Subgroups — #30827591

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22568955	C₂	1	2	yes	maximal, complemented
3	#22559095	C₂	1	2	yes	maximal, complemented
4	#22559040	C₂	1	4	yes	maximal, complemented
5	C₈×D₅₃₆	C₂	1	2	yes	maximal, complemented
6	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558609	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524051	—	2	2	no	—
10	#16863946	C₂²	1	2	yes	—
11	#16863801	C₄	1	1	yes	characteristic
12	#16863773	C₄	1	1	yes	characteristic
13	#16863743	C₄	1	2	yes	—
14	#16863632	C₄	1	2	yes	—
15	#16863632	C₄	1	2	yes	—
16	C₄×D₅₃₆	C₂²	1	2	yes	—
17	C₄×D₅₃₆	C₂²	1	2	yes	—
18	#16863475	C₂²	1	4	yes	—
19	#16863469	—	2	4	no	—
20	C₈×D₂₆₈	C₂²	1	4	yes	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	#16863364	—	2	2	no	—
23	#16863149	C₂²	1	1	yes	characteristic
24	#16863149	C₂²	1	2	yes	—
25	#16863149	C₂²	1	1	yes	characteristic
26	#16863125	C₂²	1	4	yes	—
27	#16863125	C₂²	1	4	yes	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#322×C₆₇	C₂²	1	2	yes	—
32	C₈²×C₆₇	C₂²	1	2	yes	—
33	C₂×#11761628	—	2	2	no	—
34	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
35	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
36	#14524051	—	4	8	no	—
37	C₂×#11761611	—	2	4	no	—
38	C₂²×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₈×D₆₇	—	2	4	no	—
40	#14524043	C₂×C₄	1	2	yes	—
41	#14524040	C₂×C₄	1	2	yes	—
42	#14524031	—	2	2	no	—
43	#14524017	—	2	2	no	—
44	#14524017	C₂×C₄	1	4	yes	—
45	C₄×D₂₆₈	—	2	2	no	—
46	C₄×D₂₆₈	C₂³	1	4	yes	—
47	#14523977	C₂×C₄	1	2	yes	—
48	#14523977	C₂×C₄	1	2	yes	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
52	#14523965	C₂×C₄	1	4	yes	—
53	#14523965	C₂×C₄	1	4	yes	—
54	(C₂×D₆₇)⋊C₈	—	2	4	no	—
55	(C₂×D₆₇)⋊C₈	—	2	4	no	—
56	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₂²⋊C₈×C₆₇	—	2	1	no	—
63	#97×C₆₇	D₄	1	2	yes	—
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	2	yes	—
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	2	yes	—
70	#11761628	—	4	4	no	—
71	#11761628	—	4	4	no	—
72	C₂×Dic₁₃₄	—	2	2	no	—
73	C₂×D₂₆₈	—	2	2	no	—
74	C₂×D₂₆₈	—	2	2	no	—
75	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
76	C₂×C₂²⋊D₆₇	—	2	4	no	—
77	C₂×C₄×D₆₇	—	2	4	no	—
78	D₅₃₆	—	2	4	no	—
79	D₅₃₆	—	4	2	no	—
80	#11761611	—	4	8	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	4	4	no	—
84	C₈×D₆₇	—	4	8	no	—
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
87	(C₂×D₆₇)⋊C₄	—	2	4	no	—
88	(C₂×D₆₇)⋊C₄	—	2	4	no	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
92	M₁₆×C₆₇	—	2	2	no	—
93	M₁₆×C₆₇	D₈	1	4	yes	complemented
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	4	1	no	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₇	#41	1	1	yes	characteristic
106	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂²×D₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	#135	1	2	yes	—
117	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂²⋊D₆₇	—	4	8	no	—
121	C₂×Dic₆₇	—	2	4	no	—
122	D₂₆₈	—	2	4	no	—
123	D₂₆₈	—	2	4	no	—
124	D₂₆₈	—	4	4	no	—
125	D₂₆₈	—	8	2	no	—
126	C₄×D₆₇	—	4	4	no	—
127	C₄×D₆₇	—	4	4	no	—
128	Dic₁₃₄	—	4	4	no	—
129	C₈×C₆₇	—	2	1	no	—
130	C₈×C₆₇	—	2	1	no	—
131	C₈×C₆₇	—	2	1	no	—
132	C₈×C₆₇	—	2	2	no	—
133	C₈×C₆₇	—	4	2	no	—
134	C₈×C₆₇	#131	1	2	yes	complemented
135	C₈×C₆₇	C₂×D₈	1	4	yes	—
136	C₆₇⋊C₈	—	4	2	no	—
137	C₆₇⋊C₈	—	4	2	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	C₄×D₈	1	2	yes	—
141	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×D₆₇	—	4	4	no	—
143	C₂×D₆₇	—	4	4	no	—
144	C₂×D₆₇	—	4	4	no	—
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	C₄×D₈	1	2	yes	—
149	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
150	C₄×C₆₇	#568	1	1	yes	characteristic
151	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
153	Dic₆₇	—	4	4	no	—
154	#18066	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	#2935	1	1	yes	characteristic
157	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₆₇	#2957	1	1	yes	characteristic
159	D₆₇	—	8	4	no	—
160	#3169	—	67	1	no	—
161	#2935	—	67	2	no	—
162	#1589	—	67	2	no	—
163	#1581	—	67	4	no	—
164	C₈×D₈	—	67	2	no	—
165	C₂×#335	—	67	1	no	—
166	#1480	—	67	2	no	—
167	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
168	C₆₇	#18066	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	134	2	no	—
170	#532	—	67	2	no	—
171	#500	—	67	1	no	—
172	#493	—	67	1	no	—
173	#486	—	67	2	no	—
174	#466	—	67	2	no	—
175	#466	—	67	2	no	—
176	C₄×D₈	—	67	2	no	—
177	C₄×D₈	—	67	2	no	—
178	#436	—	67	4	no	—
179	#435	—	134	4	no	—
180	C₈×D₄	—	67	4	no	—
181	#431	C₂×D₆₇	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	67	2	no	—
183	#408	—	134	2	no	—
184	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
186	#335	—	67	1	no	—
187	#335	—	67	2	no	—
188	#335	—	67	1	no	—
189	D₄⋊C₈	—	67	4	no	—
190	D₄⋊C₈	—	67	4	no	—
191	#322	C₂×D₆₇	1	2	yes	—
192	C₈²	C₂×D₆₇	1	2	yes	—
193	C₂×#41	—	134	2	no	—
194	C₂×D₈	—	67	2	no	—
195	C₂×D₈	—	67	2	no	—
196	#131	—	268	8	no	—
197	C₂×M₁₆	—	134	4	no	—
198	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
199	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
200	C₂²×C₈	—	134	2	no	—
201	C₂²×C₈	—	134	4	no	—
202	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₄⋊Q₈	—	67	2	no	—
205	#127	—	67	2	no	—
206	#124	—	134	2	no	—
207	#121	—	67	4	no	—
208	#121	—	134	2	no	—
209	C₄×D₄	—	67	4	no	—
210	C₄×D₄	—	134	2	no	—
211	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
212	#107	—	67	2	no	—
213	#107	—	67	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	67	2	no	—
216	C₄⋊C₈	—	67	2	no	—
217	C₄⋊C₈	—	134	2	no	—
218	#102	—	67	4	no	—
219	#102	—	67	4	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	134	4	no	—
222	C₂²⋊C₈	—	134	4	no	—
223	#97	D₂₆₈	1	2	yes	—
224	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
225	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₆₇	1	2	yes	—
227	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₆₇	1	2	yes	—
230	#41	—	268	4	no	—
231	#41	—	268	4	no	—
232	C₂×Q₈	—	134	2	no	—
233	C₂×D₄	—	67	4	no	—
234	C₂×D₄	—	134	4	no	—
235	C₂×D₄	—	134	2	no	—
236	C₂×D₄	—	134	2	no	—
237	C₂²×C₄	—	134	4	no	—
238	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
240	C₂²×C₄	#11761628	1	1	yes	characteristic
241	D₈	—	268	2	no	—
242	D₈	—	134	4	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	268	8	no	—
245	M₁₆	D₅₃₆	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	268	4	no	—
250	C₂×C₈	—	268	8	no	—
251	C₂×C₈	—	134	2	no	—
252	C₂×C₈	—	134	2	no	—
253	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
258	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
259	C₄⋊C₄	—	67	4	no	—
260	C₄⋊C₄	—	134	2	no	—
261	C₂²⋊C₄	—	134	4	no	—
262	C₂²⋊C₄	—	134	4	no	—
263	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
264	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
265	C₄²	#11761628	1	1	yes	characteristic
266	C₄²	C₂³×D₆₇	1	1	yes	characteristic
267	C₂³	—	134	4	no	—
268	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
269	Q₈	—	268	4	no	—
270	D₄	—	268	8	no	—
271	D₄	—	268	4	no	—
272	D₄	—	536	2	no	—
273	D₄	—	134	4	no	—
274	D₄	—	134	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	268	4	no	—
279	C₂×C₄	—	268	4	no	—
280	C₂×C₄	—	134	4	no	—
281	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
282	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
283	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
284	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
285	C₂×C₄	#14524065	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	268	2	no	—
295	C₈	—	268	2	no	—
296	C₈	#14524051	1	2	yes	complemented
297	C₈	C₂×D₅₃₆	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	268	4	no	—
301	C₂²	—	268	4	no	—
302	C₂²	—	268	4	no	—
303	C₂²	C₄×D₅₃₆	1	2	yes	—
304	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	268	4	no	—
309	C₄	C₄×D₅₃₆	1	2	yes	—
310	C₄	C₂×#14524051	1	1	yes	characteristic
311	C₄	#16864162	1	1	yes	characteristic
312	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
313	C₄	C₂×#14524065	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	536	4	no	—
316	C₂	#22568955	1	1	yes	characteristic
317	C₂	C₂×C₄×D₅₃₆	1	1	yes	characteristic
318	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#18066
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22568955	C₂	semidirect
3. #22559095	C₂	semidirect
4. #22559040	C₂	semidirect
5. C₈×D₅₃₆	C₂	semidirect
6. C₂×#16863149	C₂	semidirect
7. #22558609	C₂	semidirect
8. C₈×M₁₆×C₆₇	C₂	semidirect
93. M₁₆×C₆₇	D₈	semidirect
134. C₈×C₆₇	#131	semidirect
167. C₈×M₁₆	D₆₇	semidirect
168. C₆₇	#18066	semidirect
245. M₁₆	D₅₃₆	semidirect
296. C₈	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22568955	2
#22559095	2
#22559040	2
C₈×D₅₃₆	2
C₂×#16863149	2
#22558609	2
C₈×M₁₆×C₆₇	2
#18066	67