Subgroups — #30854434

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863514	C₂	1	2	yes	maximal
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22568634	C₂	1	2	yes	maximal, complemented
4	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560088	C₂	1	2	yes	maximal, complemented
6	#22560053	C₂	1	4	yes	maximal, complemented
7	#22560020	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524049	C₂²	1	4	yes	—
11	C₄×#11761628	C₂²	1	2	yes	—
12	#16863514	C₄	1	16	yes	complemented
13	C₄×Dic₂₆₈	C₄	1	2	yes	complemented
14	C₄×#11761617	C₄	1	4	yes	complemented
15	C₄×D₅₃₆	C₄	1	2	yes	complemented
16	C₂×#14523979	C₂²	1	2	yes	—
17	#16863434	C₂²	1	1	yes	characteristic
18	C₂×#14523971	C₂²	1	4	yes	—
19	#16863393	C₂²	1	2	yes	—
20	#16863353	—	2	4	no	—
21	#16863351	C₂²	1	4	yes	—
22	C₄×#11761612	C₂²	1	2	yes	—
23	#16863348	—	2	2	no	—
24	#97×D₆₇	—	2	2	no	—
25	#16863161	C₂²	1	2	yes	—
26	#16863156	C₂²	1	4	yes	—
27	C₂×#97×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	#376×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761628	C₂³	1	2	yes	—
33	#14524065	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#11761617	C₂×C₄	1	2	yes	—
36	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
37	#14524049	—	2	8	no	—
38	#14524049	C₂×C₄	1	16	yes	complemented
39	C₂×#11761612	C₂³	1	4	yes	—
40	C₂×#11761611	—	2	4	no	—
41	C₂×C₈×D₆₇	—	2	4	no	—
42	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₆₇	—	2	4	no	—
45	#14524002	C₂³	1	2	yes	—
46	#14524000	—	2	2	no	—
47	C₄²×D₆₇	—	2	2	no	—
48	#14523979	C₂×C₄	1	8	yes	complemented
49	#14523977	C₂×C₄	1	2	yes	—
50	#14523975	C₂×C₄	1	2	yes	—
51	#14523971	C₂×C₄	1	16	yes	complemented
52	#14523968	C₂×C₄	1	4	yes	—
53	#14523965	C₂×C₄	1	4	yes	—
54	#14523954	—	2	2	no	—
55	#14523953	—	2	4	no	—
56	C₄×C₆₇⋊C₈	—	2	2	no	—
57	C₈×Dic₆₇	—	2	4	no	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	#97×C₆₇	—	2	4	no	—
64	#97×C₆₇	D₄	1	1	yes	characteristic
65	#97×C₆₇	D₄	1	1	yes	characteristic
66	#97×C₆₇	D₄	1	2	yes	—
67	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
70	#95×C₆₇	—	2	2	no	—
71	#11761628	—	2	4	no	—
72	#11761628	C₂²×C₄	1	8	yes	—
73	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
74	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
75	C₂×C₂²⋊D₆₇	—	2	2	no	—
76	C₂×C₄×D₆₇	—	2	2	no	—
77	C₂×C₄×D₆₇	—	2	4	no	—
78	Dic₂₆₈	C₄²	1	4	yes	—
79	#11761617	C₄²	1	8	yes	—
80	D₅₃₆	C₄²	1	4	yes	—
81	#11761612	—	2	4	no	—
82	#11761612	C₂²×C₄	1	8	yes	—
83	#11761611	—	2	16	no	—
84	C₂×C₆₇⋊C₈	—	2	4	no	—
85	C₈×D₆₇	—	2	16	no	—
86	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
87	Dic₆₇⋊C₄	—	2	4	no	—
88	C₂²⋊Dic₆₇	—	2	2	no	—
89	(C₂×D₆₇)⋊C₄	—	2	4	no	—
90	C₄×Dic₆₇	—	2	4	no	—
91	C₄×Dic₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	#41	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
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	4	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	#41	1	4	yes	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
106	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂²×D₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	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	2	yes	—
120	C₂²⋊D₆₇	—	2	8	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	2	4	no	—
123	D₂₆₈	—	2	2	no	—
124	D₂₆₈	C₂×C₄²	1	4	yes	—
125	C₄×D₆₇	—	2	8	no	—
126	C₄×D₆₇	—	8	4	no	—
127	Dic₁₃₄	—	2	2	no	—
128	Dic₁₃₄	C₂×C₄²	1	4	yes	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₂×C₄²	1	2	yes	—
131	C₈×C₆₇	C₂×C₄²	1	2	yes	—
132	C₈×C₆₇	C₄×D₄	1	4	yes	—
133	C₈×C₆₇	C₄×D₄	1	4	yes	—
134	C₆₇⋊C₈	—	2	8	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₄×#41	1	2	yes	—
138	C₂×D₆₇	—	2	4	no	—
139	C₂×D₆₇	—	4	4	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
144	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
145	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
146	Dic₆₇	—	2	4	no	—
147	Dic₆₇	—	4	4	no	—
148	#20514	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
151	C₂×C₆₇	#2957	1	2	yes	—
152	D₆₇	—	4	4	no	—
153	C₂×#443	—	67	2	no	—
154	C₄×#135	—	67	1	no	—
155	#2875	—	67	2	no	—
156	C₄×#108	—	67	1	no	—
157	#1772	—	67	2	no	—
158	#1765	—	67	4	no	—
159	#1759	—	67	2	no	—
160	#1726	D₆₇	1	1	yes	characteristic, complemented
161	C₆₇	#20514	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	—	67	1	no	—
163	C₂×#131	—	67	4	no	—
164	C₄×#41	—	67	2	no	—
165	#443	—	67	16	no	—
166	C₄×Q₁₆	—	67	2	no	—
167	C₄×SD₁₆	—	67	4	no	—
168	C₄×D₈	—	67	2	no	—
169	C₂×#108	—	67	2	no	—
170	#427	—	67	1	no	—
171	C₂×#104	—	67	4	no	—
172	#416	—	67	2	no	—
173	#405	—	67	4	no	—
174	#405	—	134	4	no	—
175	C₄×M₁₆	—	67	2	no	—
176	C₄×M₁₆	—	134	2	no	—
177	C₂×#97	—	134	2	no	—
178	C₂×#97	C₂×D₆₇	1	2	yes	—
179	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
182	#376	C₂×D₆₇	1	1	yes	characteristic
183	#341	—	67	2	no	—
184	#339	—	67	4	no	—
185	C₂×#41	—	67	2	no	—
186	#135	—	67	8	no	—
187	C₂×Q₁₆	—	67	1	no	—
188	C₂×SD₁₆	—	67	2	no	—
189	C₂×D₈	—	67	1	no	—
190	#131	—	67	16	no	—
191	#131	—	134	8	no	—
192	C₂×M₁₆	—	67	4	no	—
193	C₂×M₁₆	—	134	4	no	—
194	C₂²×C₈	—	134	4	no	—
195	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
198	C₄×Q₈	—	67	4	no	—
199	C₄×D₄	—	67	4	no	—
200	C₄×D₄	—	134	4	no	—
201	#117	—	67	2	no	—
202	#117	—	134	2	no	—
203	C₂×C₄²	—	134	2	no	—
204	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	#108	—	67	8	no	—
207	#107	—	67	2	no	—
208	#106	—	67	2	no	—
209	#104	—	67	16	no	—
210	#103	—	67	4	no	—
211	#102	—	67	4	no	—
212	#97	—	2	4	no	—
213	#97	—	134	4	no	—
214	#97	—	134	2	no	—
215	#97	D₂₆₈	1	1	yes	characteristic
216	#97	D₂₆₈	1	1	yes	characteristic
217	#97	D₂₆₈	1	2	yes	—
218	C₄×C₈	—	134	4	no	—
219	C₄×C₈	—	134	2	no	—
220	C₄×C₈	C₄×D₆₇	1	2	yes	—
221	C₄×C₈	C₄×D₆₇	1	2	yes	—
222	C₄×C₈	C₄×D₆₇	1	8	yes	complemented
223	#95	—	2	2	no	—
224	#41	—	67	8	no	—
225	#41	—	134	4	no	—
226	C₂×Q₈	—	67	2	no	—
227	C₂×D₄	—	67	2	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	—	134	2	no	—
232	C₂²×C₄	#11761628	1	2	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	Q₁₆	—	67	4	no	—
235	SD₁₆	—	67	8	no	—
236	D₈	—	67	4	no	—
237	M₁₆	—	67	8	no	—
238	M₁₆	—	134	4	no	—
239	M₁₆	—	134	16	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	134	4	no	—
242	C₂×C₈	—	134	16	no	—
243	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
247	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
248	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
249	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
250	C₂×C₈	#11761628	1	4	yes	—
251	C₄⋊C₄	—	67	4	no	—
252	C₄⋊C₄	—	134	4	no	—
253	C₂²⋊C₄	—	134	2	no	—
254	C₂²⋊C₄	—	134	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	134	4	no	—
257	C₄²	—	134	2	no	—
258	C₄²	C₂×C₄×D₆₇	1	4	yes	—
259	C₄²	C₂×C₄×D₆₇	1	2	yes	—
260	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
261	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
262	C₂³	—	134	2	no	—
263	C₂³	C₂×#11761628	1	1	yes	characteristic
264	Q₈	—	67	4	no	—
265	Q₈	—	134	2	no	—
266	D₄	—	67	4	no	—
267	D₄	—	134	8	no	—
268	D₄	—	134	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	536	4	no	—
273	C₂×C₄	—	134	4	no	—
274	C₂×C₄	—	134	2	no	—
275	C₂×C₄	—	134	8	no	—
276	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
277	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
279	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
281	C₂×C₄	C₂×#11761628	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	134	8	no	—
284	C₈	C₄²×D₆₇	1	2	yes	—
285	C₈	C₄²×D₆₇	1	2	yes	—
286	C₈	C₄×D₂₆₈	1	4	yes	—
287	C₈	C₄×D₂₆₈	1	4	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	268	4	no	—
290	C₂²	—	134	4	no	—
291	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
292	C₂²	C₄×#11761628	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	268	4	no	—
297	C₄	—	134	4	no	—
298	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
299	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
300	C₄	C₄×#11761628	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	268	4	no	—
303	C₂	C₄²×D₂₆₈	1	1	yes	characteristic
304	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#20514
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
3. #22568634	C₂	semidirect
4. C₄×#14523979	C₂	semidirect
5. #22560088	C₂	semidirect
6. #22560053	C₂	semidirect
7. #22560020	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
12. #16863514	C₄	semidirect
13. C₄×Dic₂₆₈	C₄	semidirect
14. C₄×#11761617	C₄	semidirect
15. C₄×D₅₃₆	C₄	semidirect
38. #14524049	C₂×C₄	semidirect
48. #14523979	C₂×C₄	semidirect
51. #14523971	C₂×C₄	semidirect
69. C₄×C₈×C₆₇	C₂×C₄	semidirect
160. #1726	D₆₇	semidirect
161. C₆₇	#20514	semidirect
222. C₄×C₈	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×#16863514	2
C₄×#14524065	2
#22568634	2
C₄×#14523979	2
#22560088	2
#22560053	2
#22560020	2
C₆₇×#1726	2
#20514	67