Subgroups — #31010342

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523988	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569864	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569857	C₂	1	1	yes	maximal, characteristic
4	#22562322	C₂	1	4	yes	maximal, complemented
5	#22562305	C₂	1	4	yes	maximal, complemented
6	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561930	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561926	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
11	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
12	#16863967	—	2	1	no	—
13	#16863960	—	2	1	no	—
14	C₂×#14523988	—	2	4	no	—
15	C₂×#14523988	C₂²	1	2	yes	—
16	C₂×#14523988	C₂²	1	2	yes	—
17	#16863654	—	2	2	no	—
18	#16863649	—	2	2	no	—
19	#16863637	C₂²	1	4	yes	complemented
20	#16863632	C₂²	1	4	yes	—
21	#16863434	C₂²	1	1	yes	characteristic
22	#16863434	C₂²	1	1	yes	characteristic
23	#16863406	—	2	1	no	—
24	#16863402	—	2	1	no	—
25	#16863224	C₂²	1	2	yes	complemented
26	#16863222	C₂²	1	2	yes	complemented
27	#16863201	C₂²	1	4	yes	complemented
28	#16863198	C₂²	1	4	yes	—
29	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
30	#504×C₆₇	—	2	4	no	—
31	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₆₇	C₂²	1	2	yes	complemented
33	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	complemented
35	C₂²×Dic₁₃₄	—	2	1	no	—
36	C₂²×D₂₆₈	—	2	1	no	—
37	C₂×Dic₂₆₈	—	2	2	no	—
38	C₂×Dic₂₆₈	—	2	1	no	—
39	C₂×Dic₂₆₈	—	2	2	no	—
40	C₂×Dic₂₆₈	C₂³	1	2	yes	—
41	C₂×D₅₃₆	—	2	2	no	—
42	C₂×D₅₃₆	—	2	1	no	—
43	C₂×D₅₃₆	—	2	2	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	#14524041	—	4	1	no	—
46	#14524039	—	4	1	no	—
47	#14524032	—	4	1	no	—
48	#14524030	—	4	1	no	—
49	#14524024	—	4	2	no	—
50	#14524022	—	2	4	no	—
51	#14524017	—	2	4	no	—
52	#14524011	—	4	2	no	—
53	#14524002	—	2	1	no	—
54	#14524002	—	2	1	no	—
55	#14523988	—	2	4	no	—
56	#14523988	—	2	4	no	—
57	#14523977	C₂³	1	2	yes	—
58	#14523977	C₂³	1	2	yes	—
59	#14523975	—	2	1	no	—
60	#14523975	—	2	1	no	—
61	#14523968	—	2	4	no	—
62	#14523965	—	2	4	no	—
63	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
64	#131×C₆₇	—	2	2	no	—
65	#131×C₆₇	—	2	2	no	—
66	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
67	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	M₃₂×C₆₇	—	2	2	no	—
71	M₃₂×C₆₇	—	2	4	no	—
72	C₂×C₁₆×C₆₇	—	2	4	no	—
73	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
74	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×D₆₇	—	4	1	no	—
78	C₂×Dic₁₃₄	—	2	2	no	—
79	C₂×Dic₁₃₄	—	4	2	no	—
80	C₂×Dic₁₃₄	—	4	2	no	—
81	C₂×Dic₁₃₄	—	4	1	no	—
82	C₂×D₂₆₈	—	2	2	no	—
83	C₂×D₂₆₈	—	4	2	no	—
84	C₂×D₂₆₈	—	4	2	no	—
85	C₂×D₂₆₈	—	4	1	no	—
86	C₂×C₂²⋊D₆₇	—	4	4	no	—
87	C₂²×Dic₆₇	—	4	1	no	—
88	Dic₂₆₈	—	2	2	no	—
89	Dic₂₆₈	—	2	2	no	—
90	Dic₂₆₈	—	4	2	no	—
91	D₅₃₆	—	2	2	no	—
92	D₅₃₆	—	2	2	no	—
93	D₅₃₆	—	4	2	no	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	Dic₆₇⋊C₄	—	4	4	no	—
97	C₂²⋊Dic₆₇	—	4	1	no	—
98	C₂²⋊Dic₆₇	—	4	1	no	—
99	C₂²⋊Dic₆₇	—	4	4	no	—
100	(C₂×D₆₇)⋊C₄	—	4	4	no	—
101	C₄×Dic₆₇	—	4	1	no	—
102	C₄×Dic₆₇	—	4	1	no	—
103	#41×C₆₇	—	2	2	no	—
104	#41×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
106	C₂×D₄×C₆₇	D₈	1	2	yes	—
107	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₆₇	—	2	2	no	—
111	M₁₆×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	#41	1	2	yes	—
118	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₁₆×C₆₇	—	2	2	no	—
120	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
121	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
122	C₂³×C₆₇	C₂×D₈	1	2	yes	—
123	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₆₇	—	4	2	no	—
125	C₂²×D₆₇	—	8	2	no	—
126	C₂²×D₆₇	—	16	2	no	—
127	Q₈×C₆₇	—	2	2	no	—
128	D₄×C₆₇	—	2	4	no	—
129	D₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	#121	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#135	1	2	yes	—
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₆₇	—	16	4	no	—
138	C₂×Dic₆₇	—	4	2	no	—
139	C₂×Dic₆₇	—	4	2	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	—	8	2	no	—
142	D₂₆₈	—	4	2	no	—
143	D₂₆₈	—	4	2	no	—
144	D₂₆₈	—	8	2	no	—
145	Dic₁₃₄	—	4	2	no	—
146	Dic₁₃₄	—	4	2	no	—
147	Dic₁₃₄	—	8	2	no	—
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	#121	1	2	yes	—
150	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	#466	1	2	yes	—
156	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₆₇	—	8	4	no	—
158	C₂×D₆₇	—	8	2	no	—
159	C₂×D₆₇	—	16	4	no	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	#466	1	2	yes	—
162	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
164	Dic₆₇	—	8	2	no	—
165	Dic₆₇	—	8	2	no	—
166	Dic₆₇	—	8	2	no	—
167	#33609	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	—	2	2	no	—
169	C₂×C₆₇	—	4	2	no	—
170	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
171	C₂×C₆₇	#3416	1	2	yes	—
172	D₆₇	—	16	2	no	—
173	C₂²×SD₃₂	—	67	1	no	—
174	C₂×#504	D₆₇	1	1	yes	characteristic, complemented
175	#3059	—	67	1	no	—
176	#3058	—	67	1	no	—
177	#2220	—	67	4	no	—
178	#2217	—	67	4	no	—
179	#2158	—	67	1	no	—
180	#2141	—	67	1	no	—
181	#2140	—	67	1	no	—
182	C₆₇	#33609	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₁₆	—	67	1	no	—
184	C₂²×D₈	—	67	1	no	—
185	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
186	#536	—	134	1	no	—
187	#535	—	134	1	no	—
188	C₂×SD₃₂	—	67	2	no	—
189	C₂×SD₃₂	—	67	2	no	—
190	C₂×SD₃₂	—	134	4	no	—
191	#504	—	2	4	no	—
192	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₆₇	1	2	yes	complemented
194	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
195	C₂²×C₁₆	C₂×D₆₇	1	2	yes	complemented
196	#470	—	134	2	no	—
197	#469	—	134	2	no	—
198	#467	—	67	4	no	—
199	#466	—	67	4	no	—
200	#427	—	67	1	no	—
201	#427	—	67	1	no	—
202	#419	—	134	1	no	—
203	#418	—	134	1	no	—
204	#367	—	67	2	no	—
205	#366	—	67	2	no	—
206	#358	—	67	4	no	—
207	#357	—	67	4	no	—
208	C₂×#41	D₂₆₈	1	1	yes	characteristic
209	C₂²×Q₈	—	134	1	no	—
210	C₂²×D₄	—	134	1	no	—
211	C₂×Q₁₆	—	67	2	no	—
212	C₂×Q₁₆	—	134	2	no	—
213	C₂×Q₁₆	—	134	1	no	—
214	C₂×Q₁₆	—	134	2	no	—
215	C₂×D₈	—	67	2	no	—
216	C₂×D₈	—	134	2	no	—
217	C₂×D₈	—	134	1	no	—
218	C₂×D₈	—	134	2	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
222	C₂×M₁₆	D₂₆₈	1	2	yes	—
223	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
224	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	268	1	no	—
226	#127	—	268	1	no	—
227	#124	—	268	1	no	—
228	#124	—	268	1	no	—
229	#123	—	268	2	no	—
230	C₂²⋊Q₈	—	134	4	no	—
231	#121	—	134	4	no	—
232	#120	—	268	2	no	—
233	#117	—	134	1	no	—
234	#117	—	134	1	no	—
235	SD₃₂	—	134	4	no	—
236	SD₃₂	—	134	4	no	—
237	M₃₂	—	2	2	no	—
238	M₃₂	—	2	4	no	—
239	C₂×C₁₆	—	2	4	no	—
240	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	complemented
241	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
242	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
244	#107	—	67	2	no	—
245	#107	—	67	2	no	—
246	#106	—	134	1	no	—
247	#106	—	134	1	no	—
248	#103	—	134	4	no	—
249	#102	—	134	4	no	—
250	C₂⁴	—	268	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	268	1	no	—
254	C₂×Q₈	—	268	2	no	—
255	C₂×Q₈	—	268	2	no	—
256	C₂×Q₈	—	134	2	no	—
257	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
258	C₂×D₄	—	268	4	no	—
259	C₂×D₄	—	268	2	no	—
260	C₂×D₄	—	268	1	no	—
261	C₂×D₄	—	268	2	no	—
262	C₂×D₄	—	134	2	no	—
263	C₂×D₄	D₅₃₆	1	2	yes	—
264	C₂×D₄	D₅₃₆	1	1	yes	characteristic
265	C₂²×C₄	—	268	1	no	—
266	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
267	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
268	Q₁₆	—	268	2	no	—
269	Q₁₆	—	134	2	no	—
270	Q₁₆	—	134	2	no	—
271	D₈	—	268	2	no	—
272	D₈	—	134	2	no	—
273	D₈	—	134	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
279	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
280	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
281	C₂×C₈	#11761628	1	2	yes	—
282	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
283	C₄⋊C₄	—	268	4	no	—
284	C₄⋊C₄	—	134	2	no	—
285	C₄⋊C₄	—	134	2	no	—
286	C₂²⋊C₄	—	268	1	no	—
287	C₂²⋊C₄	—	268	1	no	—
288	C₂²⋊C₄	—	268	4	no	—
289	C₂²⋊C₄	—	268	4	no	—
290	C₄²	—	268	1	no	—
291	C₄²	—	268	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
294	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
295	C₂³	—	268	2	no	—
296	C₂³	—	536	2	no	—
297	C₂³	—	1072	2	no	—
298	C₂³	C₂×D₅₃₆	1	2	yes	—
299	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	268	2	no	—
302	Q₈	—	268	2	no	—
303	Q₈	—	536	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	268	2	no	—
307	D₄	—	268	2	no	—
308	D₄	—	536	2	no	—
309	D₄	—	1072	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	268	2	no	—
313	C₂×C₄	—	268	2	no	—
314	C₂×C₄	—	268	2	no	—
315	C₂×C₄	—	536	2	no	—
316	C₂×C₄	#14524017	1	2	yes	—
317	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
318	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
319	C₂×C₄	#14524065	1	2	yes	—
320	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#14524017	1	2	yes	—
323	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
324	C₈	C₂×#11761628	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	536	4	no	—
329	C₂²	—	536	2	no	—
330	C₂²	—	1072	4	no	—
331	C₂²	#16863632	1	2	yes	—
332	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	536	2	no	—
335	C₄	—	536	2	no	—
336	C₄	—	536	2	no	—
337	C₄	#16863632	1	2	yes	—
338	C₄	C₂×#14524017	1	1	yes	characteristic
339	C₄	C₂×#14524065	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	1072	2	no	—
343	C₂	C₂×#16863632	1	1	yes	characteristic
344	C₂	#22573357	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22569857	C₂
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#33609
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523988	C₂	semidirect
2. #22569864	C₂	semidirect
4. #22562322	C₂	semidirect
5. #22562305	C₂	semidirect
6. #22561993	C₂	semidirect
7. #22561930	C₂	semidirect
8. #22561926	C₂	semidirect
9. C₂×#504×C₆₇	C₂	semidirect
19. #16863637	C₂²	semidirect
25. #16863224	C₂²	semidirect
26. #16863222	C₂²	semidirect
27. #16863201	C₂²	semidirect
32. C₂×M₃₂×C₆₇	C₂²	semidirect
34. C₂²×C₁₆×C₆₇	C₂²	semidirect
73. C₂×C₁₆×C₆₇	D₄	semidirect
174. C₂×#504	D₆₇	semidirect
182. C₆₇	#33609	semidirect
193. C₂×M₃₂	C₂×D₆₇	semidirect
195. C₂²×C₁₆	C₂×D₆₇	semidirect
240. C₂×C₁₆	C₂²⋊D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523988	2
#22569864	2
#22569857	2
#22562322	2
#22562305	2
#22561993	2
#22561930	2
#22561926	2
C₂×#504×C₆₇	2
#33609	67