Subgroups — #31013510

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573183	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562433	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562431	C₂	1	2	yes	maximal, complemented
5	#22562356	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562355	C₂	1	2	yes	maximal, complemented
7	#22562249	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562247	C₂	1	2	yes	maximal, complemented
9	C₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#22562058	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₃₄	—	2	1	no	—
13	#16864067	—	2	1	no	—
14	C₂×#14523986	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863773	C₂²	1	1	yes	characteristic
18	#16863771	C₂²	1	2	yes	—
19	#16863691	—	2	1	no	—
20	#16863689	—	2	2	no	—
21	#16863591	—	2	1	no	—
22	#16863590	—	2	2	no	—
23	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
25	#16863478	C₂²	1	2	yes	—
26	#16863223	C₂²	1	2	yes	—
27	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	#16863198	C₂²	1	2	yes	—
30	#16863198	C₂²	1	2	yes	—
31	#16863197	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	—
33	C₁₆×Dic₆₇	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
37	#367×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	1	yes	characteristic
39	#357×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761628	—	4	1	no	—
42	C₂×D₅₃₆	—	2	1	no	—
43	C₂×D₅₃₆	C₂³	1	2	yes	—
44	C₂×#11761613	—	2	2	no	—
45	#14524043	—	2	1	no	—
46	#14524043	—	4	1	no	—
47	#14524042	—	2	2	no	—
48	#14524042	—	4	2	no	—
49	#14524040	—	2	1	no	—
50	#14524039	—	2	2	no	—
51	#14524017	—	4	1	no	—
52	#14524016	—	4	2	no	—
53	Q₈×Dic₆₇	—	2	2	no	—
54	C₄×Dic₁₃₄	—	4	1	no	—
55	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
56	#14524005	—	2	2	no	—
57	#14523986	—	2	4	no	—
58	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
59	#14523977	—	2	1	no	—
60	#14523977	C₂³	1	2	yes	—
61	#14523976	—	2	2	no	—
62	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
63	Dic₆₇⋊C₈	D₄	1	2	yes	—
64	#14523967	—	2	2	no	—
65	#14523963	—	2	2	no	—
66	C₈×Dic₆₇	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	2	1	no	—
71	SD₃₂×C₆₇	—	2	2	no	—
72	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	1	yes	characteristic
76	#103×C₆₇	—	2	1	no	—
77	#102×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#11761628	—	8	4	no	—
80	C₂×Dic₁₃₄	—	4	1	no	—
81	C₂×Dic₁₃₄	—	4	2	no	—
82	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
83	C₂×D₂₆₈	—	2	2	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	1	no	—
86	C₂×C₂²⋊D₆₇	—	4	2	no	—
87	C₂×C₂²⋊D₆₇	—	4	2	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	D₅₃₆	—	2	2	no	—
90	D₅₃₆	—	2	2	no	—
91	D₅₃₆	—	8	1	no	—
92	#11761613	—	8	2	no	—
93	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	—	4	2	no	—
97	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
98	Dic₆₇⋊C₄	—	4	2	no	—
99	Dic₆₇⋊C₄	D₈	1	2	yes	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
103	C₆₇⋊C₁₆	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	Q₁₆×C₆₇	—	2	1	no	—
108	Q₁₆×C₆₇	#41	1	2	yes	complemented
109	D₈×C₆₇	—	2	1	no	—
110	D₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₂²⋊C₄×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₆₇	—	2	1	no	—
120	C₁₆×C₆₇	—	2	1	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	C₂²×D₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	2	2	no	—
124	Q₈×C₆₇	—	4	1	no	—
125	D₄×C₆₇	—	4	1	no	—
126	D₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	8	2	no	—
131	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₆₇	—	8	4	no	—
135	C₂²⋊D₆₇	—	16	2	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
138	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
139	D₂₆₈	—	4	2	no	—
140	D₂₆₈	—	4	2	no	—
141	D₂₆₈	—	8	2	no	—
142	D₂₆₈	—	16	1	no	—
143	C₄×D₆₇	—	8	4	no	—
144	Dic₁₃₄	—	16	2	no	—
145	Dic₁₃₄	—	16	1	no	—
146	Dic₁₃₄	SD₃₂	1	4	yes	complemented
147	C₈×C₆₇	—	2	1	no	—
148	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₆₇⋊C₈	—	2	2	no	—
151	C₂²×C₆₇	—	8	1	no	—
152	C₂²×C₆₇	—	8	1	no	—
153	C₂²×C₆₇	—	8	1	no	—
154	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
163	C₄×C₆₇	#548	1	1	yes	characteristic
164	C₄×C₆₇	#577	1	1	yes	characteristic
165	Dic₆₇	—	2	2	no	—
166	Dic₆₇	—	8	2	no	—
167	Dic₆₇	C₂×SD₃₂	1	4	yes	—
168	#33929	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	16	1	no	—
170	C₂×C₆₇	#3358	1	1	yes	characteristic
171	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₆₇	#3415	1	1	yes	characteristic
173	D₆₇	—	16	2	no	—
174	#3391	—	67	1	no	—
175	Q₈×Q₁₆	—	67	1	no	—
176	#2240	—	67	2	no	—
177	#2240	—	67	1	no	—
178	#2226	—	67	1	no	—
179	#2226	—	67	2	no	—
180	#2207	—	67	2	no	—
181	#2207	—	67	1	no	—
182	Q₈×C₁₆	—	67	1	no	—
183	C₄×SD₃₂	—	67	2	no	—
184	C₄×SD₃₂	D₆₇	1	1	yes	characteristic, complemented
185	C₆₇	#33929	1	1	yes	characteristic, complemented, Sylow
186	Q₈²	—	134	1	no	—
187	#550	—	134	1	no	—
188	C₂×SD₃₂	—	67	2	no	—
189	C₂×SD₃₂	C₂×D₆₇	1	1	yes	characteristic
190	#500	—	67	2	no	—
191	#500	—	67	1	no	—
192	#493	—	67	2	no	—
193	#493	—	67	1	no	—
194	#477	—	134	2	no	—
195	#477	—	134	1	no	—
196	#459	—	134	2	no	—
197	#459	—	134	1	no	—
198	C₈×Q₈	—	67	1	no	—
199	C₄×Q₁₆	—	67	2	no	—
200	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
201	C₄×D₈	—	67	2	no	—
202	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
203	#367	—	67	2	no	—
204	#367	C₂×D₆₇	1	1	yes	characteristic
205	C₄⋊C₁₆	—	67	2	no	—
206	C₄⋊C₁₆	—	67	1	no	—
207	#358	C₂×D₆₇	1	1	yes	characteristic
208	#357	—	67	2	no	—
209	#357	—	67	2	no	—
210	#357	—	67	2	no	—
211	#357	—	67	2	no	—
212	#357	C₂×D₆₇	1	1	yes	characteristic
213	C₄×C₁₆	—	67	2	no	—
214	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
215	C₂×#41	—	268	1	no	—
216	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
217	C₂×D₈	—	67	2	no	—
218	C₂×D₈	—	134	2	no	—
219	C₂×D₈	—	134	1	no	—
220	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	268	1	no	—
222	C₄⋊Q₈	—	268	2	no	—
223	C₄⋊Q₈	—	134	2	no	—
224	C₄⋊Q₈	—	134	1	no	—
225	#127	—	134	2	no	—
226	#127	—	134	1	no	—
227	#121	—	268	2	no	—
228	#121	—	268	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	268	1	no	—
231	C₄×Q₈	—	67	1	no	—
232	C₄×Q₈	—	134	2	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	134	2	no	—
235	SD₃₂	—	2	2	no	—
236	SD₃₂	—	134	4	no	—
237	C₂×C₁₆	—	67	2	no	—
238	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
240	#107	—	67	2	no	—
241	#107	—	134	2	no	—
242	#107	—	134	1	no	—
243	#107	C₂²×D₆₇	1	1	yes	characteristic
244	#107	C₂²×D₆₇	1	1	yes	characteristic
245	C₄⋊C₈	—	67	2	no	—
246	C₄⋊C₈	—	67	1	no	—
247	#103	—	2	1	no	—
248	#103	—	134	2	no	—
249	#102	—	2	1	no	—
250	#102	—	134	2	no	—
251	C₄×C₈	—	67	2	no	—
252	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	#41	—	536	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	268	2	no	—
256	C₂×Q₈	—	268	1	no	—
257	C₂×Q₈	—	67	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	268	2	no	—
260	C₂×D₄	—	268	2	no	—
261	C₂×D₄	—	268	1	no	—
262	C₂×D₄	—	268	1	no	—
263	C₂×D₄	—	134	2	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	268	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	#11761628	1	2	yes	complemented
268	D₈	—	2	1	no	—
269	D₈	—	2	1	no	—
270	D₈	—	536	1	no	—
271	D₈	—	536	2	no	—
272	D₈	—	134	2	no	—
273	D₈	—	134	2	no	—
274	C₂×C₈	—	67	2	no	—
275	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	268	2	no	—
281	C₄⋊C₄	—	268	2	no	—
282	C₄⋊C₄	—	268	1	no	—
283	C₄⋊C₄	—	67	2	no	—
284	C₄⋊C₄	—	67	1	no	—
285	C₄⋊C₄	—	134	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	268	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	268	2	no	—
290	C₄²	—	67	2	no	—
291	C₄²	D₄×D₆₇	1	1	yes	characteristic
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	134	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	268	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	1072	1	no	—
300	Q₈	—	1072	2	no	—
301	Q₈	—	67	4	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	268	2	no	—
305	D₄	—	268	2	no	—
306	D₄	—	536	4	no	—
307	D₄	—	536	2	no	—
308	D₄	—	1072	1	no	—
309	D₄	—	1072	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	268	2	no	—
315	C₂×C₄	—	536	4	no	—
316	C₂×C₄	—	67	2	no	—
317	C₂×C₄	—	67	2	no	—
318	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
319	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	134	2	no	—
323	C₈	C₂×#11761628	1	1	yes	characteristic
324	C₈	#14524090	1	1	yes	characteristic
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	536	2	no	—
329	C₂²	—	536	2	no	—
330	C₂²	—	536	2	no	—
331	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	536	2	no	—
337	C₄	—	67	4	no	—
338	C₄	—	134	2	no	—
339	C₄	SD₃₂×D₆₇	1	2	yes	—
340	C₄	#16864055	1	1	yes	characteristic
341	C₄	#16864203	1	1	yes	characteristic
342	C₂	—	16	1	no	—
343	C₂	—	1072	2	no	—
344	C₂	#22572842	1	1	yes	characteristic
345	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
346	C₂	#22573347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×SD₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22573183	C₂
hypercenter	C₄×SD₃₂	D₆₇
hypocenter	C₆₇	#33929
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₃₄	C₂	semidirect
3. #22562433	C₂	semidirect
4. #22562431	C₂	semidirect
5. #22562356	C₂	semidirect
6. #22562355	C₂	semidirect
7. #22562249	C₂	semidirect
8. #22562247	C₂	semidirect
9. C₁₆×Dic₁₃₄	C₂	semidirect
10. #22562058	C₂	semidirect
11. C₄×SD₃₂×C₆₇	C₂	semidirect
108. Q₁₆×C₆₇	#41	semidirect
146. Dic₁₃₄	SD₃₂	semidirect
184. C₄×SD₃₂	D₆₇	semidirect
185. C₆₇	#33929	semidirect
267. Q₁₆	#11761628	semidirect

Maximal subgroups

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

subgroup	index
#22573183	2
Q₁₆×Dic₁₃₄	2
#22562433	2
#22562431	2
#22562356	2
#22562355	2
#22562249	2
#22562247	2
C₁₆×Dic₁₃₄	2
#22562058	2
C₄×SD₃₂×C₆₇	2
#33929	67