Subgroups — #30770839

Order 17152.

338 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	#22559895	C₂	1	1	yes	maximal, characteristic
3	#22558288	C₂	1	4	yes	maximal
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	#16863172	C₄	1	2	yes	complemented
11	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#346×C₆₇	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
18	#14523983	—	2	4	no	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14523954	—	2	2	no	—
21	#14523953	—	2	4	no	—
22	C₈×Dic₆₇	C₈	1	8	yes	—
23	C₆₇⋊#95	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	4	no	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂²×Dic₆₇	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
37	C₄×Dic₆₇	—	2	2	no	—
38	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
39	C₆₇⋊C₁₆	—	2	4	no	—
40	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	4	yes	—
48	C₄²×C₆₇	C₄²	1	1	yes	characteristic
49	C₄²×C₆₇	C₄²	1	1	yes	characteristic
50	C₄²×C₆₇	M₁₆	1	2	yes	—
51	C₁₆×C₆₇	—	2	4	no	—
52	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	#97	1	2	yes	—
55	C₂×C₄×C₆₇	#97	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#117	1	2	yes	—
58	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
60	C₂×Dic₆₇	—	2	4	no	—
61	C₂×Dic₆₇	—	4	4	no	—
62	C₈×C₆₇	C₄×C₈	1	2	yes	—
63	C₈×C₆₇	C₄×C₈	1	2	yes	—
64	C₈×C₆₇	#117	1	2	yes	—
65	C₈×C₆₇	#117	1	2	yes	—
66	C₆₇⋊C₈	—	2	4	no	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
69	C₂²×C₆₇	#431	1	2	yes	—
70	C₄×C₆₇	—	2	2	no	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	#376	1	1	yes	characteristic
73	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₆₇	#431	1	2	yes	—
75	Dic₆₇	—	4	4	no	—
76	#12593	—	67	1	no	maximal, Sylow
77	C₂×C₆₇	—	2	2	no	—
78	C₂×C₆₇	#1744	1	1	yes	characteristic
79	C₂×C₆₇	#2110	1	2	yes	—
80	C₄×M₃₂	—	67	1	no	—
81	C₄×M₃₂	D₆₇	1	1	yes	characteristic
82	#1726	—	67	1	no	—
83	#1375	—	67	4	no	—
84	C₆₇	#12593	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	67	2	no	—
86	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
87	C₂×#97	—	67	2	no	—
88	C₂×#97	—	67	1	no	—
89	C₂×C₄×C₈	—	67	2	no	—
90	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	#376	—	67	1	no	—
92	#346	—	67	2	no	—
93	#346	Dic₆₇	1	2	yes	complemented
94	C₄×C₁₆	—	67	2	no	—
95	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
96	C₂²×C₈	—	67	2	no	—
97	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	C₂×C₄²	—	67	2	no	—
100	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	134	4	no	—
103	C₂×C₁₆	—	67	4	no	—
104	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
105	#97	—	134	4	no	—
106	#97	—	134	2	no	—
107	C₄×C₈	—	67	8	no	—
108	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
109	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	#95	—	134	2	no	—
113	C₂²×C₄	—	134	2	no	—
114	C₂²×C₄	#11761611	1	2	yes	—
115	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₈	—	67	4	no	—
117	C₂×C₈	—	134	2	no	—
118	C₂×C₈	—	134	2	no	—
119	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
121	C₂×C₈	C₈×D₆₇	1	4	yes	—
122	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
123	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
124	C₂×C₈	#11761628	1	4	yes	—
125	C₄²	—	67	4	no	—
126	C₄²	—	134	2	no	—
127	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
128	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
129	C₄²	#11761611	1	2	yes	—
130	C₁₆	—	2	4	no	—
131	C₁₆	—	134	4	no	—
132	C₂³	C₂×#11761611	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	268	4	no	—
135	C₂×C₄	—	134	4	no	—
136	C₂×C₄	#14523953	1	2	yes	—
137	C₂×C₄	#14523953	1	2	yes	—
138	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
139	C₂×C₄	#14524000	1	2	yes	—
140	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
141	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
142	C₈	—	134	4	no	—
143	C₈	C₈×Dic₆₇	1	2	yes	—
144	C₈	C₈×Dic₆₇	1	2	yes	—
145	C₈	#14524000	1	2	yes	—
146	C₈	#14524002	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#14523953	1	1	yes	characteristic
149	C₂²	#16863447	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	268	4	no	—
153	C₄	#16863242	1	1	yes	characteristic
154	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
155	C₄	#16863447	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#22559954	1	1	yes	characteristic
158	C₂	#22561813	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
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	#12593
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. #16863172	C₄	semidirect
11. C₄×C₆₇⋊C₁₆	C₄	semidirect
14. #346×C₆₇	C₄	semidirect
15. C₄×C₁₆×C₆₇	C₄	semidirect
84. C₆₇	#12593	semidirect
93. #346	Dic₆₇	semidirect
95. 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
C₄×#14523983	2
#22559895	2
#22558288	2
C₄×M₃₂×C₆₇	2
#12593	67