Subgroups — #30827610

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22569156	C₂	1	2	yes	maximal
3	#22559107	C₂	1	2	yes	maximal, complemented
4	#22559061	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₆₈	C₂	1	2	yes	maximal, complemented
6	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
7	#22558613	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863953	C₂²	1	2	yes	—
10	#16863801	C₄	1	1	yes	characteristic
11	#16863776	C₄	1	1	yes	characteristic
12	#16863747	C₄	1	2	yes	—
13	#16863637	C₄	1	2	yes	—
14	#16863637	C₄	1	2	yes	—
15	#16863526	C₂²	1	4	yes	—
16	#16863524	—	2	4	no	—
17	C₈×Dic₁₃₄	C₂²	1	4	yes	—
18	C₄×Dic₂₆₈	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	2	yes	—
20	#16863424	—	2	2	no	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	#16863354	—	2	2	no	—
23	#16863149	C₂²	1	1	yes	characteristic
24	#16863149	C₂²	1	2	yes	—
25	#16863149	C₂²	1	1	yes	characteristic
26	#16863128	C₂²	1	4	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#322×C₆₇	C₂²	1	2	yes	—
32	C₈²×C₆₇	C₂²	1	2	yes	—
33	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	#14524043	C₂×C₄	1	2	yes	—
37	#14524043	C₂×C₄	1	2	yes	—
38	#14524035	—	2	2	no	—
39	#14524022	—	2	2	no	—
40	#14524022	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃₄	—	2	2	no	—
42	C₄×Dic₁₃₄	C₂³	1	4	yes	—
43	#14524002	—	2	2	no	—
44	#14523977	C₂×C₄	1	2	yes	—
45	#14523977	C₂×C₄	1	2	yes	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	Dic₆₇⋊C₈	—	2	4	no	—
50	Dic₆₇⋊C₈	—	2	4	no	—
51	#14523968	C₂×C₄	1	4	yes	—
52	#14523968	C₂×C₄	1	4	yes	—
53	#14523953	—	2	4	no	—
54	C₈×Dic₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₂²⋊C₈×C₆₇	—	2	1	no	—
62	#97×C₆₇	D₄	1	2	yes	—
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂×Dic₁₃₄	—	2	2	no	—
70	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
71	Dic₂₆₈	—	2	4	no	—
72	Dic₂₆₈	—	4	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	4	4	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	—	2	2	no	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
80	Dic₆₇⋊C₄	—	2	4	no	—
81	Dic₆₇⋊C₄	—	2	4	no	—
82	C₂²⋊Dic₆₇	—	2	4	no	—
83	C₄×Dic₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	Q₁₆	1	4	yes	complemented
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	4	1	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	#41	1	1	yes	characteristic
101	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₆₇	#135	1	2	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	2	4	no	—
116	Dic₁₃₄	—	2	4	no	—
117	Dic₁₃₄	—	2	4	no	—
118	Dic₁₃₄	—	8	2	no	—
119	C₈×C₆₇	—	2	1	no	—
120	C₈×C₆₇	—	2	1	no	—
121	C₈×C₆₇	—	2	1	no	—
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	4	2	no	—
124	C₈×C₆₇	#131	1	2	yes	—
125	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
126	C₆₇⋊C₈	—	4	2	no	—
127	C₆₇⋊C₈	—	4	2	no	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
136	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₆₇	#568	1	1	yes	characteristic
138	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
140	Dic₆₇	—	4	4	no	—
141	Dic₆₇	—	4	4	no	—
142	#18068	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	2	no	—
144	C₂×C₆₇	#2935	1	1	yes	characteristic
145	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₆₇	#2957	1	1	yes	characteristic
147	#3170	—	67	1	no	—
148	#2962	—	67	2	no	—
149	#1591	—	67	2	no	—
150	#1584	—	67	4	no	—
151	C₈×Q₁₆	—	67	2	no	—
152	C₂×#335	—	67	1	no	—
153	#1481	—	67	2	no	—
154	C₈×M₁₆	D₆₇	1	1	yes	characteristic
155	C₆₇	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	—	67	2	no	—
157	#500	—	67	1	no	—
158	#494	—	67	1	no	—
159	#487	—	67	2	no	—
160	#467	—	67	2	no	—
161	#467	—	67	2	no	—
162	#446	—	67	4	no	—
163	#446	—	134	4	no	—
164	C₈×Q₈	—	67	4	no	—
165	C₄×Q₁₆	—	67	2	no	—
166	C₄×Q₁₆	—	67	2	no	—
167	#431	C₂×D₆₇	1	1	yes	characteristic
168	#424	—	134	2	no	—
169	C₂×C₄⋊C₈	—	67	2	no	—
170	#405	—	134	2	no	—
171	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
173	#335	—	67	1	no	—
174	#335	—	67	2	no	—
175	#335	—	67	1	no	—
176	Q₈⋊C₈	—	67	4	no	—
177	Q₈⋊C₈	—	67	4	no	—
178	#322	C₂×D₆₇	1	2	yes	—
179	C₈²	C₂×D₆₇	1	2	yes	—
180	C₂×Q₁₆	—	67	2	no	—
181	C₂×Q₁₆	—	67	2	no	—
182	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
183	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
184	C₂²×C₈	—	134	2	no	—
185	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	67	2	no	—
188	C₄⋊Q₈	—	67	2	no	—
189	#125	—	134	2	no	—
190	C₂²⋊Q₈	—	67	4	no	—
191	C₂²⋊Q₈	—	134	2	no	—
192	C₄×Q₈	—	67	4	no	—
193	C₄×Q₈	—	134	2	no	—
194	#117	—	134	2	no	—
195	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
196	#107	—	67	2	no	—
197	#107	—	67	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	67	2	no	—
200	C₄⋊C₈	—	67	2	no	—
201	C₄⋊C₈	—	134	4	no	—
202	C₄⋊C₈	—	134	4	no	—
203	C₄⋊C₈	—	134	2	no	—
204	#103	—	67	4	no	—
205	#103	—	67	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	134	4	no	—
208	#97	D₂₆₈	1	2	yes	—
209	C₄×C₈	—	134	4	no	—
210	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
211	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₆₇	1	2	yes	—
213	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₆₇	1	2	yes	—
216	C₂×Q₈	—	67	4	no	—
217	C₂×Q₈	—	134	2	no	—
218	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂²×C₄	#11761628	1	1	yes	characteristic
221	Q₁₆	—	268	2	no	—
222	Q₁₆	—	134	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Dic₂₆₈	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	268	4	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
236	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
237	C₄⋊C₄	—	67	4	no	—
238	C₄⋊C₄	—	134	2	no	—
239	C₄⋊C₄	—	134	2	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₄⋊C₄	—	134	2	no	—
242	C₄⋊C₄	—	134	4	no	—
243	C₂²⋊C₄	—	134	4	no	—
244	C₄²	—	134	4	no	—
245	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
247	C₄²	#11761628	1	1	yes	characteristic
248	C₄²	C₂³×D₆₇	1	1	yes	characteristic
249	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
250	Q₈	—	536	2	no	—
251	Q₈	—	134	4	no	—
252	Q₈	—	134	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	134	4	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
262	C₂×C₄	#14524065	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	268	2	no	—
272	C₈	—	268	2	no	—
273	C₈	#14524051	1	2	yes	—
274	C₈	C₂×Dic₂₆₈	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₂₆₈	1	2	yes	—
278	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	268	4	no	—
283	C₄	—	268	4	no	—
284	C₄	C₄×Dic₂₆₈	1	2	yes	—
285	C₄	C₂×#14524051	1	1	yes	characteristic
286	C₄	#16864162	1	1	yes	characteristic
287	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
288	C₄	C₂×#14524065	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#22568955	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₂₆₈	1	1	yes	characteristic
292	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#18068
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22559107	C₂	semidirect
4. #22559061	C₂	semidirect
5. C₈×Dic₂₆₈	C₂	semidirect
88. M₁₆×C₆₇	Q₁₆	semidirect
155. C₆₇	#18068	semidirect
224. M₁₆	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22569156	2
#22559107	2
#22559061	2
C₈×Dic₂₆₈	2
C₂×#16863149	2
#22558613	2
C₈×M₁₆×C₆₇	2
#18068	67