Subgroups — #30760543

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561145	C₂	1	1	yes	maximal, characteristic
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22560215	C₂	1	2	yes	maximal
4	#22558093	C₂	1	2	yes	maximal
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₂×#14523977	—	2	1	no	—
8	C₂×#14523977	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14523968	—	2	2	no	—
11	#16863266	C₂²	1	2	yes	—
12	#16863159	—	2	2	no	—
13	#16863150	—	2	1	no	—
14	#16863150	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₄×Dic₁₃₄	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
25	C₂×Dic₆₇⋊C₄	—	2	1	no	—
26	C₂×C₄×Dic₆₇	—	2	1	no	—
27	#14523977	—	4	2	no	—
28	#14523977	—	4	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523968	—	4	4	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₂×Dic₁₃₄	—	2	2	no	—
39	C₂×Dic₁₃₄	—	2	2	no	—
40	C₂×Dic₁₃₄	—	2	1	no	—
41	C₂×Dic₁₃₄	D₈	1	2	yes	—
42	C₂²×Dic₆₇	—	2	1	no	—
43	C₂²×Dic₆₇	—	2	1	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	1	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₂×C₆₇⋊C₈	—	4	1	no	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	8	2	no	—
54	Dic₆₇⋊C₄	—	4	2	no	—
55	C₄×Dic₆₇	—	4	1	no	—
56	C₄×Dic₆₇	—	4	1	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	#41	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	4	2	no	—
65	C₂×C₈×C₆₇	—	4	2	no	—
66	C₂×C₈×C₆₇	—	4	2	no	—
67	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
69	C₄²×C₆₇	#41	1	2	yes	—
70	C₄²×C₆₇	#41	1	2	yes	—
71	C₂³×C₆₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₇	—	2	1	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	#123	1	2	yes	—
75	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#126	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
82	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#137	1	2	yes	—
84	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	4	1	no	—
87	C₂×Dic₆₇	—	4	1	no	—
88	C₂×Dic₆₇	—	4	1	no	—
89	C₂×Dic₆₇	—	4	1	no	—
90	C₂×Dic₆₇	—	4	1	no	—
91	C₂×Dic₆₇	—	8	2	no	—
92	C₂×Dic₆₇	—	8	2	no	—
93	C₂×Dic₆₇	—	8	2	no	—
94	Dic₁₃₄	—	2	4	no	—
95	Dic₁₃₄	—	4	2	no	—
96	C₈×C₆₇	—	2	2	no	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	—	8	2	no	—
99	C₆₇⋊C₈	—	8	1	no	—
100	C₂²×C₆₇	#397	1	1	yes	characteristic
101	C₂²×C₆₇	#460	1	1	yes	characteristic
102	C₂²×C₆₇	#462	1	1	yes	characteristic
103	C₂²×C₆₇	#466	1	1	yes	characteristic
104	C₂²×C₆₇	#470	1	1	yes	characteristic
105	C₂²×C₆₇	#472	1	1	yes	characteristic
106	C₂²×C₆₇	#473	1	1	yes	characteristic
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	#480	1	2	yes	—
109	C₄×C₆₇	#483	1	1	yes	characteristic
110	C₄×C₆₇	#484	1	1	yes	characteristic
111	C₄×C₆₇	#485	1	2	yes	—
112	C₄×C₆₇	#487	1	1	yes	characteristic
113	C₄×C₆₇	#489	1	1	yes	characteristic
114	Dic₆₇	—	4	2	no	—
115	Dic₆₇	—	8	1	no	—
116	Dic₆₇	—	16	2	no	—
117	#10617	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	#1668	1	1	yes	characteristic
119	C₂×C₆₇	#1674	1	1	yes	characteristic
120	C₂×C₆₇	#1690	1	1	yes	characteristic
121	C₂×C₆₇	#1696	1	1	yes	characteristic
122	C₂×C₆₇	#2072	1	1	yes	characteristic
123	C₂×C₆₇	#2075	1	1	yes	characteristic
124	C₂×C₆₇	#2078	1	1	yes	characteristic
125	#1984	—	67	1	no	—
126	#1845	—	67	1	no	—
127	#1803	—	67	2	no	—
128	#1296	—	67	2	no	—
129	#1277	D₆₇	1	1	yes	characteristic
130	C₆₇	#10617	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	—	67	1	no	—
132	C₂×#107	—	134	1	no	—
133	C₂×#107	—	134	1	no	—
134	C₂×C₄⋊C₈	—	67	1	no	—
135	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
136	C₂×#103	—	134	2	no	—
137	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	#384	—	67	2	no	—
139	#340	—	134	2	no	—
140	#336	—	134	1	no	—
141	#336	—	134	1	no	—
142	C₂²×Q₈	—	67	1	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	134	1	no	—
145	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
146	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
147	C₄×Q₈	—	134	2	no	—
148	C₄×Q₈	—	134	2	no	—
149	C₂×C₄⋊C₄	—	67	1	no	—
150	C₂×C₄⋊C₄	—	134	2	no	—
151	C₂×C₄⋊C₄	—	134	2	no	—
152	C₂×C₄⋊C₄	—	134	1	no	—
153	C₂×C₄⋊C₄	—	134	2	no	—
154	C₂×C₄²	—	134	1	no	—
155	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
156	#107	—	268	2	no	—
157	#107	—	268	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	134	2	no	—
161	#103	—	268	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	67	2	no	—
164	C₂×Q₈	—	134	2	no	—
165	C₂×Q₈	—	134	2	no	—
166	C₂×Q₈	—	134	1	no	—
167	C₂²×C₄	—	268	2	no	—
168	C₂²×C₄	—	134	1	no	—
169	C₂²×C₄	—	134	1	no	—
170	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
172	C₂²×C₄	#11761629	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	268	1	no	—
181	C₂×C₈	—	268	1	no	—
182	C₂×C₈	—	268	1	no	—
183	C₂×C₈	#11761617	1	2	yes	—
184	C₂×C₈	Dic₂₆₈	1	2	yes	—
185	C₄⋊C₄	—	268	2	no	—
186	C₄⋊C₄	—	268	2	no	—
187	C₄⋊C₄	—	268	2	no	—
188	C₄⋊C₄	—	268	2	no	—
189	C₄⋊C₄	—	268	2	no	—
190	C₄⋊C₄	—	536	2	no	—
191	C₄⋊C₄	—	134	2	no	—
192	C₄²	—	268	1	no	—
193	C₄²	—	268	1	no	—
194	C₄²	#11761627	1	2	yes	—
195	C₄²	#11761628	1	2	yes	—
196	C₂³	#14524018	1	1	yes	characteristic
197	Q₈	—	268	2	no	—
198	Q₈	—	134	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	268	1	no	—
202	C₂×C₄	—	268	1	no	—
203	C₂×C₄	—	268	1	no	—
204	C₂×C₄	—	268	1	no	—
205	C₂×C₄	—	268	1	no	—
206	C₂×C₄	—	536	2	no	—
207	C₂×C₄	—	536	2	no	—
208	C₂×C₄	—	536	2	no	—
209	C₂×C₄	—	134	2	no	—
210	C₂×C₄	#14524025	1	2	yes	—
211	C₂×C₄	#14524027	1	1	yes	characteristic
212	C₂×C₄	#14524031	1	1	yes	characteristic
213	C₂×C₄	#14524038	1	2	yes	—
214	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
217	C₂×C₄	#14524066	1	1	yes	characteristic
218	C₂×C₄	#14524071	1	1	yes	characteristic
219	C₂×C₄	#14524072	1	2	yes	—
220	C₂×C₄	#14524076	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	536	1	no	—
225	C₂²	#16863323	1	1	yes	characteristic
226	C₂²	#16863597	1	1	yes	characteristic
227	C₂²	#16863607	1	1	yes	characteristic
228	C₂²	#16863633	1	1	yes	characteristic
229	C₂²	#16863655	1	1	yes	characteristic
230	C₂²	#16863667	1	1	yes	characteristic
231	C₂²	#16863672	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	268	2	no	—
234	C₄	—	536	1	no	—
235	C₄	—	1072	2	no	—
236	C₄	#16863708	1	2	yes	—
237	C₄	#16863727	1	1	yes	characteristic
238	C₄	#16863732	1	1	yes	characteristic
239	C₄	#16863736	1	2	yes	—
240	C₄	#16863747	1	1	yes	characteristic
241	C₄	#16863756	1	1	yes	characteristic
242	C₂	#22559570	1	1	yes	characteristic
243	C₂	#22559604	1	1	yes	characteristic
244	C₂	#22559704	1	1	yes	characteristic
245	C₂	#22559738	1	1	yes	characteristic
246	C₂	#22561619	1	1	yes	characteristic
247	C₂	#22561634	1	1	yes	characteristic
248	C₂	#22561651	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#10617
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
130. C₆₇	#10617	semidirect

Maximal subgroups

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

subgroup	index
#22561145	2
#22560369	2
#22560215	2
#22558093	2
C₆₇×#1277	2
#10617	67