Subgroups — #30773859

Order 17152.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22560940	C₂	1	2	yes	maximal
4	#22558300	C₂	1	2	yes	maximal
5	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14523977	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
9	#16863292	—	2	2	no	—
10	#16863222	C₄	1	4	yes	complemented
11	C₁₆×Dic₆₇	—	2	2	no	—
12	#16863158	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	#366×C₆₇	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×Dic₆₇⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
23	#14523977	—	2	2	no	—
24	#14523977	C₂×C₄	1	4	yes	—
25	C₈×Dic₆₇	—	2	1	no	—
26	C₈×Dic₆₇	—	2	1	no	—
27	C₆₇⋊#95	—	4	2	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
32	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	2	2	no	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
37	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
38	C₂×C₆₇⋊C₈	—	2	1	no	—
39	C₂×C₆₇⋊C₈	—	2	1	no	—
40	C₂×C₆₇⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₆₇	—	2	4	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	Dic₆₇⋊C₄	—	8	4	no	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₄×Dic₆₇	—	2	1	no	—
46	C₆₇⋊C₁₆	—	2	2	no	—
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	—	8	2	no	—
61	C₂×C₄×C₆₇	—	8	2	no	—
62	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	1	no	—
69	C₂×Dic₆₇	—	2	1	no	—
70	C₂×Dic₆₇	—	2	1	no	—
71	C₂×Dic₆₇	—	2	1	no	—
72	C₂×Dic₆₇	—	4	4	no	—
73	C₂×Dic₆₇	—	8	2	no	—
74	C₂×Dic₆₇	—	8	2	no	—
75	C₂×Dic₆₇	D₁₆	1	2	yes	—
76	C₂×Dic₆₇	Q₃₂	1	2	yes	—
77	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₆₇	#125	1	1	yes	characteristic
81	C₆₇⋊C₈	—	2	2	no	—
82	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₆₇	#466	1	1	yes	characteristic
84	C₂²×C₆₇	#500	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₆₇	#508	1	1	yes	characteristic
88	C₂²×C₆₇	#508	1	1	yes	characteristic
89	C₄×C₆₇	—	16	2	no	—
90	C₄×C₆₇	#389	1	1	yes	characteristic
91	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
92	C₄×C₆₇	#467	1	1	yes	characteristic
93	C₄×C₆₇	#499	1	1	yes	characteristic
94	Dic₆₇	—	2	2	no	—
95	Dic₆₇	—	2	2	no	—
96	Dic₆₇	—	8	4	no	—
97	#13165	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	#1942	1	1	yes	characteristic
99	C₂×C₆₇	C₄×D₁₆	1	1	yes	characteristic
100	C₂×C₆₇	C₄×Q₃₂	1	1	yes	characteristic
101	C₂×C₆₇	#2215	1	1	yes	characteristic
102	C₂×C₆₇	#2216	1	1	yes	characteristic
103	C₂×C₆₇	#2252	1	1	yes	characteristic
104	C₂×C₆₇	#2253	1	1	yes	characteristic
105	C₂×#366	—	67	1	no	—
106	C₂×#366	D₆₇	1	1	yes	characteristic
107	C₂×C₄×C₁₆	—	67	1	no	—
108	#1942	—	67	2	no	—
109	#1380	—	67	2	no	—
110	C₆₇	#13165	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₁₆	—	67	1	no	—
112	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
113	C₂×#107	—	67	2	no	—
114	C₂×#107	C₂×D₆₇	1	2	yes	—
115	C₂×C₄×C₈	—	67	1	no	—
116	#389	—	134	2	no	—
117	#366	—	2	2	no	—
118	#366	—	67	4	no	—
119	C₄×C₁₆	—	134	2	no	—
120	#340	—	134	2	no	—
121	C₂²×C₈	—	67	1	no	—
122	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	268	2	no	—
125	C₂×C₄⋊C₄	—	134	2	no	—
126	C₂×C₄²	—	67	1	no	—
127	C₂×C₁₆	—	67	2	no	—
128	C₂×C₁₆	—	134	1	no	—
129	C₂×C₁₆	—	134	1	no	—
130	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
131	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	67	4	no	—
136	#107	—	134	2	no	—
137	C₄×C₈	—	134	1	no	—
138	C₄×C₈	—	134	1	no	—
139	#95	—	268	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	268	2	no	—
142	C₂²×C₄	—	67	1	no	—
143	C₂²×C₄	—	67	1	no	—
144	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
145	C₂×C₈	—	67	2	no	—
146	C₂×C₈	—	134	1	no	—
147	C₂×C₈	—	134	1	no	—
148	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
152	C₂×C₈	#11761627	1	1	yes	characteristic
153	C₂×C₈	#11761629	1	1	yes	characteristic
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	268	2	no	—
157	C₄⋊C₄	—	536	4	no	—
158	C₄⋊C₄	—	134	4	no	—
159	C₄²	—	134	1	no	—
160	C₄²	—	134	1	no	—
161	C₁₆	—	134	2	no	—
162	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
163	C₂³	D₈×D₆₇	1	1	yes	characteristic
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	268	4	no	—
168	C₂×C₄	—	536	2	no	—
169	C₂×C₄	—	536	2	no	—
170	C₂×C₄	—	67	2	no	—
171	C₂×C₄	—	67	2	no	—
172	C₂×C₄	—	134	1	no	—
173	C₂×C₄	—	134	1	no	—
174	C₂×C₄	—	134	1	no	—
175	C₂×C₄	—	134	1	no	—
176	C₂×C₄	#14524005	1	1	yes	characteristic
177	C₂×C₄	#14524014	1	1	yes	characteristic
178	C₂×C₄	#14524042	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
180	C₂×C₄	#14524062	1	1	yes	characteristic
181	C₂×C₄	#14524066	1	1	yes	characteristic
182	C₈	—	134	2	no	—
183	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
184	C₈	#14524008	1	1	yes	characteristic
185	C₈	#14524019	1	1	yes	characteristic
186	C₈	#14524034	1	1	yes	characteristic
187	C₂²	#16863478	1	1	yes	characteristic
188	C₂²	#16863629	1	1	yes	characteristic
189	C₂²	#16863800	1	1	yes	characteristic
190	C₂²	D₁₆×D₆₇	1	1	yes	characteristic
191	C₂²	Q₃₂×D₆₇	1	1	yes	characteristic
192	C₂²	#16863825	1	1	yes	characteristic
193	C₂²	#16863829	1	1	yes	characteristic
194	C₄	—	16	2	no	—
195	C₄	—	536	4	no	—
196	C₄	—	134	2	no	—
197	C₄	—	134	2	no	—
198	C₄	#16863292	1	1	yes	characteristic
199	C₄	#16863490	1	1	yes	characteristic
200	C₄	#16863634	1	1	yes	characteristic
201	C₄	#16863797	1	1	yes	characteristic
202	C₂	#22560940	1	1	yes	characteristic
203	C₂	#22562055	1	1	yes	characteristic
204	C₂	#22562067	1	1	yes	characteristic
205	C₂	#22562292	1	1	yes	characteristic
206	C₂	#22562297	1	1	yes	characteristic
207	C₂	#22562483	1	1	yes	characteristic
208	C₂	#22562486	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	#13165
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
10. #16863222	C₄	semidirect
57. C₁₆×C₆₇	C₄⋊C₄	semidirect
110. C₆₇	#13165	semidirect
162. C₁₆	Dic₆₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863222	2
C₂×C₁₆×Dic₆₇	2
#22560940	2
#22558300	2
C₂×#366×C₆₇	2
#13165	67