Subgroups — C₂×#22559896 #30853158

Order 17152.

1614 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	C₂²×#14523954	C₂	1	3	yes	maximal
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	3	yes	maximal
3	#22559896	C₂	1	8	yes	maximal, direct product factor
4	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	3	yes	—
7	C₂×#14523954	C₄	1	24	yes	complemented
8	C₂×#14523954	C₂²	1	12	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₄	1	24	yes	complemented
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	12	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
12	#376×C₆₇	C₄	1	4	yes	—
13	#376×C₆₇	C₂²	1	4	yes	—
14	C₂×#95×C₆₇	C₄	1	4	yes	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	12	yes	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	12	yes	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	24	yes	—
18	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
19	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
20	#14523954	C₂×C₄	1	48	yes	complemented
21	C₄×C₆₇⋊C₈	C₂×C₄	1	48	yes	complemented
22	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
28	#95×C₆₇	C₂×C₄	1	16	yes	—
29	C₂×C₆₇⋊C₈	C₄²	1	16	yes	—
30	C₂×C₆₇⋊C₈	C₄²	1	48	yes	—
31	C₂×C₆₇⋊C₈	C₂²×C₄	1	12	yes	—
32	C₂×C₆₇⋊C₈	C₂²×C₄	1	12	yes	—
33	C₂×C₆₇⋊C₈	C₂²×C₄	1	24	yes	—
34	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	24	no	—
36	C₂²×C₄×C₆₇	C₄²	1	24	yes	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
42	C₄²×C₆₇	C₄²	1	24	yes	—
43	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
44	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
45	C₂³×C₆₇	C₂×C₄²	1	6	yes	—
46	C₂³×C₆₇	#131	1	8	yes	—
47	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	24	no	—
49	C₂×C₄×C₆₇	—	2	48	no	—
50	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
51	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
52	C₂×C₄×C₆₇	C₂×C₄²	1	24	yes	—
53	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
54	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
55	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
56	C₆₇⋊C₈	C₂×C₄²	1	8	yes	—
57	C₆₇⋊C₈	C₂×C₄²	1	24	yes	—
58	C₂²×C₆₇	C₄³	1	4	yes	—
59	C₂²×C₆₇	#405	1	24	yes	—
60	C₂²×C₆₇	C₂²×C₄²	1	3	yes	—
61	C₂²×C₆₇	C₂×#131	1	4	yes	—
62	C₄×C₆₇	—	2	24	no	—
63	C₄×C₆₇	C₄³	1	4	yes	—
64	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
65	C₄×C₆₇	C₂²×C₄²	1	3	yes	—
66	C₂×#1726	—	67	1	no	maximal, Sylow
67	C₂×C₆₇	#1726	1	8	yes	complemented
68	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
69	C₂×C₆₇	C₂×#405	1	6	yes	—
70	C₂²×#97	—	67	3	no	—
71	C₂²×C₄×C₈	—	67	3	no	—
72	C₂×#376	D₆₇	1	1	yes	characteristic
73	#1726	—	67	8	no	—
74	C₆₇	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₈	—	67	1	no	—
76	C₂³×C₈	—	67	3	no	—
77	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
78	C₂×#97	—	67	24	no	—
79	C₂×#97	—	67	12	no	—
80	C₂×C₄×C₈	—	67	24	no	—
81	C₂×C₄×C₈	—	67	12	no	—
82	#376	Dic₆₇	1	4	yes	—
83	#376	C₂×D₆₇	1	4	yes	—
84	C₂×#95	Dic₆₇	1	4	yes	—
85	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
86	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
87	C₂²×C₈	—	67	12	no	—
88	C₂²×C₈	—	67	12	no	—
89	C₂²×C₈	—	67	24	no	—
90	C₂²×C₈	—	67	2	no	—
91	C₂²×C₈	—	67	6	no	—
92	C₂×C₄²	C₄×D₆₇	1	12	yes	—
93	C₂×C₄²	C₄×D₆₇	1	12	yes	—
94	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
95	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
96	#97	—	67	48	no	—
97	C₄×C₈	—	67	48	no	—
98	#95	C₂×Dic₆₇	1	16	yes	—
99	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
100	C₂²×C₄	—	2	24	no	—
101	C₂²×C₄	C₄×Dic₆₇	1	24	yes	—
102	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
103	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
104	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
105	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
107	C₂×C₈	—	67	16	no	—
108	C₂×C₈	—	67	48	no	—
109	C₂×C₈	—	67	12	no	—
110	C₂×C₈	—	67	12	no	—
111	C₂×C₈	—	67	24	no	—
112	C₄²	C₄×Dic₆₇	1	24	yes	—
113	C₄²	C₂×C₄×D₆₇	1	12	yes	—
114	C₄²	C₂×C₄×D₆₇	1	12	yes	—
115	C₂³	C₂×C₄×Dic₆₇	1	6	yes	—
116	C₂³	#14524050	1	8	yes	—
117	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
118	C₂×C₄	—	2	48	no	—
119	C₂×C₄	—	2	24	no	—
120	C₂×C₄	C₄²×D₆₇	1	4	yes	—
121	C₂×C₄	C₄²×D₆₇	1	12	yes	—
122	C₂×C₄	C₂×C₄×Dic₆₇	1	24	yes	—
123	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
124	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
125	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
126	C₈	—	67	8	no	—
127	C₈	—	67	24	no	—
128	C₂²	C₄²×Dic₆₇	1	4	yes	—
129	C₂²	#16863352	1	24	yes	—
130	C₂²	C₂²×C₄×Dic₆₇	1	3	yes	—
131	C₂²	C₂×#14524050	1	4	yes	—
132	C₄	—	2	24	no	—
133	C₄	C₄²×Dic₆₇	1	4	yes	—
134	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
135	C₄	C₂×C₄²×D₆₇	1	3	yes	—
136	C₂	#22559896	1	8	yes	direct product factor
137	C₂	C₂×C₄²×Dic₆₇	1	1	yes	characteristic
138	C₂	C₂×#16863352	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#376×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#376	D₆₇
hypocenter	C₆₇	C₂×#1726
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #22559896	C₂	direct (N × H)
7. C₂×#14523954	C₄	semidirect
9. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
20. #14523954	C₂×C₄	semidirect
21. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
67. C₂×C₆₇	#1726	semidirect
74. C₆₇	C₂×#1726	semidirect
136. C₂	#22559896	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×#14523954	2
C₂²×C₄×C₆₇⋊C₈	2
#22559896	2
C₂×#376×C₆₇	2
C₂×#1726	67

Subgroups — C2×#22559896 #30853158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559896 #30853158