Subgroups — #30864355

Order 17152.

1238 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863773	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863509	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₅₃₆	C₂	1	2	yes	maximal, complemented
4	#22562934	C₂	1	2	yes	maximal, complemented
5	#22561085	C₂	1	2	yes	maximal, complemented
6	#22561075	C₂	1	2	yes	maximal, complemented
7	#22561033	C₂	1	2	yes	maximal, complemented
8	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₅₃₆	C₂²	1	2	yes	—
11	C₂²×D₅₃₆	C₂²	1	2	yes	—
12	C₂×#14524040	C₂²	1	2	yes	—
13	C₂×C₄×D₂₆₈	C₂²	1	4	yes	—
14	#16863773	C₄	1	8	yes	complemented
15	#16863509	—	2	8	no	—
16	C₄×D₅₃₆	—	2	8	no	—
17	C₂×#14523977	C₂²	1	2	yes	—
18	C₂×#14523975	C₂²	1	2	yes	—
19	C₂×#14523965	C₂²	1	4	yes	—
20	C₂×#14523965	C₂²	1	4	yes	—
21	#16863298	C₂²	1	4	yes	—
22	#16863274	—	2	8	no	—
23	#16863262	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#376×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×D₂₆₈	—	2	4	no	—
32	C₂²×D₂₆₈	C₂³	1	4	yes	—
33	C₂²×C₄×D₆₇	—	2	4	no	—
34	C₂×D₅₃₆	—	2	4	no	—
35	C₂×D₅₃₆	—	2	4	no	—
36	C₂×D₅₃₆	—	2	4	no	—
37	C₂×D₅₃₆	—	2	4	no	—
38	C₂×D₅₃₆	C₂×C₄	1	8	yes	complemented
39	C₂×D₅₃₆	C₂×C₄	1	8	yes	complemented
40	#14524040	—	2	4	no	—
41	#14524040	C₂×C₄	1	8	yes	complemented
42	C₄×D₂₆₈	—	2	8	no	—
43	C₄×D₂₆₈	—	2	8	no	—
44	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
45	C₂×(C₂×D₆₇)⋊C₄	—	2	4	no	—
46	C₂×(C₂×D₆₇)⋊C₄	—	2	8	no	—
47	#14523977	—	2	4	no	—
48	#14523975	—	2	4	no	—
49	#14523965	—	2	8	no	—
50	#14523965	—	2	8	no	—
51	C₆₇⋊#95	—	2	4	no	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	#97×C₆₇	—	2	2	no	—
60	#97×C₆₇	—	2	2	no	—
61	#97×C₆₇	D₄	1	4	yes	complemented
62	C₄×C₈×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
64	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
65	#95×C₆₇	D₄	1	4	yes	—
66	C₂³×D₆₇	—	2	4	no	—
67	C₂×D₂₆₈	—	2	4	no	—
68	C₂×D₂₆₈	—	2	4	no	—
69	C₂×D₂₆₈	—	2	4	no	—
70	C₂×D₂₆₈	—	2	4	no	—
71	C₂×D₂₆₈	—	2	4	no	—
72	C₂×D₂₆₈	—	2	16	no	—
73	C₂×D₂₆₈	—	8	8	no	—
74	C₂×D₂₆₈	C₂²×C₄	1	8	yes	—
75	C₂²×Dic₆₇	—	2	4	no	—
76	C₂×C₄×D₆₇	—	4	8	no	—
77	C₂×C₄×D₆₇	—	8	8	no	—
78	D₅₃₆	—	2	16	no	—
79	D₅₃₆	—	2	16	no	—
80	C₄⋊Dic₆₇	—	2	4	no	—
81	C₄⋊Dic₆₇	—	2	4	no	—
82	(C₂×D₆₇)⋊C₄	—	4	8	no	—
83	(C₂×D₆₇)⋊C₄	—	8	8	no	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	#41	1	2	yes	—
99	C₂×C₈×C₆₇	#41	1	2	yes	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₆₇	C₂×D₄	1	2	yes	—
104	C₄²×C₆₇	C₂×D₄	1	2	yes	—
105	C₄²×C₆₇	#41	1	2	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂²×D₆₇	—	2	8	no	—
108	C₂²×D₆₇	—	4	4	no	—
109	C₂²×D₆₇	—	4	4	no	—
110	C₂²×D₆₇	—	4	8	no	—
111	C₂²×D₆₇	—	4	8	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₆₇	#127	1	4	yes	—
118	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	—	4	8	no	—
126	D₂₆₈	—	2	8	no	—
127	D₂₆₈	—	2	8	no	—
128	D₂₆₈	—	4	4	no	—
129	D₂₆₈	—	4	4	no	—
130	D₂₆₈	—	8	16	no	—
131	C₄×D₆₇	—	8	16	no	—
132	C₈×C₆₇	—	2	2	no	—
133	C₈×C₆₇	—	2	2	no	—
134	C₈×C₆₇	C₄×D₄	1	4	yes	—
135	C₈×C₆₇	C₄×D₄	1	4	yes	—
136	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
138	C₂²×C₆₇	#532	1	1	yes	characteristic
139	C₂²×C₆₇	#576	1	4	yes	—
140	C₂×D₆₇	—	4	8	no	—
141	C₂×D₆₇	—	4	16	no	—
142	C₂×D₆₇	—	8	4	no	—
143	C₂×D₆₇	—	8	4	no	—
144	C₂×D₆₇	—	8	8	no	—
145	C₄×C₆₇	—	2	4	no	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₆₇	#532	1	1	yes	characteristic
151	C₄×C₆₇	#532	1	1	yes	characteristic
152	Dic₆₇	—	8	4	no	—
153	#21697	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
155	C₂×C₆₇	#2948	1	2	yes	—
156	C₂×C₆₇	#3155	1	2	yes	—
157	C₂×C₆₇	#3176	1	2	yes	—
158	D₆₇	—	8	8	no	—
159	C₂×#493	—	67	1	no	—
160	C₂×#442	—	67	2	no	—
161	C₂×C₄×D₈	—	67	2	no	—
162	#2328	—	67	2	no	—
163	#1972	—	67	2	no	—
164	#1970	—	67	2	no	—
165	#1961	—	67	2	no	—
166	#1846	—	67	1	no	—
167	#1726	D₆₇	1	1	yes	characteristic, complemented
168	C₆₇	#21697	1	1	yes	characteristic, complemented, Sylow
169	C₂²×D₈	—	67	2	no	—
170	C₂²×D₈	—	67	2	no	—
171	C₂×#127	—	67	2	no	—
172	C₂×C₄×D₄	—	67	4	no	—
173	#493	—	67	8	no	—
174	#442	—	134	8	no	—
175	C₄×D₈	—	134	8	no	—
176	C₂×#107	—	67	2	no	—
177	C₂×#106	—	67	2	no	—
178	C₂×#102	—	67	4	no	—
179	C₂×#102	—	67	4	no	—
180	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
181	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
182	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
186	#390	—	67	4	no	—
187	#386	—	134	8	no	—
188	#382	—	67	2	no	—
189	#376	C₂×D₆₇	1	1	yes	characteristic
190	C₂²×D₄	—	67	4	no	—
191	C₂²×D₄	—	134	4	no	—
192	C₂³×C₄	—	134	4	no	—
193	C₂×D₈	—	67	8	no	—
194	C₂×D₈	—	67	8	no	—
195	C₂×D₈	—	134	4	no	—
196	C₂×D₈	—	134	4	no	—
197	C₂×D₈	—	134	4	no	—
198	C₂×D₈	—	134	4	no	—
199	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	#127	—	67	8	no	—
204	#127	—	134	4	no	—
205	C₄×D₄	—	134	8	no	—
206	C₄×D₄	—	134	8	no	—
207	C₂×C₄⋊C₄	—	67	4	no	—
208	C₂×C₂²⋊C₄	—	134	4	no	—
209	C₂×C₂²⋊C₄	—	134	8	no	—
210	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
213	#107	—	134	4	no	—
214	#106	—	134	4	no	—
215	#102	—	134	8	no	—
216	#102	—	134	8	no	—
217	#97	—	2	2	no	—
218	#97	—	2	2	no	—
219	#97	D₂₆₈	1	4	yes	complemented
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
222	C₄×C₈	D₂₆₈	1	4	yes	complemented
223	#95	—	134	4	no	—
224	#95	D₂₆₈	1	4	yes	—
225	C₂⁴	—	134	4	no	—
226	C₂×D₄	—	536	8	no	—
227	C₂×D₄	—	67	8	no	—
228	C₂×D₄	—	134	16	no	—
229	C₂×D₄	—	134	4	no	—
230	C₂×D₄	—	134	4	no	—
231	C₂×D₄	—	134	4	no	—
232	C₂×D₄	—	134	4	no	—
233	C₂×D₄	—	134	4	no	—
234	C₂²×C₄	—	268	8	no	—
235	C₂²×C₄	—	536	8	no	—
236	C₂²×C₄	—	134	4	no	—
237	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
238	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
239	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
240	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
241	D₈	—	134	16	no	—
242	D₈	—	134	16	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
249	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
250	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
251	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
252	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
253	C₂×C₈	#11761628	1	2	yes	—
254	C₂×C₈	#11761628	1	2	yes	—
255	C₄⋊C₄	—	134	4	no	—
256	C₄⋊C₄	—	134	4	no	—
257	C₂²⋊C₄	—	268	8	no	—
258	C₂²⋊C₄	—	536	8	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂×C₄×D₆₇	1	2	yes	—
262	C₄²	C₂×D₂₆₈	1	2	yes	—
263	C₄²	C₂×D₂₆₈	1	2	yes	—
264	C₄²	#11761628	1	2	yes	—
265	C₂³	—	268	4	no	—
266	C₂³	—	268	8	no	—
267	C₂³	—	268	8	no	—
268	C₂³	—	268	4	no	—
269	C₂³	—	134	8	no	—
270	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
271	D₄	—	268	4	no	—
272	D₄	—	268	4	no	—
273	D₄	—	536	16	no	—
274	D₄	—	134	8	no	—
275	D₄	—	134	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	268	4	no	—
281	C₂×C₄	—	268	8	no	—
282	C₂×C₄	—	536	16	no	—
283	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
284	C₂×C₄	#14524040	1	4	yes	—
285	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
290	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	C₄×D₂₆₈	1	4	yes	—
294	C₈	C₄×D₂₆₈	1	4	yes	—
295	C₂²	—	268	16	no	—
296	C₂²	—	268	8	no	—
297	C₂²	—	536	8	no	—
298	C₂²	—	536	4	no	—
299	C₂²	—	536	4	no	—
300	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
301	C₂²	C₂×#14524040	1	1	yes	characteristic
302	C₂²	#16863946	1	1	yes	characteristic
303	C₂²	#16864201	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	536	4	no	—
308	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
309	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
310	C₄	#16863946	1	1	yes	characteristic
311	C₄	#16863946	1	1	yes	characteristic
312	C₂	—	536	8	no	—
313	C₂	C₄×#14524040	1	1	yes	characteristic
314	C₂	#22569062	1	2	yes	—
315	C₂	#22570695	1	2	yes	—
316	C₂	#22570859	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂×#16863773	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21697
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#16863509	C₂	semidirect
3. C₂×C₄×D₅₃₆	C₂	semidirect
4. #22562934	C₂	semidirect
5. #22561085	C₂	semidirect
6. #22561075	C₂	semidirect
7. #22561033	C₂	semidirect
8. #22560373	C₂	semidirect
9. C₆₇×#1726	C₂	semidirect
14. #16863773	C₄	semidirect
38. C₂×D₅₃₆	C₂×C₄	semidirect
39. C₂×D₅₃₆	C₂×C₄	semidirect
41. #14524040	C₂×C₄	semidirect
61. #97×C₆₇	D₄	semidirect
63. C₄×C₈×C₆₇	C₂×C₄	semidirect
64. C₄×C₈×C₆₇	D₄	semidirect
167. #1726	D₆₇	semidirect
168. C₆₇	#21697	semidirect
219. #97	D₂₆₈	semidirect
221. C₄×C₈	C₄×D₆₇	semidirect
222. C₄×C₈	D₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863773	2
C₂×#16863509	2
C₂×C₄×D₅₃₆	2
#22562934	2
#22561085	2
#22561075	2
#22561033	2
#22560373	2
C₆₇×#1726	2
#21697	67