Subgroups — C₂×C₆₇×#2254 #30712976

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2254	C₂	1	8	yes	maximal, direct product factor
3	C₂×#367×C₆₇	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
6	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
7	#500×C₆₇	—	2	4	no	—
8	#500×C₆₇	C₂²	1	8	yes	—
9	C₂×#107×C₆₇	—	2	2	no	—
10	C₂×#107×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#367×C₆₇	C₂²	1	16	yes	—
13	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₇	—	4	2	no	—
15	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₇	—	2	8	no	—
18	C₄⋊Q₈×C₆₇	—	4	4	no	—
19	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
21	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₇	Q₈	1	8	yes	complemented
23	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
24	#107×C₆₇	—	2	4	no	—
25	#107×C₆₇	—	2	8	no	—
26	#107×C₆₇	C₂³	1	8	yes	—
27	C₄×C₈×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	2	yes	—
29	C₂×Q₈×C₆₇	—	4	8	no	—
30	C₂×Q₈×C₆₇	—	8	8	no	—
31	C₂²×C₄×C₆₇	—	4	4	no	—
32	C₂²×C₄×C₆₇	D₈	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
35	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
38	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄×C₆₇	—	2	8	no	—
40	C₄⋊C₄×C₆₇	—	4	4	no	—
41	C₄⋊C₄×C₆₇	—	4	8	no	—
42	C₄²×C₆₇	D₈	1	2	yes	—
43	C₄²×C₆₇	C₂×D₄	1	2	yes	—
44	C₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
45	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
46	Q₈×C₆₇	—	8	16	no	—
47	C₂×C₄×C₆₇	—	4	8	no	—
48	C₂×C₄×C₆₇	—	8	8	no	—
49	C₂×C₄×C₆₇	SD₃₂	1	8	yes	—
50	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
54	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
56	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
57	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
58	C₂²×C₆₇	#500	1	2	yes	—
59	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
60	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
61	C₄×C₆₇	—	8	8	no	—
62	C₄×C₆₇	#500	1	2	yes	—
63	C₄×C₆₇	C₂×SD₃₂	1	8	yes	—
64	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
66	C₂×#2254	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₆₇	#2254	1	4	yes	direct product factor
68	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
69	C₂×C₆₇	C₂²×SD₃₂	1	2	yes	—
70	C₂×#500	C₂×C₆₇	1	2	yes	—
71	#2254	C₂×C₆₇	1	8	yes	direct product factor
72	C₂×#367	C₂×C₆₇	1	4	yes	—
73	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
74	C₆₇	C₂×#2254	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₄⋊Q₈	—	2	2	no	—
76	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
77	#500	—	2	4	no	—
78	#500	C₂²×C₆₇	1	8	yes	—
79	C₂×#107	—	2	2	no	—
80	C₂×#107	C₂²×C₆₇	1	4	yes	—
81	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
82	#367	C₂²×C₆₇	1	16	yes	—
83	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
84	C₂²×Q₈	—	4	2	no	—
85	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
86	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	8	no	—
88	C₄⋊Q₈	—	4	4	no	—
89	C₂×C₄⋊C₄	—	2	4	no	—
90	C₂×C₄⋊C₄	—	4	2	no	—
91	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
92	C₂×C₁₆	Q₈×C₆₇	1	8	yes	complemented
93	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
94	#107	—	2	8	no	—
95	#107	—	2	4	no	—
96	#107	C₂³×C₆₇	1	8	yes	—
97	C₄×C₈	D₄×C₆₇	1	2	yes	—
98	C₄×C₈	C₂³×C₆₇	1	2	yes	—
99	C₂×Q₈	—	4	8	no	—
100	C₂×Q₈	—	8	8	no	—
101	C₂²×C₄	—	4	4	no	—
102	C₂²×C₄	D₈×C₆₇	1	2	yes	—
103	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
104	C₂×C₈	Q₁₆×C₆₇	1	4	yes	—
105	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
107	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
108	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
109	C₄⋊C₄	—	2	8	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	4	8	no	—
112	C₄²	D₈×C₆₇	1	2	yes	—
113	C₄²	C₂×D₄×C₆₇	1	2	yes	—
114	C₁₆	C₂×Q₈×C₆₇	1	8	yes	complemented
115	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
116	Q₈	—	8	16	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	—	8	8	no	—
119	C₂×C₄	SD₃₂×C₆₇	1	8	yes	—
120	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
121	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
123	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
124	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
125	C₈	C₄⋊Q₈×C₆₇	1	2	yes	—
126	C₈	C₂×Q₁₆×C₆₇	1	4	yes	—
127	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
128	C₂²	#500×C₆₇	1	2	yes	—
129	C₂²	C₂×SD₃₂×C₆₇	1	4	yes	—
130	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
131	C₄	—	8	8	no	—
132	C₄	#500×C₆₇	1	2	yes	—
133	C₄	C₂×SD₃₂×C₆₇	1	8	yes	—
134	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
135	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
136	C₂	C₆₇×#2254	1	4	yes	direct product factor
137	C₂	C₂×#500×C₆₇	1	1	yes	characteristic
138	C₂	C₂²×SD₃₂×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2254	C₂	direct (N × H)
22. C₂×C₁₆×C₆₇	Q₈	semidirect
44. C₁₆×C₆₇	C₂×Q₈	semidirect
66. C₂×#2254	C₆₇	direct (N × H)
67. C₂×C₆₇	#2254	direct (N × H)
71. #2254	C₂×C₆₇	direct (N × H)
74. C₆₇	C₂×#2254	direct (N × H)
92. C₂×C₁₆	Q₈×C₆₇	semidirect
114. C₁₆	C₂×Q₈×C₆₇	semidirect
136. C₂	C₆₇×#2254	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×#500×C₆₇	2
C₆₇×#2254	2
C₂×#367×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2254	67

Subgroups — C2×C67×#2254 #30712976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2254 #30712976