Subgroups — C₂²×#16863222 #31008852

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	2	yes	maximal
2	C₂×#16863222	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₆₇	—	2	2	no	—
5	C₂×#14523977	—	2	6	no	—
6	C₂×#14523977	C₂²	1	12	yes	—
7	#16863222	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₆₇	C₄	1	8	yes	complemented
10	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
11	C₂³×Dic₆₇	—	4	2	no	—
12	C₂×C₄⋊Dic₆₇	—	2	12	no	—
13	C₂×C₄⋊Dic₆₇	—	4	6	no	—
14	#14523977	—	2	12	no	—
15	#14523977	C₂³	1	8	yes	—
16	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₆₇	D₄	1	3	yes	—
19	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
21	C₂×C₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
22	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
23	C₂²×Dic₆₇	—	4	12	no	—
24	C₂²×Dic₆₇	—	8	8	no	—
25	C₄⋊Dic₆₇	—	2	8	no	—
26	C₄⋊Dic₆₇	—	4	12	no	—
27	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
29	C₂²×C₄×C₆₇	D₈	1	3	yes	—
30	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
31	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
32	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
33	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
34	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
35	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
36	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
37	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
38	C₂³×C₆₇	#107	1	4	yes	—
39	C₂³×C₆₇	D₁₆	1	4	yes	—
40	C₂³×C₆₇	Q₃₂	1	4	yes	—
41	C₂³×C₆₇	C₂×D₈	1	3	yes	—
42	C₂×C₄×C₆₇	#107	1	12	yes	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
44	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
45	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
46	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
47	C₂×Dic₆₇	—	4	8	no	—
48	C₂×Dic₆₇	—	8	24	no	—
49	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
50	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₆₇	#366	1	16	yes	complemented
53	C₂²×C₆₇	C₂×#107	1	6	yes	—
54	C₂²×C₆₇	C₂×D₁₆	1	6	yes	—
55	C₂²×C₆₇	C₂×Q₃₂	1	6	yes	—
56	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
57	C₄×C₆₇	C₂×#107	1	6	yes	—
58	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
60	Dic₆₇	—	8	8	no	—
61	C₂²×#366	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	C₂×#366	1	12	yes	complemented
63	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
64	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
65	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
66	C₂³×C₁₆	D₆₇	1	1	yes	characteristic
67	C₂²×#107	—	67	2	no	—
68	C₂×#366	—	67	12	no	—
69	C₆₇	C₂²×#366	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
71	C₂²×C₄⋊C₄	—	134	2	no	—
72	C₂²×C₁₆	Dic₆₇	1	8	yes	complemented
73	C₂²×C₁₆	C₂×D₆₇	1	6	yes	—
74	C₂×#107	—	67	12	no	—
75	C₂×#107	—	134	6	no	—
76	#366	—	67	16	no	—
77	C₂³×C₄	—	268	2	no	—
78	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
79	C₂²×C₈	Dic₁₃₄	1	4	yes	—
80	C₂²×C₈	D₂₆₈	1	3	yes	—
81	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
82	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
83	C₂×C₄⋊C₄	—	268	6	no	—
84	C₂×C₄⋊C₄	—	134	12	no	—
85	C₂×C₁₆	C₂×Dic₆₇	1	24	yes	complemented
86	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
87	#107	—	67	8	no	—
88	#107	—	134	12	no	—
89	C₂⁴	D₅₃₆	1	1	yes	characteristic
90	C₂²×C₄	—	268	12	no	—
91	C₂²×C₄	—	536	8	no	—
92	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
93	C₂²×C₄	D₅₃₆	1	3	yes	—
94	C₂²×C₄	Dic₂₆₈	1	4	yes	—
95	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
96	C₂×C₈	C₄⋊Dic₆₇	1	12	yes	—
97	C₂×C₈	C₂²×Dic₆₇	1	6	yes	—
98	C₂×C₈	C₂×D₂₆₈	1	3	yes	—
99	C₂×C₈	C₂×Dic₁₃₄	1	6	yes	—
100	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
101	C₄⋊C₄	—	268	12	no	—
102	C₄⋊C₄	—	134	8	no	—
103	C₁₆	C₂²×Dic₆₇	1	8	yes	complemented
104	C₂³	#14523977	1	4	yes	—
105	C₂³	D₁₀₇₂	1	4	yes	—
106	C₂³	Dic₅₃₆	1	4	yes	—
107	C₂³	C₂×D₅₃₆	1	3	yes	—
108	C₂×C₄	—	268	8	no	—
109	C₂×C₄	—	536	24	no	—
110	C₂×C₄	#14523977	1	12	yes	—
111	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
112	C₂×C₄	C₂×D₅₃₆	1	3	yes	—
113	C₂×C₄	C₂×Dic₂₆₈	1	6	yes	—
114	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
115	C₈	C₂×C₄⋊Dic₆₇	1	6	yes	—
116	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
117	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
118	C₂²	#16863222	1	16	yes	direct product factor
119	C₂²	C₂×#14523977	1	6	yes	—
120	C₂²	C₂×D₁₀₇₂	1	6	yes	—
121	C₂²	C₂×Dic₅₃₆	1	6	yes	—
122	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
123	C₄	—	536	8	no	—
124	C₄	C₂×#14523977	1	6	yes	—
125	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
126	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
127	C₂	C₂×#16863222	1	12	yes	direct product factor
128	C₂	C₂²×#14523977	1	1	yes	characteristic
129	C₂	C₂²×D₁₀₇₂	1	1	yes	characteristic
130	C₂	C₂²×Dic₅₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂³×C₁₆×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	C₂²×#366
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863222	C₂	direct (N × H)
7. #16863222	C₂²	direct (N × H)
9. C₂²×C₁₆×C₆₇	C₄	semidirect
21. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
37. C₁₆×C₆₇	C₂²×C₄	semidirect
52. C₂²×C₆₇	#366	semidirect
62. C₂×C₆₇	C₂×#366	semidirect
69. C₆₇	C₂²×#366	semidirect
72. C₂²×C₁₆	Dic₆₇	semidirect
85. C₂×C₁₆	C₂×Dic₆₇	semidirect
103. C₁₆	C₂²×Dic₆₇	semidirect
118. C₂²	#16863222	direct (N × H)
127. C₂	C₂×#16863222	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14523977	2
C₂×#16863222	2
C₂³×C₁₆×C₆₇	2
C₂²×#366	67

Subgroups — C22×#16863222 #31008852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863222 #31008852