Subgroups — #31011377

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570774	C₂	1	2	yes	maximal, complemented
2	#22562486	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562467	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562439	C₂	1	2	yes	maximal, complemented
5	#22562303	C₂	1	2	yes	maximal, complemented
6	#22562300	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562064	C₂	1	2	yes	maximal, complemented
9	D₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2215	C₂	1	1	yes	maximal, characteristic
12	#16863927	—	2	2	no	—
13	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
14	C₂×#14523987	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863797	C₂²	1	2	yes	—
17	#16863779	C₂²	1	2	yes	—
18	#16863749	—	2	2	no	—
19	#16863637	C₂²	1	2	yes	—
20	#16863623	—	2	2	no	—
21	#16863490	C₂²	1	2	yes	—
22	#16863487	—	2	2	no	—
23	D₈×Dic₆₇	C₂²	1	2	yes	—
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863223	C₂²	1	2	yes	—
26	#16863222	C₂²	1	1	yes	characteristic
27	#16863201	C₂²	1	2	yes	—
28	#16863199	C₂²	1	2	yes	—
29	#16863197	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×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#466×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₂×Dic₂₆₈	C₂³	1	2	yes	—
40	C₂×#11761616	—	2	2	no	—
41	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
42	#14524036	—	4	2	no	—
43	#14524034	—	2	2	no	—
44	#14524030	—	2	2	no	—
45	#14524024	—	4	2	no	—
46	#14524022	—	2	2	no	—
47	#14524008	—	2	2	no	—
48	D₄×Dic₆₇	—	2	2	no	—
49	D₄×Dic₆₇	—	4	2	no	—
50	#14523999	—	4	2	no	—
51	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
52	Dic₅₃₆	—	2	2	no	—
53	#14523987	—	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	#14523974	—	2	2	no	—
59	#14523968	—	2	2	no	—
60	#14523966	—	2	2	no	—
61	#14523964	—	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×D₈×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#121×C₆₇	—	2	2	no	—
69	D₁₆×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	#102×C₆₇	—	2	2	no	—
75	C₂×Dic₁₃₄	—	2	2	no	—
76	C₂²×Dic₆₇	—	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	#11761616	—	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	C₂²⋊Dic₆₇	—	4	2	no	—
91	C₂²⋊Dic₆₇	—	4	2	no	—
92	C₂²⋊Dic₆₇	—	4	2	no	—
93	C₂²⋊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×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	D₈×C₆₇	—	2	2	no	—
105	D₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	2	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	—	4	2	no	—
115	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	8	4	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	4	2	no	—
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	8	4	no	—
129	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
133	Dic₁₃₄	—	4	2	no	—
134	Dic₁₃₄	—	4	2	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₆₇⋊C₈	—	2	1	no	—
139	C₆₇⋊C₈	#135	1	2	yes	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	8	2	no	—
144	C₂²×C₆₇	—	8	2	no	—
145	C₂²×C₆₇	#508	1	2	yes	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	8	2	no	—
149	C₄×C₆₇	#532	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
151	Dic₆₇	—	2	1	no	—
152	Dic₆₇	—	2	1	no	—
153	Dic₆₇	—	2	1	no	—
154	Dic₆₇	—	8	2	no	—
155	Dic₆₇	—	8	2	no	—
156	Dic₆₇	#508	1	2	yes	—
157	#33735	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	16	2	no	—
160	C₂×C₆₇	#3169	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
162	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
163	#3168	—	67	2	no	—
164	#2253	—	67	1	no	—
165	#2249	—	67	1	no	—
166	#2242	—	67	2	no	—
167	#2217	—	67	2	no	—
168	#2216	—	67	1	no	—
169	#2215	D₆₇	1	1	yes	characteristic
170	C₄×Q₃₂	—	67	1	no	—
171	C₄×SD₃₂	—	67	2	no	—
172	C₄×D₁₆	—	67	1	no	—
173	C₂×C₄×C₁₆	—	67	1	no	—
174	C₆₇	#33735	1	1	yes	characteristic, complemented, Sylow
175	#529	—	134	2	no	—
176	C₂×Q₃₂	—	67	1	no	—
177	C₂×SD₃₂	—	67	2	no	—
178	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
179	C₂²×C₁₆	—	67	1	no	—
180	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
181	#499	—	67	2	no	—
182	#495	—	67	2	no	—
183	#488	—	134	2	no	—
184	#467	—	67	2	no	—
185	#466	C₂×D₆₇	1	2	yes	—
186	#465	—	134	2	no	—
187	C₄×Q₁₆	—	67	2	no	—
188	C₄×SD₁₆	—	134	2	no	—
189	C₄×D₈	—	67	2	no	—
190	C₂×C₄×C₈	—	67	1	no	—
191	#367	—	67	2	no	—
192	#366	—	67	1	no	—
193	#366	C₂×D₆₇	1	1	yes	characteristic
194	#358	—	67	2	no	—
195	#358	—	67	2	no	—
196	#357	—	67	2	no	—
197	#357	C₂×D₆₇	1	2	yes	—
198	C₄×C₁₆	—	67	1	no	—
199	C₄×C₁₆	—	67	1	no	—
200	C₄×C₁₆	—	67	1	no	—
201	C₄×C₁₆	—	67	1	no	—
202	C₂×Q₁₆	—	67	2	no	—
203	C₂×SD₁₆	—	134	2	no	—
204	C₂×D₈	C₂²×D₆₇	1	2	yes	—
205	C₂²×C₈	—	67	1	no	—
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	#126	—	268	2	no	—
208	#125	—	134	2	no	—
209	#124	—	134	2	no	—
210	#123	—	268	2	no	—
211	C₂²⋊Q₈	—	134	2	no	—
212	#121	—	2	2	no	—
213	C₄×Q₈	—	134	2	no	—
214	C₄×D₄	—	268	2	no	—
215	C₄×D₄	—	134	2	no	—
216	#117	—	268	2	no	—
217	C₂×C₄²	—	67	1	no	—
218	Q₃₂	—	134	2	no	—
219	SD₃₂	—	134	4	no	—
220	D₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	67	1	no	—
223	C₂×C₁₆	—	67	1	no	—
224	C₂×C₁₆	—	134	2	no	—
225	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
227	#107	—	67	2	no	—
228	#107	C₂²×D₆₇	1	2	yes	—
229	#106	—	134	2	no	—
230	#103	—	134	2	no	—
231	#103	—	134	2	no	—
232	#102	—	2	2	no	—
233	#102	—	134	2	no	—
234	C₄×C₈	—	67	1	no	—
235	C₄×C₈	—	67	1	no	—
236	C₄×C₈	—	67	1	no	—
237	C₄×C₈	—	67	1	no	—
238	C₂×Q₈	—	134	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	268	2	no	—
242	C₂²×C₄	—	67	1	no	—
243	C₂²×C₄	—	67	1	no	—
244	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
245	Q₁₆	—	134	2	no	—
246	Q₁₆	—	134	2	no	—
247	SD₁₆	—	268	4	no	—
248	D₈	—	2	2	no	—
249	D₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	67	1	no	—
253	C₂×C₈	—	67	1	no	—
254	C₂×C₈	—	134	2	no	—
255	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	268	2	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₄⋊C₄	—	268	2	no	—
261	C₄⋊C₄	—	268	2	no	—
262	C₄⋊C₄	—	134	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	268	2	no	—
265	C₂²⋊C₄	—	268	2	no	—
266	C₂²⋊C₄	—	268	2	no	—
267	C₂²⋊C₄	—	268	2	no	—
268	C₄²	—	268	2	no	—
269	C₄²	—	67	1	no	—
270	C₄²	—	67	1	no	—
271	C₄²	—	67	1	no	—
272	C₄²	—	67	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	—	67	2	no	—
275	C₁₆	—	134	1	no	—
276	C₁₆	#11761627	1	2	yes	—
277	C₂³	—	4	2	no	—
278	C₂³	D₈×D₆₇	1	1	yes	characteristic
279	Q₈	—	268	2	no	—
280	Q₈	—	268	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	268	2	no	—
288	C₂×C₄	—	268	2	no	—
289	C₂×C₄	—	536	4	no	—
290	C₂×C₄	—	67	1	no	—
291	C₂×C₄	—	67	1	no	—
292	C₂×C₄	—	67	1	no	—
293	C₂×C₄	—	67	1	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	2	no	—
296	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	67	2	no	—
300	C₈	—	134	1	no	—
301	C₈	#41×D₆₇	1	1	yes	characteristic
302	C₈	C₂×#11761627	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	#16863825	1	2	yes	—
309	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	536	2	no	—
313	C₄	—	536	2	no	—
314	C₄	—	67	2	no	—
315	C₄	—	134	1	no	—
316	C₄	—	134	1	no	—
317	C₄	—	134	1	no	—
318	C₄	#16863941	1	1	yes	characteristic
319	C₄	#135×D₆₇	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	16	2	no	—
322	C₂	#22570783	1	1	yes	characteristic
323	C₂	#508×D₆₇	1	1	yes	characteristic
324	C₂	C₂×#16863825	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₆₇×#2215	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#2215	D₆₇
hypocenter	C₆₇	#33735
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #22570774	C₂	semidirect
2. #22562486	C₂	semidirect
3. #22562467	C₂	semidirect
4. #22562439	C₂	semidirect
5. #22562303	C₂	semidirect
6. #22562300	C₂	semidirect
7. #22562067	C₂	semidirect
8. #22562064	C₂	semidirect
9. D₁₆×Dic₆₇	C₂	semidirect
10. C₂×C₁₆×Dic₆₇	C₂	semidirect
174. C₆₇	#33735	semidirect

Maximal subgroups

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

subgroup	index
#22570774	2
#22562486	2
#22562467	2
#22562439	2
#22562303	2
#22562300	2
#22562067	2
#22562064	2
D₁₆×Dic₆₇	2
C₂×C₁₆×Dic₆₇	2
C₆₇×#2215	2
#33735	67