Subgroups — #30869501

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562958	C₂	1	2	yes	maximal
2	#22561183	C₂	1	2	yes	maximal
3	#22560803	C₂	1	4	yes	maximal
4	#22560782	C₂	1	4	yes	maximal
5	#22560340	C₂	1	1	yes	maximal, characteristic
6	#22559960	C₂	1	1	yes	maximal, characteristic
7	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523954	C₂²	1	1	yes	characteristic
9	#16863303	—	2	4	no	—
10	#16863295	C₂²	1	4	yes	—
11	#16863282	—	2	4	no	—
12	#16863262	C₂²	1	2	yes	—
13	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
14	#16863242	C₂²	1	1	yes	characteristic
15	#16863159	C₂²	1	2	yes	—
16	#16863159	C₂²	1	2	yes	—
17	#16863158	C₂²	1	8	yes	—
18	#16863150	C₂²	1	2	yes	—
19	#16863150	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	#500×C₆₇	C₄	1	8	yes	complemented
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	C₂×#107×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	4	no	—
27	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
28	C₂×C₄×Dic₆₇	—	2	4	no	—
29	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
30	#14523954	—	2	2	no	—
31	C₆₇⋊#95	—	2	4	no	—
32	C₆₇⋊#95	—	2	4	no	—
33	C₆₇⋊#95	—	2	2	no	—
34	C₆₇⋊#95	—	2	2	no	—
35	C₆₇⋊#95	D₄	1	2	yes	—
36	C₆₇⋊#95	D₄	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₇	—	2	4	no	—
41	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#107×C₆₇	C₂×C₄	1	8	yes	—
46	#107×C₆₇	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×Dic₆₇	—	2	4	no	—
49	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
50	C₂×C₆₇⋊C₈	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	—	8	4	no	—
56	C₄×Dic₆₇	—	2	2	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₄×Dic₆₇	—	8	4	no	—
59	C₂×Q₈×C₆₇	—	2	8	no	—
60	C₂×Q₈×C₆₇	—	8	4	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₄⋊C₄×C₆₇	—	2	8	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	8	8	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	8	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
86	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
87	C₂×Dic₆₇	—	2	4	no	—
88	C₂×Dic₆₇	—	2	4	no	—
89	C₂×Dic₆₇	—	2	4	no	—
90	C₂×Dic₆₇	—	4	4	no	—
91	C₂×Dic₆₇	—	4	4	no	—
92	C₂×Dic₆₇	—	4	4	no	—
93	C₈×C₆₇	—	2	2	no	—
94	C₈×C₆₇	—	2	2	no	—
95	C₆₇⋊C₈	—	4	2	no	—
96	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₆₇	#536	1	2	yes	—
98	C₂²×C₆₇	#576	1	2	yes	—
99	C₂²×C₆₇	#578	1	2	yes	—
100	C₄×C₆₇	—	2	2	no	—
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	4	8	no	—
103	C₄×C₆₇	#519	1	1	yes	characteristic
104	C₄×C₆₇	#519	1	1	yes	characteristic
105	C₄×C₆₇	#541	1	2	yes	—
106	Dic₆₇	—	4	4	no	—
107	Dic₆₇	—	8	4	no	—
108	#22254	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	#2327	1	1	yes	characteristic
110	C₂×C₆₇	#2948	1	1	yes	characteristic
111	C₂×C₆₇	#2950	1	1	yes	characteristic
112	C₂×C₆₇	#3243	1	4	yes	—
113	C₂×#500	D₆₇	1	1	yes	characteristic
114	#2331	—	67	2	no	—
115	#1993	—	67	2	no	—
116	#1921	—	67	4	no	—
117	#1918	—	67	4	no	—
118	#1835	—	67	1	no	—
119	#1746	—	67	1	no	—
120	C₆₇	#22254	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
122	#500	Dic₆₇	1	8	yes	complemented
123	C₂×#107	C₂×D₆₇	1	2	yes	—
124	C₂×#107	C₂×D₆₇	1	2	yes	—
125	C₂×#97	—	67	1	no	—
126	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
127	#391	—	134	4	no	—
128	#389	—	67	4	no	—
129	#387	—	134	4	no	—
130	#382	—	67	2	no	—
131	C₄×C₄⋊C₄	—	67	4	no	—
132	#376	—	67	1	no	—
133	#340	—	67	2	no	—
134	#340	—	67	2	no	—
135	#340	—	67	8	no	—
136	#336	—	67	2	no	—
137	#336	—	67	2	no	—
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×C₈	—	67	2	no	—
140	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	67	4	no	—
146	C₂×C₄⋊C₄	—	134	4	no	—
147	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
148	C₂×C₄²	—	67	2	no	—
149	C₂×C₄²	—	134	4	no	—
150	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
151	#107	C₂×Dic₆₇	1	8	yes	—
152	#107	C₂×Dic₆₇	1	8	yes	—
153	#97	—	134	2	no	—
154	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
155	#95	—	67	2	no	—
156	#95	—	67	2	no	—
157	#95	—	134	4	no	—
158	#95	—	134	4	no	—
159	#95	—	134	2	no	—
160	#95	—	134	2	no	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	8	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	67	4	no	—
165	C₂²×C₄	—	134	4	no	—
166	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
167	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	134	2	no	—
172	C₂×C₈	—	134	2	no	—
173	C₂×C₈	—	134	2	no	—
174	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
175	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
176	C₄⋊C₄	—	2	8	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	536	4	no	—
179	C₄⋊C₄	—	134	4	no	—
180	C₄⋊C₄	—	134	4	no	—
181	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
182	C₄²	—	536	4	no	—
183	C₄²	—	134	2	no	—
184	C₄²	—	134	2	no	—
185	C₄²	C₂²×Dic₆₇	1	2	yes	—
186	C₄²	#11761627	1	2	yes	—
187	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
188	Q₈	—	8	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	268	4	no	—
195	C₂×C₄	—	268	4	no	—
196	C₂×C₄	—	268	4	no	—
197	C₂×C₄	—	134	4	no	—
198	C₂×C₄	—	134	4	no	—
199	C₂×C₄	—	134	4	no	—
200	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
201	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
202	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
203	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
204	C₂×C₄	#14524091	1	2	yes	—
205	C₂×C₄	#14524091	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	268	2	no	—
209	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
210	C₂²	#16863971	1	2	yes	—
211	C₂²	#16864197	1	2	yes	—
212	C₂²	#16864209	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	—	268	4	no	—
217	C₄	—	536	4	no	—
218	C₄	#16863874	1	1	yes	characteristic
219	C₄	#16863874	1	1	yes	characteristic
220	C₄	#16864007	1	2	yes	—
221	C₂	#22562929	1	1	yes	characteristic
222	C₂	#22569061	1	1	yes	characteristic
223	C₂	#22569077	1	1	yes	characteristic
224	C₂	#22571495	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#22254
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
21. #500×C₆₇	C₄	semidirect
120. C₆₇	#22254	semidirect
122. #500	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562958	2
#22561183	2
#22560803	2
#22560782	2
#22560340	2
#22559960	2
C₂×#500×C₆₇	2
#22254	67