Subgroups — #30864687

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22569149	C₂	1	2	yes	maximal
3	#22561190	C₂	1	2	yes	maximal
4	#22560962	C₂	1	2	yes	maximal
5	#22560957	C₂	1	2	yes	maximal
6	#22560381	C₂	1	1	yes	maximal, characteristic
7	#22560093	C₂	1	2	yes	maximal
8	#22560088	C₂	1	2	yes	maximal
9	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
10	#16863953	C₂²	1	2	yes	—
11	#16863801	C₄	1	2	yes	complemented
12	#16863798	C₄	1	2	yes	complemented
13	#16863798	C₄	1	2	yes	complemented
14	#16863794	C₄	1	2	yes	complemented
15	#16863524	—	2	8	no	—
16	C₂×#14523979	C₂²	1	2	yes	—
17	C₂×#14523979	C₂²	1	2	yes	—
18	#16863434	C₂²	1	2	yes	—
19	#16863434	C₂²	1	2	yes	—
20	#16863422	C₂²	1	4	yes	—
21	#16863420	C₂²	1	2	yes	—
22	#16863351	C₂²	1	4	yes	—
23	#16863156	C₂²	1	4	yes	—
24	#16863156	C₂²	1	4	yes	—
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#97×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	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#376×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761612	C₂³	1	4	yes	—
33	#14524043	C₂×C₄	1	4	yes	—
34	#14524035	C₂×C₄	1	4	yes	—
35	#14524022	—	2	4	no	—
36	C₄×Dic₁₃₄	—	2	4	no	—
37	#14524002	C₂³	1	4	yes	—
38	#14523979	—	2	4	no	—
39	#14523979	—	2	4	no	—
40	#14523977	C₂×C₄	1	4	yes	—
41	#14523977	C₂×C₄	1	4	yes	—
42	#14523975	C₂×C₄	1	4	yes	—
43	#14523975	C₂×C₄	1	4	yes	—
44	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
45	Dic₆₇⋊C₈	—	2	8	no	—
46	#14523953	—	2	4	no	—
47	C₈×Dic₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#97×C₆₇	—	2	2	no	—
56	#97×C₆₇	D₄	1	4	yes	—
57	#97×C₆₇	Q₈	1	4	yes	complemented
58	C₄×C₈×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	Q₈	1	4	yes	complemented
64	#95×C₆₇	—	2	2	no	—
65	C₂×Dic₁₃₄	—	2	4	no	—
66	#11761612	—	2	4	no	—
67	#11761612	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
71	Dic₆₇⋊C₄	—	2	8	no	—
72	C₂²⋊Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₂×C₈×C₆₇	#41	1	2	yes	—
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
98	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₆₇	—	2	8	no	—
114	Dic₁₃₄	—	8	4	no	—
115	C₈×C₆₇	—	2	2	no	—
116	C₈×C₆₇	C₄×Q₈	1	2	yes	—
117	C₈×C₆₇	C₄×Q₈	1	2	yes	—
118	C₈×C₆₇	C₄×Q₈	1	2	yes	—
119	C₈×C₆₇	C₄×Q₈	1	2	yes	—
120	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
121	C₆₇⋊C₈	—	4	4	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₆₇	#533	1	1	yes	characteristic
125	C₂²×C₆₇	#533	1	1	yes	characteristic
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	4	2	no	—
129	C₄×C₆₇	—	4	2	no	—
130	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₄×C₆₇	#532	1	1	yes	characteristic
134	Dic₆₇	—	4	8	no	—
135	#21733	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	—	2	1	no	—
138	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₆₇	#2957	1	2	yes	—
140	#3160	—	67	1	no	—
141	#2961	—	67	2	no	—
142	#1995	—	67	2	no	—
143	#1946	—	67	2	no	—
144	#1945	—	67	2	no	—
145	#1848	—	67	1	no	—
146	#1773	—	67	2	no	—
147	#1772	—	67	2	no	—
148	#1726	D₆₇	1	1	yes	characteristic
149	C₆₇	#21733	1	1	yes	characteristic, complemented, Sylow
150	#533	—	67	2	no	—
151	#500	—	67	2	no	—
152	#499	—	67	2	no	—
153	#499	—	67	2	no	—
154	#498	—	67	2	no	—
155	#446	—	134	8	no	—
156	C₂×#108	—	67	2	no	—
157	C₂×#108	—	67	2	no	—
158	#427	—	67	2	no	—
159	#427	—	67	2	no	—
160	#424	—	67	4	no	—
161	#423	—	67	2	no	—
162	#405	—	67	4	no	—
163	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
169	#376	C₂×D₆₇	1	1	yes	characteristic
170	#339	—	67	4	no	—
171	#339	—	67	4	no	—
172	C₂×M₁₆	—	67	4	no	—
173	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	67	4	no	—
178	#125	—	67	4	no	—
179	C₂²⋊Q₈	—	134	4	no	—
180	C₄×Q₈	—	134	4	no	—
181	#117	—	67	4	no	—
182	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
185	#108	—	134	4	no	—
186	#108	—	134	4	no	—
187	#107	—	67	4	no	—
188	#107	—	67	4	no	—
189	#106	—	67	4	no	—
190	#106	—	67	4	no	—
191	C₄⋊C₈	—	134	8	no	—
192	C₄⋊C₈	—	134	4	no	—
193	#97	—	2	2	no	—
194	#97	—	134	4	no	—
195	#97	Dic₁₃₄	1	4	yes	complemented
196	#97	D₂₆₈	1	4	yes	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	134	4	no	—
199	C₄×C₈	Dic₁₃₄	1	4	yes	complemented
200	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
201	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
202	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
203	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
204	#95	—	2	2	no	—
205	C₂×Q₈	—	134	4	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	#11761628	1	2	yes	—
209	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
210	M₁₆	—	134	4	no	—
211	M₁₆	—	134	4	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	134	4	no	—
215	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
220	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
221	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
222	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
223	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
224	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
225	C₂×C₈	#11761628	1	2	yes	—
226	C₂×C₈	#11761628	1	2	yes	—
227	C₄⋊C₄	—	67	8	no	—
228	C₄⋊C₄	—	134	8	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₂²⋊C₄	—	134	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	134	4	no	—
234	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
235	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₄²	C₂×D₂₆₈	1	2	yes	—
237	C₄²	C₂×Dic₁₃₄	1	2	yes	—
238	C₄²	C₂×Dic₁₃₄	1	2	yes	—
239	C₂³	C₂×#11761628	1	1	yes	characteristic
240	Q₈	—	536	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	134	8	no	—
247	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
248	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
251	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
252	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
253	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	268	4	no	—
257	C₈	C₄×Dic₁₃₄	1	2	yes	—
258	C₈	C₄×Dic₁₃₄	1	2	yes	—
259	C₈	C₄×Dic₁₃₄	1	2	yes	—
260	C₈	C₄×Dic₁₃₄	1	2	yes	—
261	C₈	#14524043	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
264	C₂²	#16863953	1	1	yes	characteristic
265	C₂²	#16863953	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	268	8	no	—
271	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
272	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
273	C₄	C₂×#14524043	1	1	yes	characteristic
274	C₄	#16863946	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄×#14524043	1	1	yes	characteristic
278	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21733
LCS	G ▷ C₂×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
11. #16863801	C₄	semidirect
12. #16863798	C₄	semidirect
13. #16863798	C₄	semidirect
14. #16863794	C₄	semidirect
57. #97×C₆₇	Q₈	semidirect
63. C₄×C₈×C₆₇	Q₈	semidirect
149. C₆₇	#21733	semidirect
195. #97	Dic₁₃₄	semidirect
199. C₄×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22569149	2
#22561190	2
#22560962	2
#22560957	2
#22560381	2
#22560093	2
#22560088	2
C₆₇×#1726	2
#21733	67