Subgroups — #30770663

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561814	C₂	1	1	yes	maximal, characteristic
2	#22561813	C₂	1	2	yes	maximal
3	#22561805	C₂	1	2	yes	maximal
4	#22559964	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	#16863351	C₂²	1	2	yes	—
8	#16863174	C₄	1	4	yes	complemented
9	#16863173	C₄	1	8	yes	complemented
10	#16863171	C₄	1	4	yes	complemented
11	C₁₆×Dic₆₇	C₄	1	4	yes	complemented
12	C₆₇⋊#338	C₄	1	1	yes	characteristic
13	#16863153	C₄	1	1	yes	characteristic
14	#16863152	C₄	1	2	yes	—
15	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#405×C₆₇	C₂²	1	1	yes	characteristic
18	#346×C₆₇	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
20	C₂×#11761612	C₂×C₄	1	2	yes	—
21	#14524002	C₂×C₄	1	2	yes	—
22	#14523983	C₂×C₄	1	4	yes	—
23	#14523983	C₂×C₄	1	4	yes	—
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
25	#14523953	C₂×C₄	1	4	yes	—
26	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	#97×C₆₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	#11761612	—	2	2	no	—
39	#11761612	—	2	2	no	—
40	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
41	C₄⋊Dic₆₇	—	2	2	no	—
42	C₂²⋊Dic₆₇	—	2	2	no	—
43	C₄×Dic₆₇	C₄²	1	4	yes	—
44	C₆₇⋊C₁₆	C₄²	1	8	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	C₄²	1	2	yes	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂²⋊C₄×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	C₄²	1	2	yes	—
55	C₁₆×C₆₇	C₄²	1	2	yes	—
56	C₁₆×C₆₇	C₄²	1	1	yes	characteristic
57	C₁₆×C₆₇	C₄²	1	4	yes	—
58	C₁₆×C₆₇	C₄²	1	1	yes	characteristic
59	C₂³×C₆₇	#131	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#131	1	2	yes	—
65	C₂×Dic₆₇	—	2	4	no	—
66	C₈×C₆₇	—	2	2	no	—
67	C₈×C₆₇	C₂×C₄²	1	2	yes	—
68	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
70	C₆₇⋊C₈	—	2	4	no	—
71	C₂²×C₆₇	—	4	1	no	—
72	C₂²×C₆₇	#405	1	2	yes	—
73	C₂²×C₆₇	#405	1	1	yes	characteristic
74	C₄×C₆₇	—	4	2	no	—
75	C₄×C₆₇	C₄³	1	1	yes	characteristic
76	C₄×C₆₇	#405	1	2	yes	—
77	C₄×C₆₇	#405	1	1	yes	characteristic
78	Dic₆₇	—	4	4	no	—
79	#12542	—	67	1	no	maximal, Sylow
80	C₂×C₆₇	—	2	2	no	—
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	#1726	1	1	yes	characteristic
83	#2110	—	67	2	no	—
84	#2110	—	67	1	no	—
85	#2108	—	67	2	no	—
86	#2108	D₆₇	1	1	yes	characteristic
87	#1747	—	67	1	no	—
88	C₆₇	#12542	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	67	2	no	—
90	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
91	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
92	#405	—	67	2	no	—
93	#405	C₂×D₆₇	1	1	yes	characteristic
94	#347	—	67	8	no	—
95	#347	—	67	4	no	—
96	#346	—	67	4	no	—
97	#346	Dic₆₇	1	2	yes	complemented
98	C₄×C₁₆	—	67	4	no	—
99	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
100	#338	—	67	1	no	—
101	H(3,ℤ/4)	—	67	1	no	—
102	H(3,ℤ/4)	—	67	2	no	—
103	C₂×M₁₆	—	67	2	no	—
104	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
106	#117	—	67	2	no	—
107	#117	C₄×D₆₇	1	1	yes	characteristic
108	M₃₂	—	67	4	no	—
109	M₃₂	—	67	4	no	—
110	M₃₂	C₄×D₆₇	1	4	yes	—
111	C₂×C₁₆	—	67	4	no	—
112	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
113	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
117	#97	—	67	4	no	—
118	#97	C₄×D₆₇	1	2	yes	—
119	C₄×C₈	—	67	4	no	—
120	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	134	2	no	—
124	M₁₆	—	134	2	no	—
125	C₂×C₈	—	67	4	no	—
126	C₂×C₈	C₄×Dic₆₇	1	2	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	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	134	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	134	2	no	—
135	C₄²	—	67	4	no	—
136	C₄²	C₄×Dic₆₇	1	2	yes	—
137	C₁₆	—	67	8	no	—
138	C₁₆	C₄×Dic₆₇	1	1	yes	characteristic
139	C₁₆	C₄×Dic₆₇	1	2	yes	—
140	C₁₆	C₄×Dic₆₇	1	4	yes	—
141	C₁₆	C₄×Dic₆₇	1	1	yes	characteristic
142	C₂³	#14524049	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	134	4	no	—
145	C₂×C₄	C₄²×D₆₇	1	2	yes	—
146	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
147	C₂×C₄	#14524049	1	1	yes	characteristic
148	C₂×C₄	#14524049	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	134	4	no	—
151	C₈	C₄²×D₆₇	1	2	yes	—
152	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	#16863350	1	2	yes	—
156	C₂²	#16863351	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	268	4	no	—
159	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
160	C₄	#16863350	1	2	yes	—
161	C₄	#16863351	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#22559895	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₆₇
commutator	C₄×C₆₇	C₄³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂×C₆₇	#1726
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#12542
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
8. #16863174	C₄	semidirect
9. #16863173	C₄	semidirect
10. #16863171	C₄	semidirect
11. C₁₆×Dic₆₇	C₄	semidirect
18. #346×C₆₇	C₄	semidirect
19. C₄×C₁₆×C₆₇	C₄	semidirect
88. C₆₇	#12542	semidirect
97. #346	Dic₆₇	semidirect
99. 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
#22561814	2
#22561813	2
#22561805	2
#22559964	2
C₆₇×#2108	2
#12542	67