Subgroups — #30760094

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#22561132	C₂	1	2	yes	maximal
3	C₂×#16863141	C₂	1	2	yes	maximal
4	#22558044	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863801	—	2	4	no	—
8	C₂×#14523977	—	2	1	no	—
9	C₂×#14523977	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×#14523966	—	2	2	no	—
12	C₂×#14523966	—	2	2	no	—
13	#16863141	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	1	no	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	D₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	#14523977	—	2	4	no	—
28	#14523977	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	#14523966	—	4	4	no	—
32	#14523966	—	4	4	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	4	2	no	—
42	C₄×C₈×C₆₇	—	2	2	no	—
43	#95×C₆₇	—	2	1	no	—
44	C₂×Dic₁₃₄	—	2	4	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	—	4	2	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂²×Dic₆₇	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	2	1	no	—
59	C₂²×C₄×C₆₇	—	2	1	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	1	no	—
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	4	1	no	—
68	C₂×C₈×C₆₇	—	4	1	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
71	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
75	C₄²×C₆₇	—	4	1	no	—
76	C₄²×C₆₇	—	4	1	no	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	#41	1	2	yes	—
79	C₂³×C₆₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	—	4	1	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#121	1	2	yes	—
90	C₂×C₄×C₆₇	#123	1	2	yes	—
91	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	Dic₁₃₄	—	4	4	no	—
103	Dic₁₃₄	—	4	4	no	—
104	Dic₁₃₄	—	8	2	no	—
105	Dic₁₃₄	—	8	2	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	8	1	no	—
109	C₆₇⋊C₈	—	8	2	no	—
110	C₂²×C₆₇	#394	1	1	yes	characteristic
111	C₂²×C₆₇	#461	1	1	yes	characteristic
112	C₂²×C₆₇	#462	1	1	yes	characteristic
113	C₂²×C₆₇	#466	1	1	yes	characteristic
114	C₂²×C₆₇	#469	1	1	yes	characteristic
115	C₂²×C₆₇	#473	1	1	yes	characteristic
116	C₂²×C₆₇	#473	1	1	yes	characteristic
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	—	8	1	no	—
120	C₄×C₆₇	#448	1	1	yes	characteristic
121	C₄×C₆₇	#451	1	1	yes	characteristic
122	C₄×C₆₇	#459	1	2	yes	—
123	C₄×C₆₇	#463	1	2	yes	—
124	C₄×C₆₇	#487	1	1	yes	characteristic
125	C₄×C₆₇	#491	1	1	yes	characteristic
126	Dic₆₇	—	8	4	no	—
127	#10545	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	#1673	1	1	yes	characteristic
129	C₂×C₆₇	#1674	1	1	yes	characteristic
130	C₂×C₆₇	#1693	1	1	yes	characteristic
131	C₂×C₆₇	#1694	1	1	yes	characteristic
132	C₂×C₆₇	#2043	1	1	yes	characteristic
133	C₂×C₆₇	#2047	1	1	yes	characteristic
134	C₂×C₆₇	#2054	1	1	yes	characteristic
135	C₂×#500	—	67	1	no	—
136	#1981	—	67	2	no	—
137	#1939	D₆₇	1	1	yes	characteristic
138	C₂×#332	—	67	2	no	—
139	#1277	—	67	1	no	—
140	C₆₇	#10545	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄⋊Q₈	—	67	2	no	—
142	#500	—	134	4	no	—
143	C₂×#107	—	134	1	no	—
144	C₂×#107	—	134	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₄⋊C₈	—	67	2	no	—
147	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
148	C₂×#103	—	134	2	no	—
149	C₂×#103	—	134	2	no	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	#332	—	134	4	no	—
154	C₂²×Q₈	—	134	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	67	4	no	—
160	C₄⋊Q₈	—	134	2	no	—
161	C₄⋊Q₈	—	134	2	no	—
162	C₄⋊Q₈	—	134	2	no	—
163	C₂×C₄⋊C₄	—	134	2	no	—
164	C₂×C₄⋊C₄	—	134	2	no	—
165	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
166	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
169	#107	—	134	4	no	—
170	#107	—	134	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	4	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	#103	—	268	4	no	—
176	#103	—	268	4	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	2	1	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	2	no	—
181	C₂×Q₈	—	268	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	268	2	no	—
188	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
190	C₂²×C₄	#11761628	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	268	2	no	—
199	C₂×C₈	—	268	2	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	#11761616	1	2	yes	—
202	C₂×C₈	#11761618	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	268	2	no	—
207	C₄⋊C₄	—	134	4	no	—
208	C₄⋊C₄	—	134	4	no	—
209	C₄⋊C₄	D₅₃₆	1	2	yes	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	D₄×D₆₇	1	2	yes	—
213	C₄²	#11761629	1	2	yes	—
214	C₂³	#14524017	1	1	yes	characteristic
215	Q₈	—	268	4	no	—
216	Q₈	—	268	4	no	—
217	Q₈	—	536	2	no	—
218	Q₈	—	536	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	268	4	no	—
228	C₂×C₄	—	536	2	no	—
229	C₂×C₄	—	536	2	no	—
230	C₂×C₄	#14524011	1	1	yes	characteristic
231	C₂×C₄	#14524018	1	2	yes	—
232	C₂×C₄	#14524028	1	2	yes	—
233	C₂×C₄	#14524032	1	1	yes	characteristic
234	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
235	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
237	C₂×C₄	#14524065	1	1	yes	characteristic
238	C₂×C₄	#14524071	1	2	yes	—
239	C₂×C₄	#14524073	1	1	yes	characteristic
240	C₂×C₄	#14524073	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	1	no	—
244	C₈	—	536	2	no	—
245	C₂²	#16863313	1	1	yes	characteristic
246	C₂²	#16863601	1	1	yes	characteristic
247	C₂²	#16863606	1	1	yes	characteristic
248	C₂²	#16863632	1	1	yes	characteristic
249	C₂²	#16863649	1	1	yes	characteristic
250	C₂²	#16863671	1	1	yes	characteristic
251	C₂²	#16863671	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	8	1	no	—
255	C₄	—	536	4	no	—
256	C₄	#16863534	1	1	yes	characteristic
257	C₄	#16863551	1	1	yes	characteristic
258	C₄	#16863592	1	2	yes	—
259	C₄	#16863612	1	2	yes	—
260	C₄	#16863748	1	1	yes	characteristic
261	C₄	#16863766	1	1	yes	characteristic
262	C₂	#22559598	1	1	yes	characteristic
263	C₂	#22559603	1	1	yes	characteristic
264	C₂	#22559722	1	1	yes	characteristic
265	C₂	#22559727	1	1	yes	characteristic
266	C₂	#22561455	1	1	yes	characteristic
267	C₂	#22561477	1	1	yes	characteristic
268	C₂	#22561520	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1939	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1939	D₆₇
hypocenter	C₆₇	#10545
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
140. C₆₇	#10545	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
#22561132	2
C₂×#16863141	2
#22558044	2
C₆₇×#1939	2
#10545	67