Subgroups — #30864679

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569144	C₂	1	2	yes	maximal
2	#22561187	C₂	1	2	yes	maximal
3	#22560958	C₂	1	4	yes	maximal
4	#22560381	C₂	1	1	yes	maximal, characteristic
5	#22560090	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
8	#16863527	—	2	4	no	—
9	Q₈×C₆₇⋊C₈	—	2	4	no	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	#16863423	C₂²	1	4	yes	—
12	#16863420	C₂²	1	2	yes	—
13	#16863352	C₂²	1	4	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863155	C₂²	1	8	yes	—
19	#533×C₆₇	C₂²	1	2	yes	—
20	#500×C₆₇	C₄	1	2	yes	complemented
21	#499×C₆₇	C₄	1	4	yes	complemented
22	#498×C₆₇	C₄	1	2	yes	complemented
23	#427×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₆₇	—	2	2	no	—
29	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
30	#14523979	—	2	8	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
33	#14523954	—	2	4	no	—
34	C₄×C₆₇⋊C₈	—	2	4	no	—
35	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
36	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
37	C₈×Dic₆₇	—	2	4	no	—
38	C₈×Dic₆₇	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
41	#125×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²⋊Q₈×C₆₇	—	2	4	no	—
43	C₄×Q₈×C₆₇	—	2	4	no	—
44	#117×C₆₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	#107×C₆₇	C₂×C₄	1	8	yes	—
47	#106×C₆₇	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₆₇	—	2	2	no	—
52	#11761612	—	2	8	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
56	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
59	C₄×Dic₆₇	—	2	8	no	—
60	C₄×Dic₆₇	—	2	2	no	—
61	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	#41	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₆₇	#41	1	4	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	C₂²⋊C₄×C₆₇	—	2	4	no	—
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
78	Q₈×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	8	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	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
87	C₂×Dic₆₇	—	2	8	no	—
88	C₂×Dic₆₇	—	2	4	no	—
89	C₈×C₆₇	C₄×Q₈	1	4	yes	—
90	C₈×C₆₇	C₄×Q₈	1	4	yes	—
91	C₆₇⋊C₈	—	2	2	no	—
92	C₆₇⋊C₈	—	4	4	no	—
93	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
94	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₆₇	#533	1	2	yes	—
98	C₄×C₆₇	—	2	2	no	—
99	C₄×C₆₇	—	4	8	no	—
100	C₄×C₆₇	#443	1	4	yes	—
101	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
102	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₆₇	#532	1	1	yes	characteristic
104	Dic₆₇	—	2	8	no	—
105	#21732	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	—	2	2	no	—
107	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#443	1	2	yes	—
109	#3160	D₆₇	1	1	yes	characteristic
110	#2960	—	67	2	no	—
111	#1994	—	67	2	no	—
112	#1945	—	67	4	no	—
113	#1848	—	67	1	no	—
114	#1772	—	67	4	no	—
115	C₄²×C₈	—	67	1	no	—
116	C₆₇	#21732	1	1	yes	characteristic, complemented, Sylow
117	#533	C₂×D₆₇	1	2	yes	—
118	#500	Dic₆₇	1	2	yes	complemented
119	#499	Dic₆₇	1	4	yes	complemented
120	#498	Dic₆₇	1	2	yes	complemented
121	#446	—	134	4	no	—
122	C₈×Q₈	—	134	4	no	—
123	C₂×#108	—	67	4	no	—
124	#427	C₂×D₆₇	1	4	yes	—
125	#424	—	67	4	no	—
126	#423	—	67	2	no	—
127	#405	—	67	4	no	—
128	C₂×C₄×C₈	—	67	2	no	—
129	C₂×C₄×C₈	—	67	1	no	—
130	C₂×C₄×C₈	—	67	2	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄³	—	67	1	no	—
133	#339	—	67	8	no	—
134	C₂×M₁₆	—	67	4	no	—
135	C₂²×C₈	—	67	1	no	—
136	C₂²×C₈	—	67	1	no	—
137	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
138	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
139	#125	C₂×Dic₆₇	1	4	yes	—
140	C₂²⋊Q₈	—	2	4	no	—
141	C₄×Q₈	—	2	4	no	—
142	#117	C₂²×D₆₇	1	4	yes	—
143	C₂×C₄²	—	67	2	no	—
144	C₂×C₄²	—	134	2	no	—
145	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
146	#108	—	134	8	no	—
147	#107	C₂×Dic₆₇	1	8	yes	—
148	#106	C₂×Dic₆₇	1	8	yes	—
149	C₄⋊C₈	—	134	8	no	—
150	C₄⋊C₈	—	134	4	no	—
151	#97	—	134	4	no	—
152	C₄×C₈	—	67	8	no	—
153	C₄×C₈	—	67	2	no	—
154	C₄×C₈	—	67	2	no	—
155	C₄×C₈	—	134	4	no	—
156	C₄×C₈	—	134	4	no	—
157	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
159	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂×Q₈	—	2	4	no	—
161	C₂²×C₄	—	134	2	no	—
162	C₂²×C₄	#11761627	1	2	yes	—
163	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
164	M₁₆	—	134	8	no	—
165	C₂×C₈	—	67	1	no	—
166	C₂×C₈	—	67	1	no	—
167	C₂×C₈	—	67	4	no	—
168	C₂×C₈	—	67	2	no	—
169	C₂×C₈	—	134	4	no	—
170	C₂×C₈	—	134	2	no	—
171	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
172	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
173	C₂×C₈	Q₈×D₆₇	1	4	yes	—
174	C₂×C₈	#11761629	1	4	yes	—
175	C₄⋊C₄	—	2	8	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	67	4	no	—
181	C₄²	—	134	8	no	—
182	C₄²	—	134	2	no	—
183	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
184	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
185	C₄²	D₄×D₆₇	1	2	yes	—
186	C₂³	C₂×#11761627	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	134	8	no	—
191	C₂×C₄	—	134	4	no	—
192	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
193	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
194	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
197	C₂×C₄	C₂×#11761629	1	2	yes	—
198	C₈	—	268	4	no	—
199	C₈	—	67	2	no	—
200	C₈	—	67	2	no	—
201	C₈	—	134	2	no	—
202	C₈	Q₈×Dic₆₇	1	4	yes	—
203	C₈	Q₈×Dic₆₇	1	4	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
206	C₂²	#16863951	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	4	8	no	—
209	C₄	—	134	8	no	—
210	C₄	#16863513	1	4	yes	—
211	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
212	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
213	C₄	#16863947	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
216	C₂	C₂×#16863513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#21732
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
20. #500×C₆₇	C₄	semidirect
21. #499×C₆₇	C₄	semidirect
22. #498×C₆₇	C₄	semidirect
38. C₈×Dic₆₇	Q₈	semidirect
116. C₆₇	#21732	semidirect
118. #500	Dic₆₇	semidirect
119. #499	Dic₆₇	semidirect
120. #498	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569144	2
#22561187	2
#22560958	2
#22560381	2
#22560090	2
C₄×C₈×Dic₆₇	2
C₆₇×#3160	2
#21732	67