Subgroups — #31013495

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573199	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573078	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562442	C₂	1	2	yes	maximal, complemented
4	#22562375	C₂	1	2	yes	maximal, complemented
5	#22562353	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562267	C₂	1	2	yes	maximal
7	#22562251	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562058	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864120	—	2	1	no	—
13	#16864107	—	2	1	no	—
14	C₂×#14523986	C₂²	1	2	yes	—
15	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
16	#16863797	C₂²	1	2	yes	—
17	#16863781	C₂²	1	2	yes	—
18	#16863699	—	2	2	no	—
19	#16863683	—	2	1	no	—
20	#16863620	—	2	2	no	—
21	#16863598	—	2	1	no	—
22	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
24	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
25	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
26	#16863478	C₂²	1	2	yes	—
27	#16863223	C₂²	1	2	yes	—
28	#16863222	C₂²	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
30	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
31	#16863200	C₂²	1	2	yes	—
32	#16863198	C₂²	1	1	yes	characteristic
33	#16863198	C₂²	1	1	yes	characteristic
34	#16863196	C₂²	1	2	yes	—
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	#494×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	#358×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761615	—	2	2	no	—
42	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
43	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
44	#14524038	—	4	2	no	—
45	#14524034	—	2	2	no	—
46	#14524032	—	2	2	no	—
47	#14524023	—	4	1	no	—
48	#14524023	—	4	1	no	—
49	#14524019	—	4	2	no	—
50	Q₈×Dic₆₇	—	2	2	no	—
51	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
52	C₄×D₂₆₈	—	2	1	no	—
53	C₄×D₂₆₈	—	2	1	no	—
54	#14524005	—	2	2	no	—
55	#14524001	—	4	1	no	—
56	C₄⋊C₄×D₆₇	—	4	1	no	—
57	#14523986	—	2	4	no	—
58	D₁₀₇₂	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
60	#14523977	C₂³	1	1	yes	characteristic
61	#14523977	C₂³	1	1	yes	characteristic
62	#14523974	—	2	2	no	—
63	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
64	Dic₆₇⋊C₈	D₄	1	2	yes	—
65	#14523967	—	2	2	no	—
66	#14523965	—	2	1	no	—
67	#14523965	—	2	1	no	—
68	#14523963	—	2	2	no	—
69	C₈×Dic₆₇	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₇	—	2	1	no	—
71	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₆₇	—	2	1	no	—
73	C₄⋊Q₈×C₆₇	—	2	1	no	—
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	2	1	no	—
77	#107×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
80	C₂×D₂₆₈	—	2	1	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	C₂×C₄×D₆₇	—	4	1	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	#11761615	—	8	2	no	—
86	D₅₃₆	—	2	1	no	—
87	D₅₃₆	—	2	1	no	—
88	D₅₃₆	—	2	1	no	—
89	D₅₃₆	#41	1	2	yes	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₆₇	—	2	1	no	—
92	C₄⋊Dic₆₇	—	2	1	no	—
93	C₄⋊Dic₆₇	—	4	2	no	—
94	C₄⋊Dic₆₇	—	4	2	no	—
95	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	D₈	1	2	yes	—
98	(C₂×D₆₇)⋊C₄	—	4	1	no	—
99	(C₂×D₆₇)⋊C₄	—	4	1	no	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	2	no	—
102	(C₂×D₆₇)⋊C₄	—	4	2	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
105	C₆₇⋊C₁₆	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	4	1	no	—
108	Q₁₆×C₆₇	—	2	2	no	—
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	—	8	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	2	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₆₇	—	2	1	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₂²×D₆₇	—	4	1	no	—
121	C₂²×D₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	4	2	no	—
124	Q₈×C₆₇	—	16	1	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	—	4	1	no	—
132	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
133	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
134	D₂₆₈	—	2	2	no	—
135	D₂₆₈	—	4	1	no	—
136	D₂₆₈	—	4	1	no	—
137	D₂₆₈	—	4	1	no	—
138	C₄×D₆₇	—	8	2	no	—
139	C₄×D₆₇	—	8	2	no	—
140	C₄×D₆₇	—	16	2	no	—
141	Dic₁₃₄	—	2	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₆₇⋊C₈	—	2	2	no	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₆₇	—	4	2	no	—
148	C₂×D₆₇	—	8	1	no	—
149	C₂×D₆₇	—	8	1	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₂×D₆₇	—	8	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	#508	1	2	yes	—
157	C₄×C₆₇	#548	1	1	yes	characteristic
158	C₄×C₆₇	#577	1	1	yes	characteristic
159	Dic₆₇	—	2	2	no	—
160	Dic₆₇	—	2	2	no	—
161	Dic₆₇	—	8	1	no	—
162	Dic₆₇	—	8	1	no	—
163	#33927	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	#3358	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
166	C₂×C₆₇	#3416	1	1	yes	characteristic
167	D₆₇	—	8	2	no	—
168	D₆₇	—	16	1	no	—
169	#3393	—	67	1	no	—
170	#3380	—	67	1	no	—
171	#2242	—	67	2	no	—
172	#2241	D₆₇	1	1	yes	characteristic, complemented
173	#2229	—	67	2	no	—
174	#2226	—	67	1	no	—
175	#2210	—	67	2	no	—
176	#2208	—	67	1	no	—
177	Q₈×C₁₆	—	67	1	no	—
178	C₄×SD₃₂	—	67	2	no	—
179	C₄×D₁₆	—	67	1	no	—
180	C₆₇	#33927	1	1	yes	characteristic, complemented, Sylow
181	#556	—	134	1	no	—
182	#554	—	134	1	no	—
183	C₂×SD₃₂	—	67	2	no	—
184	C₂×D₁₆	—	67	1	no	—
185	#500	C₂×D₆₇	1	1	yes	characteristic
186	#499	—	67	2	no	—
187	#495	—	67	2	no	—
188	#494	C₂×D₆₇	1	1	yes	characteristic
189	#478	—	134	2	no	—
190	#476	—	134	1	no	—
191	#464	—	134	2	no	—
192	#461	—	134	1	no	—
193	C₈×Q₈	—	67	1	no	—
194	C₄×Q₁₆	—	67	2	no	—
195	C₄×D₈	—	67	2	no	—
196	C₄×D₈	—	67	1	no	—
197	C₄×D₈	—	67	1	no	—
198	#367	—	67	2	no	—
199	#366	—	67	1	no	—
200	C₄⋊C₁₆	—	67	2	no	—
201	C₄⋊C₁₆	—	67	1	no	—
202	#358	—	67	2	no	—
203	#358	C₂×D₆₇	1	2	yes	—
204	#358	C₂×D₆₇	1	2	yes	—
205	#357	—	67	1	no	—
206	#357	—	67	1	no	—
207	#357	—	67	2	no	—
208	C₄×C₁₆	—	67	2	no	—
209	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
212	C₂×SD₁₆	—	134	2	no	—
213	C₂×D₈	—	67	1	no	—
214	C₂×D₈	—	67	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#126	—	268	2	no	—
218	#125	—	134	2	no	—
219	#124	—	134	2	no	—
220	C₂²⋊Q₈	—	268	1	no	—
221	C₂²⋊Q₈	—	268	1	no	—
222	C₂²⋊Q₈	—	268	2	no	—
223	C₄×Q₈	—	67	1	no	—
224	C₄×Q₈	—	134	2	no	—
225	C₄×D₄	—	134	2	no	—
226	C₄×D₄	—	134	1	no	—
227	C₄×D₄	—	134	1	no	—
228	#117	—	268	1	no	—
229	C₂×C₄⋊C₄	—	268	1	no	—
230	SD₃₂	—	134	4	no	—
231	D₁₆	—	134	2	no	—
232	C₂×C₁₆	—	67	2	no	—
233	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	#107	—	2	1	no	—
236	#107	—	67	1	no	—
237	#107	—	67	1	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#106	—	134	2	no	—
240	C₄⋊C₈	—	67	2	no	—
241	C₄⋊C₈	—	67	1	no	—
242	#103	—	134	2	no	—
243	#102	—	134	2	no	—
244	#102	—	134	1	no	—
245	#102	—	134	1	no	—
246	C₄×C₈	—	67	2	no	—
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	67	1	no	—
251	C₂×D₄	—	134	1	no	—
252	C₂×D₄	—	134	1	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	268	1	no	—
255	C₂²×C₄	—	268	1	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	536	2	no	—
260	D₈	—	67	2	no	—
261	D₈	—	134	1	no	—
262	D₈	—	134	1	no	—
263	D₈	—	134	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	67	1	no	—
275	C₄⋊C₄	—	67	2	no	—
276	C₄⋊C₄	—	134	1	no	—
277	C₄⋊C₄	—	134	1	no	—
278	C₂²⋊C₄	—	268	2	no	—
279	C₂²⋊C₄	—	268	2	no	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₂²⋊C₄	—	268	1	no	—
282	C₂²⋊C₄	—	268	1	no	—
283	C₄²	—	268	2	no	—
284	C₄²	—	67	2	no	—
285	C₄²	D₄×D₆₇	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	134	2	no	—
289	C₂³	—	268	1	no	—
290	C₂³	—	268	1	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	16	1	no	—
294	Q₈	—	134	2	no	—
295	D₄	—	268	1	no	—
296	D₄	—	268	1	no	—
297	D₄	—	268	1	no	—
298	D₄	—	134	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	268	1	no	—
302	C₂×C₄	—	268	1	no	—
303	C₂×C₄	—	536	2	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	—	1072	2	no	—
306	C₂×C₄	—	67	2	no	—
307	C₂×C₄	—	67	2	no	—
308	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	134	2	no	—
313	C₈	#41×D₆₇	1	1	yes	characteristic
314	C₈	#14524090	1	1	yes	characteristic
315	C₂²	—	268	2	no	—
316	C₂²	—	536	1	no	—
317	C₂²	—	536	1	no	—
318	C₂²	—	536	1	no	—
319	C₂²	—	536	1	no	—
320	C₂²	—	536	1	no	—
321	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	536	1	no	—
326	C₄	—	536	1	no	—
327	C₄	—	134	2	no	—
328	C₄	—	134	2	no	—
329	C₄	#16863829	1	2	yes	—
330	C₄	#16864054	1	1	yes	characteristic
331	C₄	#16864202	1	1	yes	characteristic
332	C₂	—	536	2	no	—
333	C₂	—	1072	1	no	—
334	C₂	#22572841	1	1	yes	characteristic
335	C₂	C₂×#16863829	1	1	yes	characteristic
336	C₂	#22573351	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×#2241	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×D₁₀₇₂	C₂²
hypercenter	#2241	D₆₇
hypocenter	C₆₇	#33927
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #22573199	C₂	semidirect
2. #22573078	C₂	semidirect
3. #22562442	C₂	semidirect
4. #22562375	C₂	semidirect
5. #22562353	C₂	semidirect
7. #22562251	C₂	semidirect
8. C₁₆×Dic₁₃₄	C₂	semidirect
9. #22562058	C₂	semidirect
11. C₆₇×#2241	C₂	semidirect
172. #2241	D₆₇	semidirect
180. C₆₇	#33927	semidirect

Maximal subgroups

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

subgroup	index
#22573199	2
#22573078	2
#22562442	2
#22562375	2
#22562353	2
#22562267	2
#22562251	2
C₁₆×Dic₁₃₄	2
#22562058	2
C₄×D₁₀₇₂	2
C₆₇×#2241	2
#33927	67