Subgroups — #31011370

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570790	C₂	1	2	yes	maximal
2	#22562483	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562460	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562435	C₂	1	2	yes	maximal, complemented
5	#22562300	C₂	1	1	yes	maximal, characteristic
6	#22562298	C₂	1	2	yes	maximal
7	Q₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562065	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2216	C₂	1	1	yes	maximal, characteristic
12	#16863948	—	2	2	no	—
13	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
14	C₂×#14523989	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863774	C₂²	1	2	yes	—
18	#16863745	—	2	2	no	—
19	#16863637	C₂²	1	2	yes	—
20	#16863635	—	2	2	no	—
21	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
22	#16863490	C₂²	1	2	yes	—
23	#16863488	—	2	2	no	—
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863222	C₂²	1	1	yes	characteristic
26	#16863221	C₂²	1	2	yes	—
27	#16863201	C₂²	1	2	yes	—
28	#16863200	C₂²	1	2	yes	—
29	#16863199	C₂²	1	2	yes	—
30	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#467×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	C₂×Dic₂₆₈	C₂³	1	2	yes	—
40	C₂×#11761618	—	2	2	no	—
41	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
42	#14524042	—	2	2	no	—
43	#14524041	—	2	2	no	—
44	#14524033	—	4	2	no	—
45	#14524022	—	2	2	no	—
46	#14524020	—	4	2	no	—
47	Q₈×Dic₆₇	—	2	2	no	—
48	#14524008	—	2	2	no	—
49	#14524008	—	4	2	no	—
50	#14523999	—	4	2	no	—
51	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
52	Dic₅₃₆	—	2	2	no	—
53	#14523989	—	2	4	no	—
54	C₂×C₆₇⋊C₁₆	—	2	2	no	—
55	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14523977	C₂³	1	2	yes	—
58	#14523976	—	2	2	no	—
59	#14523968	—	2	2	no	—
60	#14523967	—	2	2	no	—
61	#14523966	—	2	2	no	—
62	C₈×Dic₆₇	D₄	1	1	yes	characteristic
63	C₈×Dic₆₇	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₆₇	—	2	2	no	—
69	Q₃₂×C₆₇	—	2	2	no	—
70	C₂×C₁₆×C₆₇	—	2	2	no	—
71	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	C₂³	1	2	yes	—
74	#103×C₆₇	—	2	2	no	—
75	C₂×Dic₁₃₄	—	2	2	no	—
76	C₂×Dic₁₃₄	—	4	2	no	—
77	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
78	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
79	Dic₂₆₈	—	2	2	no	—
80	Dic₂₆₈	—	2	2	no	—
81	#11761618	—	4	4	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	—	4	2	no	—
87	Dic₆₇⋊C₄	—	4	2	no	—
88	Dic₆₇⋊C₄	—	4	2	no	—
89	Dic₆₇⋊C₄	—	4	2	no	—
90	Dic₆₇⋊C₄	—	4	2	no	—
91	Dic₆₇⋊C₄	—	4	2	no	—
92	C₂²⋊Dic₆₇	—	4	2	no	—
93	C₄×Dic₆₇	—	4	2	no	—
94	C₄×Dic₆₇	—	4	2	no	—
95	C₄×Dic₆₇	D₈	1	1	yes	characteristic
96	C₄×Dic₆₇	D₈	1	1	yes	characteristic
97	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
99	C₆₇⋊C₁₆	—	2	1	no	—
100	C₆₇⋊C₁₆	#41	1	2	yes	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₆₇	—	2	2	no	—
104	Q₁₆×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₁₆×C₆₇	—	2	1	no	—
113	C₁₆×C₆₇	#41	1	2	yes	—
114	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₆₇	—	4	2	no	—
116	Q₈×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	4	2	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	Q₃₂	1	4	yes	complemented
127	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
131	Dic₁₃₄	—	4	2	no	—
132	Dic₁₃₄	—	4	2	no	—
133	Dic₁₃₄	—	8	4	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₆₇⋊C₈	—	2	1	no	—
138	C₆₇⋊C₈	#135	1	2	yes	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
142	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	8	2	no	—
145	C₄×C₆₇	—	8	2	no	—
146	C₄×C₆₇	#532	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
148	Dic₆₇	—	2	1	no	—
149	Dic₆₇	—	8	2	no	—
150	Dic₆₇	—	8	2	no	—
151	Dic₆₇	C₂×Q₃₂	1	2	yes	—
152	Dic₆₇	C₂×Q₃₂	1	2	yes	—
153	Dic₆₇	#508	1	2	yes	—
154	#33734	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	#3169	1	1	yes	characteristic
157	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
158	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
159	#3170	—	67	2	no	—
160	#2252	—	67	1	no	—
161	#2247	—	67	1	no	—
162	#2241	—	67	2	no	—
163	#2216	—	67	1	no	—
164	#2216	—	67	2	no	—
165	#2216	D₆₇	1	1	yes	characteristic
166	C₄×Q₃₂	—	67	1	no	—
167	C₄×Q₃₂	—	67	1	no	—
168	C₄×Q₃₂	—	67	2	no	—
169	C₂×C₄×C₁₆	—	67	1	no	—
170	C₆₇	#33734	1	1	yes	characteristic, complemented, Sylow
171	#533	—	134	2	no	—
172	C₂×Q₃₂	—	67	2	no	—
173	C₂×Q₃₂	—	67	1	no	—
174	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
175	C₂²×C₁₆	—	67	1	no	—
176	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
177	#500	—	67	2	no	—
178	#494	—	67	2	no	—
179	#487	—	134	2	no	—
180	#467	—	67	2	no	—
181	#467	—	134	2	no	—
182	#467	C₂×D₆₇	1	2	yes	—
183	C₄×Q₁₆	—	67	2	no	—
184	C₄×Q₁₆	—	67	2	no	—
185	C₄×Q₁₆	—	134	2	no	—
186	C₂×C₄×C₈	—	67	1	no	—
187	#366	—	67	1	no	—
188	#366	—	67	2	no	—
189	#366	C₂×D₆₇	1	1	yes	characteristic
190	#358	—	67	2	no	—
191	#358	—	67	2	no	—
192	#358	—	67	2	no	—
193	#358	C₂×D₆₇	1	2	yes	—
194	C₄×C₁₆	—	67	1	no	—
195	C₄×C₁₆	—	67	1	no	—
196	C₄×C₁₆	—	67	1	no	—
197	C₄×C₁₆	—	67	1	no	—
198	C₂×Q₁₆	—	67	2	no	—
199	C₂×Q₁₆	—	134	2	no	—
200	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
201	C₂²×C₈	—	67	1	no	—
202	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	134	2	no	—
204	C₄⋊Q₈	—	134	2	no	—
205	#125	—	268	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	268	2	no	—
208	C₂²⋊Q₈	—	134	2	no	—
209	C₄×Q₈	—	268	2	no	—
210	C₄×Q₈	—	134	2	no	—
211	C₄×Q₈	—	134	2	no	—
212	#117	—	268	2	no	—
213	C₂×C₄²	—	67	1	no	—
214	Q₃₂	—	2	2	no	—
215	Q₃₂	—	134	4	no	—
216	Q₃₂	—	134	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	67	1	no	—
219	C₂×C₁₆	—	67	1	no	—
220	C₂×C₁₆	—	134	2	no	—
221	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
223	#107	—	67	2	no	—
224	#107	—	134	2	no	—
225	#107	C₂²×D₆₇	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	134	2	no	—
228	#103	—	134	2	no	—
229	#103	—	134	2	no	—
230	C₄×C₈	—	67	1	no	—
231	C₄×C₈	—	67	1	no	—
232	C₄×C₈	—	67	1	no	—
233	C₄×C₈	—	67	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	268	2	no	—
236	C₂×Q₈	—	134	2	no	—
237	C₂²×C₄	—	67	1	no	—
238	C₂²×C₄	—	67	1	no	—
239	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	268	4	no	—
243	Q₁₆	—	134	2	no	—
244	Q₁₆	—	134	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	67	1	no	—
248	C₂×C₈	—	67	1	no	—
249	C₂×C₈	—	134	2	no	—
250	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	268	2	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	268	2	no	—
258	C₄⋊C₄	—	268	2	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₄⋊C₄	—	134	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	268	2	no	—
263	C₄²	—	268	2	no	—
264	C₄²	—	268	2	no	—
265	C₄²	—	67	1	no	—
266	C₄²	—	67	1	no	—
267	C₄²	—	67	1	no	—
268	C₄²	—	67	1	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	67	2	no	—
271	C₁₆	—	134	1	no	—
272	C₁₆	#11761627	1	2	yes	—
273	C₂³	D₈×D₆₇	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	268	2	no	—
277	Q₈	—	268	2	no	—
278	Q₈	—	536	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	268	2	no	—
285	C₂×C₄	—	67	4	no	—
286	C₂×C₄	—	67	1	no	—
287	C₂×C₄	—	67	1	no	—
288	C₂×C₄	—	67	1	no	—
289	C₂×C₄	—	67	1	no	—
290	C₂×C₄	—	134	2	no	—
291	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	67	2	no	—
295	C₈	—	134	1	no	—
296	C₈	#41×D₆₇	1	1	yes	characteristic
297	C₈	C₂×#11761627	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	Q₃₂×D₆₇	1	2	yes	—
301	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	536	2	no	—
306	C₄	—	536	2	no	—
307	C₄	—	67	2	no	—
308	C₄	—	67	2	no	—
309	C₄	—	67	2	no	—
310	C₄	—	134	1	no	—
311	C₄	#16863941	1	1	yes	characteristic
312	C₄	#135×D₆₇	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#22570783	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic
316	C₂	#508×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2216	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#2216	D₆₇
hypocenter	C₆₇	#33734
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #22562483	C₂	semidirect
3. #22562460	C₂	semidirect
4. #22562435	C₂	semidirect
7. Q₃₂×Dic₆₇	C₂	semidirect
8. #22562067	C₂	semidirect
9. #22562065	C₂	semidirect
126. C₂×Dic₆₇	Q₃₂	semidirect
170. C₆₇	#33734	semidirect

Maximal subgroups

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

subgroup	index
#22570790	2
#22562483	2
#22562460	2
#22562435	2
#22562300	2
#22562298	2
Q₃₂×Dic₆₇	2
#22562067	2
#22562065	2
C₂×C₁₆×Dic₆₇	2
C₆₇×#2216	2
#33734	67