄×C₈	D₂₆₈	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
104	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
105	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	268	4	no	—
108	C₂×C₈	—	536	4	no	—
109	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
110	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
111	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
112	C₂×C₈	#11761612	1	2	yes	—
113	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
115	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
116	C₄²	D₅₃₆	1	1	yes	characteristic
117	C₄²	Dic₂₆₈	1	1	yes	characteristic
118	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
119	C₁₆	—	2	2	no	—
120	C₁₆	C₂×C₆₇⋊C₈	1	4	yes	complemented
121	C₂³	#14523977	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
124	C₂×C₄	#14523977	1	1	yes	characteristic
125	C₂×C₄	#14523977	1	1	yes	characteristic
126	C₂×C₄	#14523977	1	1	yes	characteristic
127	C₂×C₄	#14523979	1	2	yes	—
128	C₂×C₄	#14523988	1	4	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	536	4	no	—
135	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
136	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
137	C₈	C₂×#11761612	1	1	yes	characteristic
138	C₂²	#16863149	1	2	yes	—
139	C₂²	#16863224	1	2	yes	—
140	C₂²	#16863226	1	2	yes	—
141	C₂²	C₂×#14523977	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	#16863149	1	2	yes	—
145	C₄	#16863224	1	2	yes	—
146	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
147	C₄	C₂×#14523979	1	1	yes	characteristic
148	C₄	C₂×#14523988	1	2	yes	—
149	C₂	#22558283	1	4	yes	direct product factor
150	C₂	C₂×#16863149	1	1	yes	characteristic
151	C₂	C₂×#16863224	1	1	yes	characteristic
152	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×#1372
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. #22558283	C₂	direct (N × H)
19. C₂×C₁₆×C₆₇	C₈	semidirect
43. C₁₆×C₆₇	C₂×C₈	semidirect
73. C₂×C₆₇	#1372	semidirect
80. C₆₇	C₂×#1372	semidirect
94. C₂×C₁₆	C₆₇⋊C₈	semidirect
120. C₁₆	C₂×C₆₇⋊C₈	semidirect
149. C₂	#22558283	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
#22558283	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#1372	67

Subgroups — C2×#22558283 #30767412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558283 #30767412