Subgroups — #30770661

Order 17152.

402 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863173	C₂	1	4	yes	maximal
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
3	#22559895	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523954	C₄	1	1	yes	characteristic
7	C₂×#14523953	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
9	#16863242	C₄	1	1	yes	characteristic
10	#16863173	C₄	1	16	yes	complemented
11	#16863172	C₄	1	2	yes	complemented
12	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#346×C₆₇	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	#14523983	C₂×C₄	1	8	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
21	#14523954	—	2	2	no	—
22	#14523953	C₂×C₄	1	8	yes	—
23	C₈×Dic₆₇	—	2	4	no	—
24	C₆₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₇	C₂×C₄	1	8	yes	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₆₇	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
38	C₄×Dic₆₇	—	2	2	no	—
39	C₄×Dic₆₇	C₄²	1	4	yes	—
40	C₆₇⋊C₁₆	C₄²	1	8	yes	—
41	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₆₇	C₄²	1	1	yes	characteristic
50	C₄²×C₆₇	C₄²	1	1	yes	characteristic
51	C₄²×C₆₇	M₁₆	1	2	yes	—
52	C₁₆×C₆₇	C₄²	1	8	yes	—
53	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	4	2	no	—
55	C₂×C₄×C₆₇	#97	1	2	yes	—
56	C₂×C₄×C₆₇	#97	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
58	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₆₇	—	2	4	no	—
62	C₂×Dic₆₇	—	4	4	no	—
63	C₈×C₆₇	#97	1	2	yes	—
64	C₈×C₆₇	#97	1	2	yes	—
65	C₈×C₆₇	C₂×C₄²	1	2	yes	—
66	C₈×C₆₇	C₂×C₄²	1	2	yes	—
67	C₆₇⋊C₈	—	2	4	no	—
68	C₂²×C₆₇	—	2	2	no	—
69	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
70	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	C₄³	1	1	yes	characteristic
74	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
75	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
76	Dic₆₇	—	4	4	no	—
77	#12541	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	—	2	2	no	—
79	C₂×C₆₇	C₄×#97	1	1	yes	characteristic
80	C₂×C₆₇	#2110	1	2	yes	—
81	C₂×#347	—	67	4	no	—
82	C₄×M₃₂	—	67	1	no	—
83	C₄×M₃₂	D₆₇	1	1	yes	characteristic
84	#1726	—	67	1	no	—
85	C₆₇	#12541	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	67	2	no	—
87	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
88	C₂×#97	—	67	2	no	—
89	C₂×#97	—	67	1	no	—
90	C₂×C₄×C₈	—	67	2	no	—
91	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
92	#376	—	67	1	no	—
93	#347	—	67	16	no	—
94	#346	—	67	2	no	—
95	#346	Dic₆₇	1	2	yes	complemented
96	C₄×C₁₆	—	67	2	no	—
97	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
98	C₂²×C₈	—	67	2	no	—
99	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
101	C₂×C₄²	—	67	2	no	—
102	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
103	M₃₂	—	67	8	no	—
104	M₃₂	C₄×D₆₇	1	8	yes	—
105	C₂×C₁₆	—	67	4	no	—
106	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
107	#97	—	67	8	no	—
108	#97	—	134	2	no	—
109	C₄×C₈	—	134	4	no	—
110	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
111	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
114	#95	—	134	2	no	—
115	C₂²×C₄	—	134	2	no	—
116	C₂²×C₄	#11761611	1	2	yes	—
117	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
118	C₂×C₈	—	67	4	no	—
119	C₂×C₈	—	134	2	no	—
120	C₂×C₈	—	134	2	no	—
121	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
123	C₂×C₈	#11761611	1	4	yes	—
124	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
126	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
127	C₄²	—	67	4	no	—
128	C₄²	—	134	2	no	—
129	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
130	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
131	C₄²	#11761611	1	2	yes	—
132	C₁₆	—	67	8	no	—
133	C₁₆	C₄×Dic₆₇	1	8	yes	—
134	C₂³	C₂×#11761611	1	1	yes	characteristic
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	268	4	no	—
137	C₂×C₄	—	134	4	no	—
138	C₂×C₄	#14523953	1	2	yes	—
139	C₂×C₄	#14523953	1	2	yes	—
140	C₂×C₄	C₄²×D₆₇	1	2	yes	—
141	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
142	C₂×C₄	C₂×#11761611	1	2	yes	—
143	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
144	C₈	—	134	4	no	—
145	C₈	#14523953	1	2	yes	—
146	C₈	#14523953	1	2	yes	—
147	C₈	C₄²×D₆₇	1	2	yes	—
148	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₂×#14523953	1	1	yes	characteristic
151	C₂²	C₄×#11761611	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	268	4	no	—
155	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
156	C₄	C₂×#14523953	1	1	yes	characteristic
157	C₄	C₄×#11761611	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	C₄×#14523953	1	1	yes	characteristic
160	C₂	#22561813	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₄³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#12541
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16863173	C₄	semidirect
11. #16863172	C₄	semidirect
12. C₄×C₆₇⋊C₁₆	C₄	semidirect
15. #346×C₆₇	C₄	semidirect
16. C₄×C₁₆×C₆₇	C₄	semidirect
85. C₆₇	#12541	semidirect
95. #346	Dic₆₇	semidirect
97. C₄×C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863173	2
C₄×#14523983	2
#22559895	2
C₄×M₃₂×C₆₇	2
#12541	67