Subgroups — #30770656

Order 17152.

478 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863171	C₂	1	2	yes	maximal
2	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₆₇	C₂	1	2	yes	maximal
4	#22559895	C₂	1	1	yes	maximal, characteristic
5	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14523954	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
11	#16863242	C₄	1	1	yes	characteristic
12	#16863171	C₄	1	8	yes	complemented
13	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
14	C₁₆×Dic₆₇	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#346×C₆₇	C₄	1	4	yes	complemented
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	#14523954	C₈	1	4	yes	—
25	#14523953	C₈	1	8	yes	—
26	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
27	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
28	C₆₇⋊#95	C₈	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
38	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
39	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
40	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
41	C₄×Dic₆₇	C₄²	1	4	yes	—
42	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
43	C₆₇⋊C₁₆	C₄²	1	4	yes	—
44	C₆₇⋊C₁₆	C₄²	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄²	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₄²	1	2	yes	—
54	C₄²×C₆₇	C₂×C₈	1	2	yes	—
55	C₁₆×C₆₇	C₄²	1	8	yes	—
56	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
59	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
63	C₂×Dic₆₇	—	2	8	no	—
64	C₂×Dic₆₇	C₄×C₈	1	8	yes	—
65	C₈×C₆₇	C₄×C₈	1	4	yes	—
66	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
68	C₈×C₆₇	C₂×C₄²	1	2	yes	—
69	C₆₇⋊C₈	C₄×C₈	1	8	yes	—
70	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
71	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂²×C₆₇	#504	1	4	yes	—
73	C₄×C₆₇	—	2	4	no	—
74	C₄×C₆₇	C₄³	1	1	yes	characteristic
75	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
76	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
77	Dic₆₇	—	2	8	no	—
78	#12539	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
80	C₂×C₆₇	#2108	1	4	yes	—
81	C₂×C₆₇	#2108	1	2	yes	—
82	C₂×#346	—	67	2	no	—
83	C₂×#346	D₆₇	1	1	yes	characteristic
84	C₂×C₄×C₁₆	—	67	2	no	—
85	C₂×C₄×C₁₆	—	67	1	no	—
86	#1726	—	67	1	no	—
87	C₆₇	#12539	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	67	1	no	—
89	C₂²×C₁₆	—	67	1	no	—
90	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
91	C₂×#97	—	67	2	no	—
92	C₂×#97	—	67	1	no	—
93	C₂×C₄×C₈	—	67	2	no	—
94	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	#376	—	67	1	no	—
96	#346	—	67	8	no	—
97	#346	Dic₆₇	1	4	yes	complemented
98	C₄×C₁₆	—	67	8	no	—
99	C₄×C₁₆	—	67	4	no	—
100	C₂²×C₈	—	67	2	no	—
101	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	C₂×C₄²	—	67	2	no	—
104	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
105	C₂×C₁₆	—	67	2	no	—
106	C₂×C₁₆	—	67	2	no	—
107	C₂×C₁₆	—	67	4	no	—
108	C₂×C₁₆	—	67	4	no	—
109	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
110	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
111	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
112	#97	—	67	8	no	—
113	#97	—	67	4	no	—
114	C₄×C₈	—	67	4	no	—
115	C₄×C₈	—	67	4	no	—
116	C₄×C₈	C₄×D₆₇	1	2	yes	—
117	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
118	#95	—	67	4	no	—
119	C₂²×C₄	—	67	4	no	—
120	C₂²×C₄	C₈×D₆₇	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
122	C₂×C₈	—	67	4	no	—
123	C₂×C₈	—	67	4	no	—
124	C₂×C₈	—	67	4	no	—
125	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
126	C₂×C₈	C₈×D₆₇	1	4	yes	—
127	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
130	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
131	C₄²	—	67	4	no	—
132	C₄²	—	67	4	no	—
133	C₄²	C₄×Dic₆₇	1	2	yes	—
134	C₄²	C₈×D₆₇	1	2	yes	—
135	C₁₆	—	67	4	no	—
136	C₁₆	—	67	4	no	—
137	C₁₆	C₄×Dic₆₇	1	8	yes	—
138	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	67	8	no	—
141	C₂×C₄	—	134	8	no	—
142	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
143	C₂×C₄	C₄²×D₆₇	1	2	yes	—
144	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
147	C₈	—	67	8	no	—
148	C₈	C₈×Dic₆₇	1	4	yes	—
149	C₈	C₄²×D₆₇	1	1	yes	characteristic
150	C₈	C₄²×D₆₇	1	1	yes	characteristic
151	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
152	C₂²	C₄×C₈×D₆₇	1	2	yes	—
153	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
154	C₂²	#16863814	1	4	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	134	8	no	—
157	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
158	C₄	C₄×C₈×D₆₇	1	2	yes	—
159	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
160	C₂	C₄×C₈×Dic₆₇	1	1	yes	characteristic
161	C₂	#22561807	1	4	yes	—
162	C₂	#22561808	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₄²×C₈
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#12539
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. #16863171	C₄	semidirect
13. C₄×C₆₇⋊C₁₆	C₄	semidirect
14. C₁₆×Dic₆₇	C₄	semidirect
17. #346×C₆₇	C₄	semidirect
87. C₆₇	#12539	semidirect
97. #346	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863171	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₁₆×Dic₆₇	2
#22559895	2
C₂×#346×C₆₇	2
#12539	67