Subgroups — #30746991

Order 17152.

414 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558493	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×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
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	C₈×C₆₇⋊C₈	—	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	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
29	C₈×Dic₆₇	—	2	4	no	—
30	C₈×Dic₆₇	C₈	1	8	yes	complemented
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²⋊C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	—	2	2	no	—
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	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₇	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
46	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
47	C₄×Dic₆₇	—	2	4	no	—
48	C₄×Dic₆₇	—	2	4	no	—
49	C₄×Dic₆₇	—	2	2	no	—
50	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	C₂³×C₆₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
76	C₂×Dic₆₇	—	2	4	no	—
77	C₂×Dic₆₇	—	2	4	no	—
78	C₂×Dic₆₇	—	2	4	no	—
79	C₈×C₆₇	—	2	1	no	—
80	C₈×C₆₇	—	2	1	no	—
81	C₈×C₆₇	—	4	4	no	—
82	C₈×C₆₇	C₄×C₈	1	2	yes	—
83	C₈×C₆₇	C₄×C₈	1	2	yes	—
84	C₆₇⋊C₈	—	4	4	no	—
85	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	—
86	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₆₇	#431	1	2	yes	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
96	C₄×C₆₇	#443	1	2	yes	—
97	C₄×C₆₇	#443	1	2	yes	—
98	Dic₆₇	—	2	8	no	—
99	#8266	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₆₇	#1772	1	1	yes	characteristic
103	C₂×C₆₇	#1772	1	1	yes	characteristic
104	C₄²×C₈	—	67	1	no	—
105	#1564	—	67	1	no	—
106	#1564	D₆₇	1	1	yes	characteristic
107	#1450	—	67	2	no	—
108	#1276	—	67	2	no	—
109	C₆₇	#8266	1	1	yes	characteristic, complemented, Sylow
110	#431	—	67	2	no	—
111	#431	C₂×D₆₇	1	2	yes	—
112	C₂×C₄×C₈	—	67	2	no	—
113	C₂×C₄×C₈	—	67	1	no	—
114	C₂×C₄×C₈	—	67	2	no	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	C₄³	—	67	1	no	—
117	#335	—	67	2	no	—
118	#335	Dic₆₇	1	2	yes	complemented
119	#334	—	67	2	no	—
120	#334	Dic₆₇	1	2	yes	complemented
121	#322	—	134	2	no	—
122	C₈²	—	134	2	no	—
123	C₂²×C₈	—	67	2	no	—
124	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₄²	—	67	2	no	—
127	C₂×C₄²	—	134	1	no	—
128	C₂×C₄²	—	134	1	no	—
129	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₄⋊C₈	—	67	4	no	—
131	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	134	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	67	8	no	—
136	C₄×C₈	—	67	1	no	—
137	C₄×C₈	—	67	1	no	—
138	C₄×C₈	—	67	2	no	—
139	C₄×C₈	—	134	4	no	—
140	C₄×C₈	—	134	2	no	—
141	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
142	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
143	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
145	C₂²×C₄	—	134	2	no	—
146	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
147	C₂²×C₄	#11761627	1	1	yes	characteristic
148	C₂²×C₄	#11761629	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	67	2	no	—
152	C₂×C₈	—	67	2	no	—
153	C₂×C₈	—	67	4	no	—
154	C₂×C₈	—	67	4	no	—
155	C₂×C₈	—	134	4	no	—
156	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
159	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
160	C₂×C₈	C₈×D₆₇	1	4	yes	—
161	C₄²	—	67	4	no	—
162	C₄²	—	134	4	no	—
163	C₄²	—	134	4	no	—
164	C₄²	—	134	2	no	—
165	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
166	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
167	C₄²	D₄×D₆₇	1	1	yes	characteristic
168	C₄²	Q₈×D₆₇	1	1	yes	characteristic
169	C₂³	#14524001	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	134	4	no	—
173	C₂×C₄	—	134	4	no	—
174	C₂×C₄	—	134	4	no	—
175	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄	#14524005	1	1	yes	characteristic
178	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	#14524008	1	1	yes	characteristic
180	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
182	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	4	no	—
186	C₈	—	268	4	no	—
187	C₈	—	67	4	no	—
188	C₈	—	67	4	no	—
189	C₈	C₈×Dic₆₇	1	2	yes	—
190	C₈	C₈×Dic₆₇	1	2	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
194	C₂²	#16863448	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	134	8	no	—
198	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
199	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
200	C₄	C₄⋊C₈×D₆₇	1	2	yes	—
201	C₄	#16863512	1	2	yes	—
202	C₄	#16863513	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₄⋊C₈×Dic₆₇	1	1	yes	characteristic
205	C₂	#22560090	1	1	yes	characteristic
206	C₂	#22560091	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
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	#8266
LCS	G ▷ 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
30. C₈×Dic₆₇	C₈	semidirect
109. C₆₇	#8266	semidirect
118. #335	Dic₆₇	semidirect
120. #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×C₈×Dic₆₇	2
#22558957	2
#22558493	2
#22558041	2
C₆₇×#1564	2
#8266	67