Subgroups — #30869521

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#22569143	C₂	1	2	yes	maximal
3	#22561186	C₂	1	2	yes	maximal
4	#22560809	C₂	1	8	yes	maximal
5	#22560347	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₂²	1	2	yes	—
8	#16863801	C₄	1	1	yes	characteristic, complemented
9	#16863801	C₄	1	1	yes	characteristic, complemented
10	#16863798	C₄	1	4	yes	complemented
11	#16863794	C₄	1	2	yes	complemented
12	#16863526	—	2	2	no	—
13	#16863526	—	2	2	no	—
14	C₈×Dic₁₃₄	—	2	2	no	—
15	C₈×Dic₁₃₄	—	2	2	no	—
16	#16863434	C₂²	1	4	yes	—
17	#16863422	C₂²	1	4	yes	—
18	#16863420	C₂²	1	2	yes	—
19	#16863351	C₂²	1	4	yes	—
20	#16863161	C₂²	1	4	yes	—
21	#16863156	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	C₂²	1	4	yes	—
26	C₂×#11761612	C₂³	1	4	yes	—
27	#14524043	C₂×C₄	1	2	yes	—
28	#14524043	C₂×C₄	1	2	yes	—
29	#14524035	C₂×C₄	1	4	yes	—
30	#14524022	—	2	2	no	—
31	#14524022	—	2	2	no	—
32	C₄×Dic₁₃₄	—	2	2	no	—
33	C₄×Dic₁₃₄	—	2	2	no	—
34	#14524002	C₂³	1	4	yes	—
35	#14523977	C₂×C₄	1	4	yes	—
36	#14523977	C₂×C₄	1	4	yes	—
37	#14523975	C₂×C₄	1	8	yes	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	Dic₆₇⋊C₈	—	2	4	no	—
41	Dic₆₇⋊C₈	—	2	4	no	—
42	#14523953	—	2	4	no	—
43	C₈×Dic₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂×Dic₁₃₄	—	2	2	no	—
57	C₂×Dic₁₃₄	—	2	2	no	—
58	#11761612	—	2	8	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
63	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
64	Dic₆₇⋊C₄	—	2	4	no	—
65	Dic₆₇⋊C₄	—	2	4	no	—
66	C₂²⋊Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	4	2	no	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
98	C₂×Dic₆₇	—	2	4	no	—
99	C₂×Dic₆₇	—	2	4	no	—
100	Dic₁₃₄	—	4	4	no	—
101	Dic₁₃₄	—	4	4	no	—
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₆₇⋊C₈	—	4	4	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂²×C₆₇	#541	1	2	yes	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	—	4	4	no	—
114	C₄×C₆₇	#443	1	2	yes	—
115	C₄×C₆₇	#444	1	2	yes	—
116	C₄×C₆₇	#519	1	2	yes	—
117	C₄×C₆₇	#536	1	2	yes	—
118	Dic₆₇	—	4	4	no	—
119	Dic₆₇	—	4	4	no	—
120	#22256	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	2	no	—
122	C₂×C₆₇	#2327	1	1	yes	characteristic
123	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
124	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
125	#3160	—	67	1	no	—
126	#2960	—	67	2	no	—
127	#1994	—	67	2	no	—
128	#1922	—	67	8	no	—
129	#1837	—	67	1	no	—
130	#1744	D₆₇	1	1	yes	characteristic
131	C₆₇	#22256	1	1	yes	characteristic, complemented, Sylow
132	#533	—	67	2	no	—
133	#500	—	67	1	no	—
134	#500	—	67	1	no	—
135	#499	—	67	4	no	—
136	#498	—	67	2	no	—
137	#446	—	134	2	no	—
138	#446	—	134	2	no	—
139	C₈×Q₈	—	134	2	no	—
140	C₈×Q₈	—	134	2	no	—
141	#427	—	67	4	no	—
142	#424	—	67	4	no	—
143	#423	—	67	2	no	—
144	#405	—	67	4	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#341	—	67	4	no	—
149	#339	—	67	8	no	—
150	#336	C₂×D₆₇	1	4	yes	—
151	C₂×M₁₆	—	67	4	no	—
152	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
154	C₄⋊Q₈	—	67	2	no	—
155	C₄⋊Q₈	—	67	2	no	—
156	#125	—	67	4	no	—
157	C₂²⋊Q₈	—	134	2	no	—
158	C₂²⋊Q₈	—	134	2	no	—
159	C₄×Q₈	—	134	2	no	—
160	C₄×Q₈	—	134	2	no	—
161	#117	—	67	4	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	#107	—	67	4	no	—
167	#107	—	67	4	no	—
168	#106	—	67	8	no	—
169	C₄⋊C₈	—	134	4	no	—
170	C₄⋊C₈	—	134	4	no	—
171	C₄⋊C₈	—	134	2	no	—
172	C₄⋊C₈	—	134	2	no	—
173	#97	—	134	4	no	—
174	C₄×C₈	—	134	4	no	—
175	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
176	C₄×C₈	C₄×D₆₇	1	2	yes	—
177	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
178	C₄×C₈	D₂₆₈	1	1	yes	characteristic
179	C₄×C₈	D₂₆₈	1	2	yes	—
180	C₄×C₈	D₂₆₈	1	1	yes	characteristic
181	C₂×Q₈	—	134	2	no	—
182	C₂×Q₈	—	134	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	#11761628	1	1	yes	characteristic
185	C₂²×C₄	#11761628	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
187	M₁₆	—	134	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	134	4	no	—
191	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
193	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
194	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
195	C₄⋊C₄	—	67	4	no	—
196	C₄⋊C₄	—	67	4	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₄⋊C₄	—	134	4	no	—
199	C₄⋊C₄	—	134	2	no	—
200	C₄⋊C₄	—	134	2	no	—
201	C₂²⋊C₄	—	134	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	134	4	no	—
206	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
209	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
210	C₂³	C₂×#11761628	1	1	yes	characteristic
211	Q₈	—	268	4	no	—
212	Q₈	—	268	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	134	4	no	—
219	C₂×C₄	—	134	4	no	—
220	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
221	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
222	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
223	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
226	C₂×C₄	#14524090	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	268	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
235	C₂²	#16864008	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	268	4	no	—
240	C₄	—	268	4	no	—
241	C₄	#16863514	1	2	yes	—
242	C₄	#16863519	1	2	yes	—
243	C₄	#16863873	1	2	yes	—
244	C₄	#16863972	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	#22562928	1	1	yes	characteristic
247	C₂	C₂×#16863514	1	1	yes	characteristic
248	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1744	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#22256
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
8. #16863801	C₄	semidirect
9. #16863801	C₄	semidirect
10. #16863798	C₄	semidirect
11. #16863794	C₄	semidirect
131. C₆₇	#22256	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22569143	2
#22561186	2
#22560809	2
#22560347	2
C₆₇×#1744	2
#22256	67