Subgroups — #30747003

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558502	C₂	1	2	yes	maximal
4	#22558041	C₂	1	2	yes	maximal
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523953	C₄	1	2	yes	—
8	C₂×#14523953	C₄	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
11	#16863149	C₄	1	2	yes	complemented
12	#16863147	C₄	1	2	yes	complemented
13	#16863118	—	2	2	no	—
14	#16863117	—	2	2	no	—
15	#431×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#335×C₆₇	C₄	1	2	yes	complemented
18	#334×C₆₇	C₄	1	2	yes	complemented
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₇	—	2	1	no	—
21	C₂×C₄×Dic₆₇	—	2	1	no	—
22	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	#14523953	—	2	4	no	—
26	#14523953	—	2	4	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₇	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₄×Dic₆₇	—	2	2	no	—
46	C₄×Dic₆₇	—	4	2	no	—
47	C₄×Dic₆₇	—	4	2	no	—
48	C₄×Dic₆₇	M₁₆	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
59	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂³×C₆₇	#117	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
74	C₂×Dic₆₇	—	2	4	no	—
75	C₂×Dic₆₇	—	4	2	no	—
76	C₂×Dic₆₇	—	4	2	no	—
77	C₈×C₆₇	—	2	1	no	—
78	C₈×C₆₇	—	2	1	no	—
79	C₈×C₆₇	—	4	4	no	—
80	C₈×C₆₇	#97	1	2	yes	—
81	C₈×C₆₇	#97	1	2	yes	—
82	C₆₇⋊C₈	—	2	4	no	—
83	C₆₇⋊C₈	—	4	4	no	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₆₇	#432	1	2	yes	—
88	C₄×C₆₇	—	2	1	no	—
89	C₄×C₆₇	—	2	1	no	—
90	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
91	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₆₇	#423	1	2	yes	—
93	C₄×C₆₇	#444	1	2	yes	—
94	C₄×C₆₇	#444	1	2	yes	—
95	Dic₆₇	—	4	4	no	—
96	#8269	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	#1767	1	1	yes	characteristic
99	C₂×C₆₇	#1773	1	1	yes	characteristic
100	C₂×C₆₇	#1773	1	1	yes	characteristic
101	C₄×#97	—	67	1	no	—
102	#1564	—	67	1	no	—
103	#1564	D₆₇	1	1	yes	characteristic
104	#1452	—	67	2	no	—
105	#1276	—	67	2	no	—
106	C₆₇	#8269	1	1	yes	characteristic, complemented, Sylow
107	#431	—	67	2	no	—
108	#431	C₂×D₆₇	1	2	yes	—
109	C₂×#97	—	67	2	no	—
110	C₂×#97	—	67	2	no	—
111	C₂×C₄×C₈	—	67	1	no	—
112	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
113	C₄³	—	67	1	no	—
114	#335	—	67	2	no	—
115	#335	Dic₆₇	1	2	yes	complemented
116	#334	—	67	2	no	—
117	#334	Dic₆₇	1	2	yes	complemented
118	#322	—	134	2	no	—
119	#322	—	134	2	no	—
120	C₂²×C₈	—	67	2	no	—
121	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
123	C₂×C₄²	—	67	2	no	—
124	C₂×C₄²	—	134	1	no	—
125	C₂×C₄²	—	134	1	no	—
126	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊C₈	—	67	4	no	—
128	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	134	2	no	—
131	#97	—	134	4	no	—
132	#97	—	134	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	67	2	no	—
135	C₄×C₈	—	67	2	no	—
136	C₄×C₈	—	134	2	no	—
137	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
139	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
141	C₂²×C₄	—	134	2	no	—
142	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₂²×C₄	#11761627	1	1	yes	characteristic
144	C₂²×C₄	#11761629	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	67	4	no	—
148	C₂×C₈	—	134	4	no	—
149	C₂×C₈	—	134	2	no	—
150	C₂×C₈	—	134	2	no	—
151	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
155	C₂×C₈	#11761611	1	4	yes	—
156	C₄²	—	268	2	no	—
157	C₄²	—	268	2	no	—
158	C₄²	—	67	4	no	—
159	C₄²	—	134	2	no	—
160	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
161	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₄²	D₄×D₆₇	1	1	yes	characteristic
163	C₄²	Q₈×D₆₇	1	1	yes	characteristic
164	C₂³	#14524001	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	268	2	no	—
168	C₂×C₄	—	268	2	no	—
169	C₂×C₄	—	134	4	no	—
170	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
172	C₂×C₄	#14524005	1	1	yes	characteristic
173	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
174	C₂×C₄	#14524008	1	1	yes	characteristic
175	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
177	C₂×C₄	C₂×#11761611	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	4	no	—
181	C₈	—	268	4	no	—
182	C₈	—	134	4	no	—
183	C₈	#14523953	1	2	yes	—
184	C₈	#14523953	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
188	C₂²	#16863452	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	268	4	no	—
192	C₄	C₂×#14523953	1	1	yes	characteristic
193	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
194	C₄	#16863418	1	2	yes	—
195	C₄	#16863517	1	2	yes	—
196	C₄	#16863518	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#22560064	1	1	yes	characteristic
199	C₂	#22560095	1	1	yes	characteristic
200	C₂	#22560096	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₄×C₄⋊C₄
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#8269
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
11. #16863149	C₄	semidirect
12. #16863147	C₄	semidirect
17. #335×C₆₇	C₄	semidirect
18. #334×C₆₇	C₄	semidirect
106. C₆₇	#8269	semidirect
115. #335	Dic₆₇	semidirect
117. #334	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523953	2
#22558957	2
#22558502	2
#22558041	2
C₆₇×#1564	2
#8269	67