Subgroups — #30749578

Order 17152.

418 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	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	C₂×#16863146	C₂	1	2	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14523954	C₄	1	2	yes	—
9	C₂×#14523954	C₄	1	1	yes	characteristic
10	C₂×#14523954	C₄	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	#16863151	—	2	2	no	—
13	#16863148	—	2	4	no	—
14	#16863146	—	2	4	no	—
15	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	#14523954	—	2	4	no	—
25	#14523954	—	2	2	no	—
26	#14523954	C₈	1	4	yes	complemented
27	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
28	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₄⋊C₈×C₆₇	—	4	4	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	—	4	1	no	—
41	C₂×C₆₇⋊C₈	—	4	1	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
47	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	—	4	4	no	—
55	C₄²×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	—	4	2	no	—
59	C₄²×C₆₇	C₂×C₈	1	2	yes	—
60	C₄²×C₆₇	M₁₆	1	2	yes	—
61	C₄²×C₆₇	C₂×D₄	1	2	yes	—
62	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₆₇	#137	1	2	yes	—
80	C₈×C₆₇	—	8	2	no	—
81	C₆₇⋊C₈	—	2	4	no	—
82	C₆₇⋊C₈	—	8	1	no	—
83	C₆₇⋊C₈	—	8	1	no	—
84	C₆₇⋊C₈	C₄⋊C₈	1	8	yes	complemented
85	C₂²×C₆₇	#384	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₇	#423	1	1	yes	characteristic
88	C₂²×C₆₇	#428	1	2	yes	—
89	C₂²×C₆₇	#430	1	2	yes	—
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₆₇	#423	1	2	yes	—
95	C₄×C₆₇	#496	1	1	yes	characteristic
96	C₄×C₆₇	#497	1	1	yes	characteristic
97	C₄×C₆₇	#501	1	2	yes	—
98	#8808	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	#1565	1	2	yes	—
100	C₂×C₆₇	#1569	1	2	yes	—
101	C₂×C₆₇	#1842	1	1	yes	characteristic
102	C₂×C₆₇	#1847	1	1	yes	characteristic
103	C₂×C₆₇	#1849	1	1	yes	characteristic
104	#1842	D₆₇	1	1	yes	characteristic
105	#1831	—	67	1	no	—
106	C₄×#97	—	67	1	no	—
107	C₂×#335	—	67	2	no	—
108	C₂×#334	—	67	2	no	—
109	C₆₇	#8808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	2	2	no	—
111	C₂×C₄⋊C₈	—	67	1	no	—
112	C₂×C₄⋊C₈	—	67	1	no	—
113	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
114	C₂×#97	—	67	1	no	—
115	C₂×#97	—	67	2	no	—
116	C₂×#97	—	67	1	no	—
117	C₂×C₄×C₈	—	67	2	no	—
118	C₄³	C₂×D₆₇	1	1	yes	characteristic
119	#336	—	134	2	no	—
120	#335	—	134	4	no	—
121	#334	—	134	4	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	67	2	no	—
124	C₂²×C₈	—	67	2	no	—
125	C₂²×C₈	—	134	1	no	—
126	C₂²×C₈	—	134	1	no	—
127	C₂×C₄²	Dic₁₃₄	1	2	yes	—
128	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
131	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	4	4	no	—
135	C₄⋊C₈	—	134	2	no	—
136	C₄⋊C₈	—	134	2	no	—
137	#97	—	67	4	no	—
138	#97	—	134	4	no	—
139	#97	—	134	2	no	—
140	C₄×C₈	—	67	4	no	—
141	C₄×C₈	—	67	4	no	—
142	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
143	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	—	268	1	no	—
150	C₂×C₈	—	268	1	no	—
151	C₂×C₈	—	268	2	no	—
152	C₂×C₈	—	268	2	no	—
153	C₂×C₈	—	67	4	no	—
154	C₂×C₈	—	67	4	no	—
155	C₂×C₈	—	67	4	no	—
156	C₂×C₈	—	67	4	no	—
157	C₂×C₈	—	134	2	no	—
158	C₂×C₈	—	134	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₈×D₆₇	1	2	yes	—
164	C₄²	#11761611	1	2	yes	—
165	C₄²	D₄×D₆₇	1	2	yes	—
166	C₄²	Q₈×D₆₇	1	2	yes	—
167	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
175	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
176	C₂×C₄	#14524039	1	1	yes	characteristic
177	C₂×C₄	#14524041	1	1	yes	characteristic
178	C₂×C₄	#14524042	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
181	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
182	C₂×C₄	#14524069	1	2	yes	—
183	C₂×C₄	#14524074	1	2	yes	—
184	C₈	—	8	2	no	—
185	C₈	—	536	1	no	—
186	C₈	—	536	1	no	—
187	C₈	—	67	8	no	—
188	C₈	—	134	4	no	—
189	C₂²	#16863265	1	1	yes	characteristic
190	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
191	C₂²	#16863419	1	1	yes	characteristic
192	C₂²	#16863436	1	2	yes	—
193	C₂²	#16863444	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	C₄⋊C₈×D₆₇	1	2	yes	—
198	C₄	#16863418	1	2	yes	—
199	C₄	#16863783	1	1	yes	characteristic
200	C₄	#16863788	1	1	yes	characteristic
201	C₄	#16863802	1	2	yes	—
202	C₂	#22558959	1	2	yes	—
203	C₂	#22558974	1	2	yes	—
204	C₂	#22560360	1	1	yes	characteristic
205	C₂	#22560375	1	1	yes	characteristic
206	C₂	#22560383	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#8808
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
26. #14523954	C₈	semidirect
84. C₆₇⋊C₈	C₄⋊C₈	semidirect
109. C₆₇	#8808	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
C₄×#14523954	2
C₂×#16863148	2
C₂×#16863146	2
C₆₇×#1842	2
#8808	67