Subgroups — #31009216

Order 17152.

674 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	C₂×#16863829	C₂	1	1	yes	maximal, characteristic
2	#22569084	C₂	1	2	yes	maximal, complemented
3	#22562076	C₂	1	2	yes	maximal, complemented
4	#22562071	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562055	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561920	C₂	1	2	yes	maximal, complemented
10	#22561804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524066	C₂²	1	2	yes	—
13	#16863878	—	2	2	no	—
14	#16863829	C₄	1	8	yes	complemented
15	C₂×#14523986	C₂²	1	2	yes	—
16	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
17	C₂×#14523982	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	#16863499	—	2	2	no	—
20	#16863494	—	2	2	no	—
21	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
22	C₄×D₅₃₆	C₂²	1	2	yes	—
23	#16863478	C₂²	1	2	yes	—
24	#107×D₆₇	C₂²	1	2	yes	—
25	#16863396	—	2	2	no	—
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863222	C₂²	1	1	yes	characteristic
28	#16863221	C₂²	1	2	yes	—
29	#16863200	C₂²	1	2	yes	—
30	#16863198	C₂²	1	2	yes	—
31	#16863196	C₂²	1	2	yes	—
32	#16863172	C₂²	1	1	yes	characteristic
33	#16863171	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761629	—	2	2	no	—
41	#14524066	—	2	4	no	—
42	#14524066	C₂×C₄	1	8	yes	complemented
43	C₂×#11761615	—	2	2	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	C₂×#11761611	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	Q₈×Dic₆₇	—	2	2	no	—
48	C₄×D₂₆₈	—	2	2	no	—
49	C₄×D₂₆₈	—	4	2	no	—
50	#14524005	—	2	2	no	—
51	#14524005	—	4	2	no	—
52	#14524000	—	4	2	no	—
53	#14524000	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₆₇	—	2	2	no	—
55	C₄⋊C₄×D₆₇	—	4	2	no	—
56	#14523986	C₂×C₄	1	8	yes	complemented
57	D₁₀₇₂	C₂×C₄	1	4	yes	complemented
58	#14523982	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	complemented
62	#14523977	C₂³	1	2	yes	—
63	#14523976	—	2	2	no	—
64	#14523967	—	2	2	no	—
65	#14523965	—	2	2	no	—
66	#14523963	—	2	2	no	—
67	#14523954	D₄	1	1	yes	characteristic
68	#14523953	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	Q₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#11761629	—	2	8	no	—
79	#11761629	—	4	4	no	—
80	C₂×D₂₆₈	—	2	2	no	—
81	C₂×D₂₆₈	—	4	2	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	C₂×C₄×D₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	C₂×C₄×D₆₇	—	4	2	no	—
86	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
87	#11761615	—	2	8	no	—
88	D₅₃₆	—	2	2	no	—
89	D₅₃₆	C₂²×C₄	1	4	yes	—
90	#11761611	—	2	2	no	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₆₇	C₂²×C₄	1	2	yes	—
94	C₈×D₆₇	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
99	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	2	no	—
102	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
103	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
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	2	no	—
115	C₄²×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×D₆₇	—	4	2	no	—
120	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₆₇	—	2	4	no	—
122	Q₈×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	2	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	D₂₆₈	—	2	4	no	—
133	D₂₆₈	—	4	8	no	—
134	D₂₆₈	—	4	2	no	—
135	C₄×D₆₇	—	4	8	no	—
136	C₄×D₆₇	—	16	2	no	—
137	C₄×D₆₇	—	16	2	no	—
138	C₄×D₆₇	—	16	2	no	—
139	C₄×D₆₇	C₄×D₄	1	2	yes	—
140	C₄×D₆₇	#135	1	2	yes	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₆₇	—	2	1	no	—
148	C₂×D₆₇	—	2	1	no	—
149	C₂×D₆₇	—	4	4	no	—
150	C₂×D₆₇	—	8	2	no	—
151	C₂×D₆₇	—	8	2	no	—
152	C₂×D₆₇	C₄×D₈	1	2	yes	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	4	4	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
159	Dic₆₇	—	2	1	no	—
160	Dic₆₇	—	2	1	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	C₄×D₈	1	2	yes	—
163	#33459	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
165	C₂×C₆₇	#3415	1	1	yes	characteristic
166	C₂×C₆₇	#3417	1	1	yes	characteristic
167	D₆₇	—	2	2	no	—
168	D₆₇	—	8	4	no	—
169	C₂×#508	—	67	1	no	—
170	#2951	—	67	2	no	—
171	#2175	—	67	2	no	—
172	#2175	—	67	2	no	—
173	C₄×Q₃₂	—	67	1	no	—
174	C₄×Q₃₂	D₆₇	1	1	yes	characteristic, complemented
175	C₄×D₁₆	—	67	1	no	—
176	C₄×D₁₆	—	67	1	no	—
177	C₂×#366	—	67	1	no	—
178	#2139	—	67	2	no	—
179	#2108	—	67	1	no	—
180	C₆₇	#33459	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	67	2	no	—
182	#520	—	134	2	no	—
183	#508	—	67	8	no	—
184	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
185	C₂×SD₃₂	—	67	2	no	—
186	C₂×D₁₆	—	67	1	no	—
187	C₂×M₃₂	—	67	1	no	—
188	C₂²×C₁₆	—	67	1	no	—
189	#440	—	134	2	no	—
190	#440	—	134	2	no	—
191	C₄×Q₁₆	—	67	2	no	—
192	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
193	C₄×D₈	—	67	2	no	—
194	C₄×D₈	—	67	2	no	—
195	C₂×#107	—	67	2	no	—
196	#417	—	134	2	no	—
197	#405	—	67	1	no	—
198	#366	—	67	1	no	—
199	#366	—	67	2	no	—
200	#366	C₂×D₆₇	1	1	yes	characteristic
201	#358	—	67	2	no	—
202	#358	C₂×D₆₇	1	2	yes	—
203	#357	—	67	2	no	—
204	#357	—	67	2	no	—
205	#346	—	67	1	no	—
206	#346	—	67	1	no	—
207	C₄×C₁₆	—	67	1	no	—
208	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#41	—	134	2	no	—
210	#135	—	67	8	no	—
211	#135	—	134	4	no	—
212	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
213	C₂×SD₁₆	—	134	2	no	—
214	C₂×D₈	—	67	2	no	—
215	C₂×M₁₆	—	67	1	no	—
216	C₂²×C₈	—	67	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	134	2	no	—
219	C₄×D₄	—	268	2	no	—
220	C₄×D₄	—	268	2	no	—
221	C₄×D₄	—	134	2	no	—
222	C₄×D₄	—	134	2	no	—
223	#117	—	268	2	no	—
224	#117	—	67	1	no	—
225	C₂×C₄⋊C₄	—	268	2	no	—
226	C₂×C₄⋊C₄	—	134	2	no	—
227	Q₃₂	C₄×D₆₇	1	4	yes	complemented
228	SD₃₂	—	67	8	no	—
229	D₁₆	—	67	4	no	—
230	M₃₂	—	134	2	no	—
231	C₂×C₁₆	—	67	1	no	—
232	C₂×C₁₆	—	67	4	no	—
233	C₂×C₁₆	—	67	1	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	#107	—	67	2	no	—
237	#107	—	134	2	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#103	—	2	2	no	—
240	#103	—	134	2	no	—
241	#102	—	134	2	no	—
242	#102	—	134	2	no	—
243	#97	—	67	1	no	—
244	#97	—	67	1	no	—
245	C₄×C₈	—	67	1	no	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	#41	—	268	4	no	—
248	#41	—	134	8	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	268	2	no	—
251	C₂×D₄	—	134	2	no	—
252	C₂²×C₄	—	268	2	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	—	268	2	no	—
256	C₂²×C₄	—	67	1	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
259	SD₁₆	—	134	8	no	—
260	D₈	—	67	4	no	—
261	D₈	—	134	2	no	—
262	M₁₆	—	134	2	no	—
263	C₂×C₈	—	67	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	1	no	—
267	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	67	1	no	—
274	C₄⋊C₄	—	67	1	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₂²⋊C₄	—	67	1	no	—
279	C₂²⋊C₄	—	67	1	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	268	2	no	—
282	C₄²	—	67	1	no	—
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	67	2	no	—
286	C₁₆	—	134	1	no	—
287	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
288	C₂³	—	268	2	no	—
289	C₂³	—	67	1	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	268	8	no	—
293	D₄	—	268	2	no	—
294	D₄	—	134	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	268	8	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	1072	2	no	—
300	C₂×C₄	—	1072	2	no	—
301	C₂×C₄	—	1072	2	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	67	1	no	—
306	C₂×C₄	—	67	1	no	—
307	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	67	2	no	—
312	C₈	—	134	1	no	—
313	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
314	C₈	#41×D₆₇	1	1	yes	characteristic
315	C₂²	—	268	4	no	—
316	C₂²	—	536	2	no	—
317	C₂²	—	536	2	no	—
318	C₂²	—	67	2	no	—
319	C₂²	—	134	1	no	—
320	C₂²	—	134	1	no	—
321	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	536	2	no	—
327	C₄	—	67	2	no	—
328	C₄	—	134	1	no	—
329	C₄	—	134	1	no	—
330	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
331	C₄	#135×D₆₇	1	1	yes	characteristic
332	C₂	—	536	4	no	—
333	C₂	—	134	2	no	—
334	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
335	C₂	#22573347	1	1	yes	characteristic
336	C₂	#22573358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#16863829	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×Q₃₂	D₆₇
hypocenter	C₆₇	#33459
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22569084	C₂	semidirect
3. #22562076	C₂	semidirect
4. #22562071	C₂	semidirect
5. Q₃₂×Dic₆₇	C₂	semidirect
6. C₄×D₁₀₇₂	C₂	semidirect
7. #22562055	C₂	semidirect
8. #366×D₆₇	C₂	semidirect
9. #22561920	C₂	semidirect
10. #22561804	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
14. #16863829	C₄	semidirect
42. #14524066	C₂×C₄	semidirect
56. #14523986	C₂×C₄	semidirect
57. D₁₀₇₂	C₂×C₄	semidirect
61. C₁₆×D₆₇	C₂×C₄	semidirect
72. Q₃₂×C₆₇	C₂×C₄	semidirect
174. C₄×Q₃₂	D₆₇	semidirect
180. C₆₇	#33459	semidirect
227. Q₃₂	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863829	2
#22569084	2
#22562076	2
#22562071	2
Q₃₂×Dic₆₇	2
C₄×D₁₀₇₂	2
#22562055	2
#366×D₆₇	2
#22561920	2
#22561804	2
C₄×Q₃₂×C₆₇	2
#33459	67