Subgroups — #322×Dic₆₇ #30745178

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₆₇	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863118	C₂	1	2	yes	maximal
4	C₂×#16863117	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₇	C₄	1	8	yes	—
9	C₂×C₈×Dic₆₇	C₄	1	4	yes	—
10	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
11	#16863118	C₄	1	8	yes	complemented
12	#16863117	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	#322×C₆₇	C₄	1	4	yes	complemented
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
20	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₈×Dic₆₇	—	2	16	no	—
24	C₈×Dic₆₇	C₈	1	16	yes	complemented
25	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×Dic₆₇	C₄²	1	4	yes	—
33	C₂×C₆₇⋊C₈	—	2	8	no	—
34	C₂×C₆₇⋊C₈	C₄²	1	8	yes	—
35	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
36	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	—
37	C₄×Dic₆₇	C₂×C₈	1	8	yes	—
38	C₄×Dic₆₇	M₁₆	1	16	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₆₇	—	2	8	no	—
42	C₂×C₈×C₆₇	C₄²	1	8	yes	—
43	C₂×C₈×C₆₇	C₄²	1	4	yes	—
44	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	—
45	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
53	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
54	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
55	C₂×Dic₆₇	C₄×C₈	1	8	yes	—
56	C₂×Dic₆₇	#97	1	16	yes	—
57	C₈×C₆₇	—	2	8	no	—
58	C₈×C₆₇	C₄×C₈	1	8	yes	complemented
59	C₆₇⋊C₈	—	2	8	no	—
60	C₆₇⋊C₈	C₄×C₈	1	8	yes	complemented
61	C₂²×C₆₇	C₄³	1	1	yes	characteristic
62	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
63	C₂²×C₆₇	C₂×#97	1	4	yes	—
64	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
65	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
66	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
67	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
68	Dic₆₇	#322	1	16	yes	direct product factor
69	C₄×#322	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	C₂×#322	1	4	yes	—
71	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
72	C₂×C₆₇	C₄×#97	1	2	yes	—
73	C₄²×C₈	—	67	1	no	—
74	C₄²×C₈	—	67	2	no	—
75	C₂×#322	—	67	1	no	—
76	C₂×#322	—	67	2	no	—
77	C₂×#322	D₆₇	1	1	yes	characteristic
78	C₆₇	C₄×#322	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄×C₈	—	67	4	no	—
80	C₂×C₄×C₈	—	67	1	no	—
81	C₂×C₄×C₈	—	67	2	no	—
82	C₂×C₄×C₈	—	67	8	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
85	C₄³	—	67	1	no	—
86	#322	—	67	4	no	—
87	#322	—	67	8	no	—
88	#322	Dic₆₇	1	4	yes	direct product factor
89	C₂²×C₈	—	67	2	no	—
90	C₂²×C₈	—	67	4	no	—
91	C₂²×C₈	C₄×D₆₇	1	4	yes	—
92	C₂²×C₈	C₄×D₆₇	1	2	yes	—
93	C₂×C₄²	—	67	2	no	—
94	C₂×C₄²	—	67	4	no	—
95	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
96	C₄×C₈	—	67	16	no	—
97	C₄×C₈	—	67	4	no	—
98	C₄×C₈	—	67	4	no	—
99	C₄×C₈	—	67	4	no	—
100	C₄×C₈	—	134	16	no	—
101	C₄×C₈	C₄×D₆₇	1	4	yes	—
102	C₄×C₈	C₄×D₆₇	1	2	yes	—
103	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
104	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
105	C₂²×C₄	—	67	4	no	—
106	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	67	4	no	—
110	C₂×C₈	—	67	8	no	—
111	C₂×C₈	—	67	8	no	—
112	C₂×C₈	—	134	8	no	—
113	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
114	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
115	C₂×C₈	C₈×D₆₇	1	8	yes	—
116	C₄²	—	67	8	no	—
117	C₄²	—	67	16	no	—
118	C₄²	C₂×C₄×D₆₇	1	2	yes	—
119	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
120	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
121	C₂³	C₄²×D₆₇	1	1	yes	characteristic
122	C₂×C₄	—	67	8	no	—
123	C₂×C₄	—	67	16	no	—
124	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
125	C₂×C₄	C₄²×D₆₇	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
129	C₂×C₄	M₁₆×D₆₇	1	8	yes	—
130	C₈	—	2	8	no	—
131	C₈	—	67	8	no	—
132	C₈	—	134	8	no	—
133	C₈	C₈×Dic₆₇	1	8	yes	complemented
134	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
135	C₂²	C₄×C₈×D₆₇	1	2	yes	—
136	C₂²	#97×D₆₇	1	4	yes	—
137	C₄	—	67	16	no	—
138	C₄	C₄×C₈×D₆₇	1	2	yes	—
139	C₄	C₂×C₈×Dic₆₇	1	2	yes	—
140	C₄	M₁₆×Dic₆₇	1	4	yes	—
141	C₄	#16863348	1	4	yes	—
142	C₂	#322×D₆₇	1	4	yes	—
143	C₂	C₄×C₈×Dic₆₇	1	1	yes	characteristic
144	C₂	#97×Dic₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂×C₆₇	C₄²×C₈
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	C₄×#322
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
11. #16863118	C₄	semidirect
12. #16863117	C₄	semidirect
15. #322×C₆₇	C₄	semidirect
24. C₈×Dic₆₇	C₈	semidirect
58. C₈×C₆₇	C₄×C₈	semidirect
60. C₆₇⋊C₈	C₄×C₈	semidirect
68. Dic₆₇	#322	direct (N × H)
78. C₆₇	C₄×#322	semidirect
88. #322	Dic₆₇	direct (N × H)
133. C₈	C₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₆₇	2
C₂×#16863118	2
C₂×#16863117	2
C₂×#322×C₆₇	2
C₄×#322	67

Subgroups — #322×Dic67 #30745178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #322×Dic₆₇ #30745178