Subgroups — #30748930

Order 17152.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560369	C₂	1	1	yes	maximal, characteristic
2	#22560214	C₂	1	2	yes	maximal
3	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
4	#22558093	C₂	1	2	yes	maximal
5	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523967	—	2	2	no	—
10	#16863266	C₂²	1	2	yes	—
11	#16863158	—	2	2	no	—
12	#16863147	—	2	2	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#446×C₆₇	—	2	2	no	—
15	#446×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	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	#14523977	—	4	2	no	—
25	#14523977	—	4	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	#14523967	—	4	4	no	—
29	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₆₇	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	—	2	1	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
44	#97×C₆₇	—	2	4	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×Dic₆₇	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	8	2	no	—
56	C₂×Q₈×C₆₇	—	2	1	no	—
57	C₂×Q₈×C₆₇	—	2	2	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	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	#41	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	4	1	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₂³×C₆₇	#121	1	1	yes	characteristic
85	Q₈×C₆₇	—	2	4	no	—
86	Q₈×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	—	4	1	no	—
93	C₂×C₄×C₆₇	—	4	1	no	—
94	C₂×C₄×C₆₇	—	4	1	no	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#137	1	2	yes	—
107	C₂×Dic₆₇	—	8	2	no	—
108	C₂×Dic₆₇	—	8	2	no	—
109	C₂×Dic₆₇	—	8	2	no	—
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	4	2	no	—
112	C₈×C₆₇	#103	1	4	yes	complemented
113	C₆₇⋊C₈	—	8	2	no	—
114	C₂²×C₆₇	#380	1	1	yes	characteristic
115	C₂²×C₆₇	#460	1	1	yes	characteristic
116	C₂²×C₆₇	#462	1	1	yes	characteristic
117	C₂²×C₆₇	#469	1	1	yes	characteristic
118	C₂²×C₆₇	#470	1	1	yes	characteristic
119	C₂²×C₆₇	#472	1	1	yes	characteristic
120	C₂²×C₆₇	#473	1	1	yes	characteristic
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	4	2	no	—
124	C₄×C₆₇	—	4	2	no	—
125	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
126	C₄×C₆₇	#417	1	1	yes	characteristic
127	C₄×C₆₇	#428	1	2	yes	—
128	C₄×C₆₇	#468	1	2	yes	—
129	C₄×C₆₇	#475	1	1	yes	characteristic
130	C₄×C₆₇	#477	1	1	yes	characteristic
131	Dic₆₇	—	16	2	no	—
132	#8653	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	#1688	1	1	yes	characteristic
134	C₂×C₆₇	#1690	1	1	yes	characteristic
135	C₂×C₆₇	#1694	1	1	yes	characteristic
136	C₂×C₆₇	#1696	1	1	yes	characteristic
137	C₂×C₆₇	#1803	1	1	yes	characteristic
138	C₂×C₆₇	#1809	1	1	yes	characteristic
139	C₂×C₆₇	#1816	1	1	yes	characteristic
140	C₂×#446	D₆₇	1	1	yes	characteristic
141	#1845	—	67	1	no	—
142	#1803	—	67	2	no	—
143	C₂×#334	—	67	1	no	—
144	#1296	—	67	2	no	—
145	C₆₇	#8653	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
147	#446	—	2	2	no	—
148	#446	Dic₆₇	1	4	yes	complemented
149	C₂×#107	—	134	1	no	—
150	C₂×#107	—	134	1	no	—
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄⋊C₈	—	67	2	no	—
153	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₂×#103	—	134	2	no	—
155	C₂×#97	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#384	—	67	2	no	—
158	#340	—	134	2	no	—
159	#334	—	134	2	no	—
160	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	134	2	no	—
163	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄×Q₈	—	2	1	no	—
166	C₄×Q₈	—	2	1	no	—
167	C₄×Q₈	Dic₁₃₄	1	2	yes	—
168	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	134	2	no	—
171	C₂×C₄⋊C₄	—	134	2	no	—
172	C₂×C₄⋊C₄	—	134	2	no	—
173	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	#107	—	268	2	no	—
177	#107	—	268	2	no	—
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	134	2	no	—
181	C₄⋊C₈	—	134	2	no	—
182	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
183	#103	—	268	4	no	—
184	#97	—	2	4	no	—
185	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
186	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	268	2	no	—
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
196	C₂²×C₄	#11761628	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	268	2	no	—
203	C₂×C₈	—	268	2	no	—
204	C₂×C₈	—	268	2	no	—
205	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
206	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
207	C₂×C₈	#11761615	1	2	yes	—
208	C₂×C₈	#11761618	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	268	2	no	—
213	C₄⋊C₄	—	268	2	no	—
214	C₄⋊C₄	—	268	2	no	—
215	C₄⋊C₄	—	268	2	no	—
216	C₄⋊C₄	—	536	2	no	—
217	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
223	C₄²	#11761628	1	1	yes	characteristic
224	C₂³	#14524017	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	536	2	no	—
236	C₂×C₄	—	536	2	no	—
237	C₂×C₄	—	536	2	no	—
238	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
240	C₂×C₄	#14524002	1	1	yes	characteristic
241	C₂×C₄	#14524017	1	1	yes	characteristic
242	C₂×C₄	#14524022	1	1	yes	characteristic
243	C₂×C₄	#14524022	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
246	C₂×C₄	#14524068	1	1	yes	characteristic
247	C₂×C₄	#14524071	1	2	yes	—
248	C₂×C₄	#14524073	1	1	yes	characteristic
249	C₂×C₄	#14524076	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	536	2	no	—
253	C₈	#14523967	1	4	yes	complemented
254	C₂²	#16863256	1	1	yes	characteristic
255	C₂²	#16863596	1	1	yes	characteristic
256	C₂²	#16863606	1	1	yes	characteristic
257	C₂²	#16863649	1	1	yes	characteristic
258	C₂²	#16863654	1	1	yes	characteristic
259	C₂²	#16863666	1	1	yes	characteristic
260	C₂²	#16863671	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	1072	2	no	—
266	C₄	C₂×#14523967	1	1	yes	characteristic
267	C₄	#16863397	1	1	yes	characteristic
268	C₄	#16863438	1	2	yes	—
269	C₄	#16863642	1	2	yes	—
270	C₄	#16863681	1	1	yes	characteristic
271	C₄	#16863691	1	1	yes	characteristic
272	C₂	#22559693	1	1	yes	characteristic
273	C₂	#22559703	1	1	yes	characteristic
274	C₂	#22559727	1	1	yes	characteristic
275	C₂	#22559737	1	1	yes	characteristic
276	C₂	#22560214	1	1	yes	characteristic
277	C₂	#22560242	1	1	yes	characteristic
278	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#446×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#446	D₆₇
hypocenter	C₆₇	#8653
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
15. #446×C₆₇	C₄	semidirect
112. C₈×C₆₇	#103	semidirect
145. C₆₇	#8653	semidirect
148. #446	Dic₆₇	semidirect
253. C₈	#14523967	semidirect

Maximal subgroups

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

subgroup	index
#22560369	2
#22560214	2
C₂×#16863147	2
#22558093	2
C₂×#446×C₆₇	2
#8653	67