Subgroups — #30748931

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	C₂×#16863526	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	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	#16863526	—	2	2	no	—
8	#16863526	C₄	1	4	yes	complemented
9	C₂×#14523977	—	2	1	no	—
10	C₂×#14523977	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×#14523968	—	2	2	no	—
14	C₂×#14523953	—	2	1	no	—
15	#16863266	C₂²	1	2	yes	—
16	#16863159	—	2	2	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#334×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₃₄	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	#14523977	—	4	2	no	—
33	#14523977	—	4	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₆₇⋊C₈	—	2	4	no	—
37	#14523968	—	4	4	no	—
38	#14523953	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	C₂×Dic₁₃₄	—	2	2	no	—
49	C₂×Dic₁₃₄	—	2	1	no	—
50	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₆₇	—	2	1	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	4	1	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₄⋊Dic₆₇	—	2	1	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	8	2	no	—
63	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	Dic₆₇⋊C₄	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
79	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
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	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²⋊Q₈	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	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#137	1	2	yes	—
98	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	4	1	no	—
102	C₂×Dic₆₇	—	4	1	no	—
103	C₂×Dic₆₇	—	4	1	no	—
104	C₂×Dic₆₇	—	4	1	no	—
105	C₂×Dic₆₇	—	8	2	no	—
106	C₂×Dic₆₇	—	8	2	no	—
107	C₂×Dic₆₇	—	8	2	no	—
108	Dic₁₃₄	—	2	4	no	—
109	Dic₁₃₄	—	4	2	no	—
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	8	2	no	—
112	C₈×C₆₇	#103	1	4	yes	complemented
113	C₆₇⋊C₈	—	4	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	C₂×#103	1	1	yes	characteristic
124	C₄×C₆₇	#417	1	1	yes	characteristic
125	C₄×C₆₇	#428	1	2	yes	—
126	C₄×C₆₇	#468	1	2	yes	—
127	C₄×C₆₇	#475	1	1	yes	characteristic
128	C₄×C₆₇	#477	1	1	yes	characteristic
129	Dic₆₇	—	4	2	no	—
130	Dic₆₇	—	4	2	no	—
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	—	67	1	no	—
141	#1845	—	67	1	no	—
142	#1803	—	67	2	no	—
143	C₂×#334	D₆₇	1	1	yes	characteristic
144	#1296	—	67	2	no	—
145	C₆₇	#8653	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄×Q₈	—	67	1	no	—
147	#446	—	67	4	no	—
148	#446	—	134	2	no	—
149	C₂×#107	—	134	1	no	—
150	C₂×#107	—	134	1	no	—
151	C₂×C₄⋊C₈	—	67	1	no	—
152	C₂×C₄⋊C₈	—	134	1	no	—
153	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
154	C₂×#103	—	134	2	no	—
155	C₂×#97	—	134	1	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#384	—	67	2	no	—
158	#340	—	134	2	no	—
159	#334	—	2	2	no	—
160	C₂²×Q₈	—	67	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	134	1	no	—
163	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
164	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
165	C₄×Q₈	—	67	2	no	—
166	C₄×Q₈	—	67	2	no	—
167	C₄×Q₈	—	134	1	no	—
168	C₄×Q₈	—	134	1	no	—
169	C₂×C₄⋊C₄	—	67	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	—	134	1	no	—
174	C₂×C₄²	—	134	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	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	67	2	no	—
181	C₄⋊C₈	—	134	4	no	—
182	C₄⋊C₈	—	134	1	no	—
183	#103	—	268	4	no	—
184	#97	—	134	4	no	—
185	C₄×C₈	C₄×D₆₇	1	2	yes	—
186	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
187	C₂×Q₈	—	67	2	no	—
188	C₂×Q₈	—	134	2	no	—
189	C₂×Q₈	—	134	2	no	—
190	C₂×Q₈	—	134	1	no	—
191	C₂²×C₄	—	268	2	no	—
192	C₂²×C₄	—	134	1	no	—
193	C₂²×C₄	—	134	1	no	—
194	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
196	C₂²×C₄	#11761629	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	268	1	no	—
203	C₂×C₈	—	268	1	no	—
204	C₂×C₈	—	134	2	no	—
205	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
206	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
207	C₂×C₈	#11761617	1	2	yes	—
208	C₂×C₈	Dic₂₆₈	1	2	yes	—
209	C₄⋊C₄	—	268	2	no	—
210	C₄⋊C₄	—	268	2	no	—
211	C₄⋊C₄	—	268	2	no	—
212	C₄⋊C₄	—	268	2	no	—
213	C₄⋊C₄	—	536	2	no	—
214	C₄⋊C₄	—	67	2	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₄⋊C₄	—	134	1	no	—
217	C₄⋊C₄	—	134	2	no	—
218	C₄²	—	134	2	no	—
219	C₄²	—	134	2	no	—
220	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₄²	Q₈×D₆₇	1	1	yes	characteristic
223	C₄²	#11761627	1	1	yes	characteristic
224	C₂³	#14524018	1	1	yes	characteristic
225	Q₈	—	268	2	no	—
226	Q₈	—	134	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	268	1	no	—
230	C₂×C₄	—	268	1	no	—
231	C₂×C₄	—	268	1	no	—
232	C₂×C₄	—	268	1	no	—
233	C₂×C₄	—	536	2	no	—
234	C₂×C₄	—	536	2	no	—
235	C₂×C₄	—	536	2	no	—
236	C₂×C₄	—	134	2	no	—
237	C₂×C₄	—	134	2	no	—
238	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
240	C₂×C₄	#14524001	1	1	yes	characteristic
241	C₂×C₄	#14524014	1	1	yes	characteristic
242	C₂×C₄	#14524019	1	1	yes	characteristic
243	C₂×C₄	#14524023	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
246	C₂×C₄	#14524070	1	2	yes	—
247	C₂×C₄	#14524071	1	1	yes	characteristic
248	C₂×C₄	#14524072	1	2	yes	—
249	C₂×C₄	#14524076	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	—	268	2	no	—
253	C₈	#14523968	1	4	yes	complemented
254	C₂²	#16863253	1	1	yes	characteristic
255	C₂²	#16863597	1	1	yes	characteristic
256	C₂²	#16863607	1	1	yes	characteristic
257	C₂²	#16863650	1	1	yes	characteristic
258	C₂²	#16863655	1	1	yes	characteristic
259	C₂²	#16863667	1	1	yes	characteristic
260	C₂²	#16863672	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	268	2	no	—
264	C₄	—	268	2	no	—
265	C₄	—	1072	2	no	—
266	C₄	C₂×#14523968	1	1	yes	characteristic
267	C₄	#16863398	1	1	yes	characteristic
268	C₄	#16863435	1	2	yes	—
269	C₄	#16863639	1	2	yes	—
270	C₄	#16863682	1	1	yes	characteristic
271	C₄	#16863692	1	1	yes	characteristic
272	C₂	#22559694	1	1	yes	characteristic
273	C₂	#22559704	1	1	yes	characteristic
274	C₂	#22559728	1	1	yes	characteristic
275	C₂	#22559738	1	1	yes	characteristic
276	C₂	#22560215	1	1	yes	characteristic
277	C₂	#22560239	1	1	yes	characteristic
278	C₂	#22560270	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#8653
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16863526	C₄	semidirect
112. C₈×C₆₇	#103	semidirect
145. C₆₇	#8653	semidirect
253. C₈	#14523968	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
#22560369	2
#22560215	2
#22558093	2
C₂×#334×C₆₇	2
#8653	67