Subgroups — #30746270

Order 17152.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559895	C₂	1	1	yes	maximal, characteristic
2	#22558492	C₂	1	1	yes	maximal, characteristic
3	#22558039	C₂	1	2	yes	maximal
4	#22558037	C₂	1	2	yes	maximal
5	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
10	#16863242	C₂²	1	1	yes	characteristic
11	#16863117	C₄	1	2	yes	complemented
12	C₈×C₆₇⋊C₈	C₄	1	2	yes	complemented
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	C₄	1	2	yes	complemented
16	C₈²×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	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
21	#14523954	—	2	2	no	—
22	#14523953	—	2	4	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₈×Dic₆₇	C₈	1	8	yes	complemented
25	C₆₇⋊#95	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₂²⋊C₈×C₆₇	—	2	2	no	—
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	4	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₆₇	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
41	C₄×Dic₆₇	—	2	2	no	—
42	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	#41	1	2	yes	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
52	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₂³×C₆₇	#117	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	4	2	no	—
57	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#117	1	2	yes	—
59	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#117	1	2	yes	—
61	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₆₇	—	2	4	no	—
64	C₂×Dic₆₇	—	4	4	no	—
65	C₈×C₆₇	—	4	4	no	—
66	C₈×C₆₇	C₄×C₈	1	2	yes	—
67	C₈×C₆₇	C₄×C₈	1	2	yes	—
68	C₈×C₆₇	#97	1	2	yes	—
69	C₈×C₆₇	#97	1	2	yes	—
70	C₆₇⋊C₈	—	2	4	no	—
71	C₆₇⋊C₈	—	4	4	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	#376	1	1	yes	characteristic
74	C₂²×C₆₇	#431	1	2	yes	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
79	C₄×C₆₇	#431	1	2	yes	—
80	Dic₆₇	—	4	4	no	—
81	#8137	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	2	no	—
83	C₂×C₆₇	#1744	1	1	yes	characteristic
84	C₂×C₆₇	#1747	1	2	yes	—
85	#1726	—	67	1	no	—
86	#1450	—	67	1	no	—
87	#1450	D₆₇	1	1	yes	characteristic
88	#1275	—	67	2	no	—
89	#1274	—	67	2	no	—
90	C₆₇	#8137	1	1	yes	characteristic, complemented, Sylow
91	#431	—	67	2	no	—
92	#431	C₂×D₆₇	1	2	yes	—
93	C₂×#97	—	67	2	no	—
94	C₂×#97	—	67	1	no	—
95	C₂×C₄×C₈	—	67	2	no	—
96	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
97	#376	—	67	1	no	—
98	#322	—	67	2	no	—
99	#322	Dic₆₇	1	2	yes	complemented
100	C₈²	—	67	2	no	—
101	C₈²	Dic₆₇	1	2	yes	complemented
102	C₂²×C₈	—	67	2	no	—
103	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
105	C₂×C₄²	—	67	2	no	—
106	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	134	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	134	2	no	—
111	#97	—	134	4	no	—
112	#97	—	134	2	no	—
113	C₄×C₈	—	67	8	no	—
114	C₄×C₈	—	67	4	no	—
115	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
116	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
118	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
120	#95	—	134	2	no	—
121	C₂²×C₄	—	134	2	no	—
122	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
123	C₂²×C₄	#11761628	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	67	4	no	—
126	C₂×C₈	—	134	4	no	—
127	C₂×C₈	—	134	2	no	—
128	C₂×C₈	—	134	2	no	—
129	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
130	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
133	C₂×C₈	C₈×D₆₇	1	4	yes	—
134	C₂×C₈	#11761611	1	4	yes	—
135	C₄²	—	67	4	no	—
136	C₄²	—	134	2	no	—
137	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
138	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
139	C₄²	#11761628	1	2	yes	—
140	C₂³	#14524000	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	268	4	no	—
143	C₂×C₄	—	134	4	no	—
144	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
145	C₂×C₄	#14524000	1	2	yes	—
146	C₂×C₄	#14524000	1	1	yes	characteristic
147	C₂×C₄	#14524002	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
149	C₂×C₄	C₂×#11761611	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	268	4	no	—
152	C₈	—	134	4	no	—
153	C₈	C₈×Dic₆₇	1	2	yes	—
154	C₈	C₈×Dic₆₇	1	2	yes	—
155	C₈	#14523953	1	2	yes	—
156	C₈	#14523953	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#16863242	1	1	yes	characteristic
159	C₂²	#16863447	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	268	4	no	—
163	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
164	C₄	C₂×#14523953	1	1	yes	characteristic
165	C₄	#16863447	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#22559954	1	1	yes	characteristic
168	C₂	#22559964	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1450	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#8137
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #16863117	C₄	semidirect
12. C₈×C₆₇⋊C₈	C₄	semidirect
15. #322×C₆₇	C₄	semidirect
16. C₈²×C₆₇	C₄	semidirect
24. C₈×Dic₆₇	C₈	semidirect
90. C₆₇	#8137	semidirect
99. #322	Dic₆₇	semidirect
101. C₈²	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559895	2
#22558492	2
#22558039	2
#22558037	2
C₆₇×#1450	2
#8137	67