Subgroups — #30755588

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22561169	C₂	1	2	yes	maximal
3	#22561010	C₂	1	2	yes	maximal
4	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863801	—	2	2	no	—
8	#16863801	C₄	1	4	yes	complemented
9	C₂×#14523977	—	2	2	no	—
10	C₂×#14523966	—	2	4	no	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863158	—	2	4	no	—
14	C₈×C₆₇⋊C₈	—	2	2	no	—
15	C₂×#107×C₆₇	—	2	2	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	C₂²	1	2	yes	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	#14523977	—	2	8	no	—
28	#14523966	—	4	8	no	—
29	C₄×C₆₇⋊C₈	—	2	1	no	—
30	C₄×C₆₇⋊C₈	—	2	1	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#107×C₆₇	—	4	4	no	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	C₂×Dic₁₃₄	—	2	4	no	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	4	4	no	—
44	C₂×Dic₁₃₄	—	4	2	no	—
45	C₂²×Dic₆₇	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
51	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
62	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	#41	1	1	yes	characteristic
69	C₄²×C₆₇	#41	1	1	yes	characteristic
70	C₂³×C₆₇	#127	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	8	2	no	—
72	C₂×C₄×C₆₇	—	8	2	no	—
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₆₇	#135	1	2	yes	—
85	C₂×C₄×C₆₇	#135	1	2	yes	—
86	C₂×Dic₆₇	—	4	4	no	—
87	C₂×Dic₆₇	—	8	2	no	—
88	C₂×Dic₆₇	—	8	2	no	—
89	Dic₁₃₄	—	4	8	no	—
90	Dic₁₃₄	—	8	4	no	—
91	C₈×C₆₇	#103	1	8	yes	complemented
92	C₆₇⋊C₈	—	2	4	no	—
93	C₆₇⋊C₈	—	2	4	no	—
94	C₂²×C₆₇	#390	1	1	yes	characteristic
95	C₂²×C₆₇	#492	1	1	yes	characteristic
96	C₂²×C₆₇	#493	1	1	yes	characteristic
97	C₂²×C₆₇	#494	1	1	yes	characteristic
98	C₂²×C₆₇	#494	1	1	yes	characteristic
99	C₂²×C₆₇	#495	1	1	yes	characteristic
100	C₂²×C₆₇	#495	1	1	yes	characteristic
101	C₄×C₆₇	—	16	2	no	—
102	C₄×C₆₇	C₂×#103	1	2	yes	—
103	C₄×C₆₇	C₄×D₈	1	2	yes	—
104	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
105	C₄×C₆₇	#466	1	2	yes	—
106	C₄×C₆₇	#467	1	2	yes	—
107	C₄×C₆₇	#487	1	2	yes	—
108	Dic₆₇	—	8	4	no	—
109	#9823	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	#1706	1	1	yes	characteristic
111	C₂×C₆₇	#1707	1	1	yes	characteristic
112	C₂×C₆₇	#1710	1	1	yes	characteristic
113	C₂×C₆₇	#1713	1	1	yes	characteristic
114	C₂×C₆₇	#1956	1	1	yes	characteristic
115	C₂×C₆₇	#1969	1	1	yes	characteristic
116	C₂×C₆₇	#1970	1	1	yes	characteristic
117	C₂×#500	—	67	1	no	—
118	#1990	—	67	2	no	—
119	#1956	—	67	2	no	—
120	#1845	D₆₇	1	1	yes	characteristic
121	C₂×C₈²	—	67	1	no	—
122	C₆₇	#9823	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄⋊Q₈	—	67	2	no	—
124	#500	—	67	4	no	—
125	#500	—	134	2	no	—
126	C₂×#107	—	2	2	no	—
127	C₂×#107	—	134	2	no	—
128	C₂×#103	—	134	4	no	—
129	C₂×C₄×C₈	—	67	1	no	—
130	C₂×C₄×C₈	—	67	1	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	#384	C₂×D₆₇	1	2	yes	—
133	#340	—	134	4	no	—
134	C₈²	—	134	2	no	—
135	C₂²×Q₈	—	134	2	no	—
136	C₂²×C₈	—	67	2	no	—
137	C₂²×C₈	—	67	2	no	—
138	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
139	C₄⋊Q₈	—	67	4	no	—
140	C₄⋊Q₈	—	134	2	no	—
141	C₄⋊Q₈	—	134	2	no	—
142	C₄⋊Q₈	—	134	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	134	2	no	—
147	C₂×C₄⋊C₄	—	134	2	no	—
148	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
149	#107	—	4	4	no	—
150	#107	—	134	8	no	—
151	#103	—	268	8	no	—
152	C₄×C₈	—	134	1	no	—
153	C₄×C₈	—	134	1	no	—
154	C₄×C₈	—	134	1	no	—
155	C₄×C₈	—	134	1	no	—
156	C₄×C₈	C₄×D₆₇	1	2	yes	—
157	C₄×C₈	D₂₆₈	1	2	yes	—
158	C₂×Q₈	—	268	2	no	—
159	C₂×Q₈	—	268	4	no	—
160	C₂×Q₈	—	268	4	no	—
161	C₂×Q₈	—	134	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₈	—	67	4	no	—
168	C₂×C₈	—	67	4	no	—
169	C₂×C₈	—	134	2	no	—
170	C₂×C₈	—	134	2	no	—
171	C₂×C₈	—	134	2	no	—
172	C₂×C₈	—	134	2	no	—
173	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
174	C₂×C₈	#11761616	1	4	yes	—
175	C₂×C₈	#11761618	1	4	yes	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	268	2	no	—
180	C₄⋊C₄	—	268	2	no	—
181	C₄⋊C₄	—	134	4	no	—
182	C₄⋊C₄	—	134	4	no	—
183	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
184	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
185	C₄²	#11761629	1	1	yes	characteristic
186	C₄²	#11761629	1	1	yes	characteristic
187	C₂³	#14524039	1	1	yes	characteristic
188	Q₈	—	268	8	no	—
189	Q₈	—	536	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	268	4	no	—
194	C₂×C₄	—	536	2	no	—
195	C₂×C₄	—	536	2	no	—
196	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
197	C₂×C₄	#14524005	1	1	yes	characteristic
198	C₂×C₄	#14524005	1	1	yes	characteristic
199	C₂×C₄	#14524018	1	1	yes	characteristic
200	C₂×C₄	#14524018	1	1	yes	characteristic
201	C₂×C₄	#14524032	1	1	yes	characteristic
202	C₂×C₄	D₈×D₆₇	1	2	yes	—
203	C₂×C₄	C₂×#11761616	1	2	yes	—
204	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
205	C₂×C₄	C₂×#11761618	1	2	yes	—
206	C₂×C₄	#14524062	1	2	yes	—
207	C₂×C₄	#14524066	1	2	yes	—
208	C₈	—	134	4	no	—
209	C₈	—	134	4	no	—
210	C₈	#14523966	1	8	yes	complemented
211	C₂²	#16863297	1	1	yes	characteristic
212	C₂²	#16863768	1	1	yes	characteristic
213	C₂²	#16863771	1	1	yes	characteristic
214	C₂²	#16863774	1	1	yes	characteristic
215	C₂²	#16863775	1	1	yes	characteristic
216	C₂²	#16863779	1	1	yes	characteristic
217	C₂²	#16863781	1	1	yes	characteristic
218	C₄	—	16	2	no	—
219	C₄	—	536	4	no	—
220	C₄	C₂×#14523966	1	2	yes	—
221	C₄	#16863478	1	2	yes	—
222	C₄	#16863490	1	2	yes	—
223	C₄	#16863633	1	2	yes	—
224	C₄	#16863638	1	2	yes	—
225	C₄	#16863748	1	2	yes	—
226	C₂	#22559798	1	1	yes	characteristic
227	C₂	#22559803	1	1	yes	characteristic
228	C₂	#22559818	1	1	yes	characteristic
229	C₂	#22559826	1	1	yes	characteristic
230	C₂	#22561010	1	1	yes	characteristic
231	C₂	#22561068	1	1	yes	characteristic
232	C₂	#22561073	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#9823
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
8. #16863801	C₄	semidirect
91. C₈×C₆₇	#103	semidirect
122. C₆₇	#9823	semidirect
210. C₈	#14523966	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863801	2
#22561169	2
#22561010	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1845	2
#9823	67