Subgroups — #30864674

Order 17152.

702 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	#22570719	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic
3	#22561187	C₂	1	1	yes	maximal, characteristic
4	#22561185	C₂	1	1	yes	maximal, characteristic
5	#22560954	C₂	1	4	yes	maximal
6	#22560380	C₂	1	1	yes	maximal, characteristic
7	#22560089	C₂	1	4	yes	maximal
8	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
10	#16863953	C₂²	1	1	yes	characteristic
11	#16863799	C₄	1	2	yes	complemented
12	#16863795	C₄	1	4	yes	complemented
13	#16863792	C₄	1	2	yes	complemented
14	#16863526	—	2	2	no	—
15	C₈×Dic₁₃₄	—	2	2	no	—
16	C₂×#14523978	C₂²	1	4	yes	—
17	#16863432	C₂²	1	4	yes	—
18	#16863422	C₂²	1	2	yes	—
19	#16863420	C₂²	1	1	yes	characteristic
20	#16863351	C₂²	1	2	yes	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
24	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863157	C₂²	1	4	yes	—
26	#16863155	C₂²	1	4	yes	—
27	#533×C₆₇	C₂²	1	1	yes	characteristic
28	#446×C₆₇	—	2	2	no	—
29	C₈×Q₈×C₆₇	—	2	2	no	—
30	#424×C₆₇	C₂²	1	2	yes	—
31	#423×C₆₇	C₂²	1	1	yes	characteristic
32	#405×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	C₂³	1	2	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
36	#14524043	C₂×C₄	1	2	yes	—
37	#14524035	C₂×C₄	1	2	yes	—
38	#14524022	—	2	2	no	—
39	C₄×Dic₁₃₄	—	2	2	no	—
40	#14524002	C₂³	1	2	yes	—
41	C₂×C₄×Dic₆₇	—	2	2	no	—
42	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
43	#14523978	—	2	8	no	—
44	#14523976	C₂×C₄	1	8	yes	—
45	#14523974	C₂×C₄	1	8	yes	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	Dic₆₇⋊C₈	—	2	4	no	—
48	#14523953	—	2	2	no	—
49	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
51	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₈×Dic₆₇	—	2	4	no	—
53	C₈×Dic₆₇	—	2	2	no	—
54	C₈×Dic₆₇	Q₈	1	8	yes	complemented
55	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
59	#125×C₆₇	C₂×C₄	1	2	yes	—
60	C₂²⋊Q₈×C₆₇	—	2	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	#117×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₆₇	—	2	4	no	—
65	C₄⋊C₈×C₆₇	—	2	2	no	—
66	#97×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	D₄	1	2	yes	—
69	C₄×C₈×C₆₇	D₄	1	2	yes	—
70	C₂×Dic₁₃₄	—	2	2	no	—
71	C₂²×Dic₆₇	—	2	2	no	—
72	#11761612	—	2	4	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
77	C₂×C₆₇⋊C₈	#41	1	4	yes	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
80	Dic₆₇⋊C₄	—	2	4	no	—
81	C₂²⋊Dic₆₇	—	2	2	no	—
82	C₄×Dic₆₇	—	2	8	no	—
83	C₄×Dic₆₇	—	2	2	no	—
84	C₄×Dic₆₇	—	2	2	no	—
85	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	2	yes	—
104	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
105	Q₈×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
114	C₂×Dic₆₇	—	2	8	no	—
115	C₂×Dic₆₇	—	2	4	no	—
116	C₂×Dic₆₇	—	2	4	no	—
117	Dic₁₃₄	—	4	4	no	—
118	C₈×C₆₇	—	2	2	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
121	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
124	C₆₇⋊C₈	C₄×Q₈	1	4	yes	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂²×C₆₇	#533	1	2	yes	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	4	4	no	—
130	C₄×C₆₇	#443	1	4	yes	—
131	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
132	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₄×C₆₇	#532	1	1	yes	characteristic
134	Dic₆₇	—	2	8	no	—
135	Dic₆₇	—	4	4	no	—
136	#21732	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#443	1	2	yes	—
140	#3160	—	67	1	no	—
141	#2960	—	67	1	no	—
142	#2960	D₆₇	1	1	yes	characteristic
143	#1994	—	67	1	no	—
144	#1994	—	67	1	no	—
145	#1945	—	67	4	no	—
146	#1848	—	67	1	no	—
147	#1772	—	67	4	no	—
148	C₄²×C₈	—	67	1	no	—
149	C₆₇	#21732	1	1	yes	characteristic, complemented, Sylow
150	#533	—	67	1	no	—
151	#533	C₂×D₆₇	1	1	yes	characteristic
152	#500	—	67	2	no	—
153	#499	—	67	4	no	—
154	#498	—	67	2	no	—
155	#446	—	2	2	no	—
156	#446	—	134	2	no	—
157	C₈×Q₈	—	2	2	no	—
158	C₈×Q₈	—	134	2	no	—
159	C₂×#108	—	67	4	no	—
160	#427	—	67	4	no	—
161	#424	—	67	2	no	—
162	#424	C₂×D₆₇	1	2	yes	—
163	#423	—	67	1	no	—
164	#423	C₂×D₆₇	1	1	yes	characteristic
165	#405	—	67	2	no	—
166	#405	C₂×D₆₇	1	2	yes	—
167	C₂×C₄×C₈	—	67	2	no	—
168	C₂×C₄×C₈	—	67	1	no	—
169	C₂×C₄×C₈	—	67	2	no	—
170	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
171	C₄³	—	67	1	no	—
172	#339	—	67	4	no	—
173	#339	—	67	4	no	—
174	C₂×M₁₆	—	67	2	no	—
175	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
176	C₂²×C₈	—	67	2	no	—
177	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	67	2	no	—
180	C₄⋊Q₈	C₄×D₆₇	1	2	yes	—
181	#125	—	67	2	no	—
182	#125	C₄×D₆₇	1	2	yes	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	134	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	134	2	no	—
187	#117	—	67	2	no	—
188	#117	C₂²×D₆₇	1	2	yes	—
189	C₂×C₄²	—	67	2	no	—
190	C₂×C₄²	—	134	2	no	—
191	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
192	#108	—	134	8	no	—
193	#107	—	67	8	no	—
194	#106	—	67	8	no	—
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	134	4	no	—
198	C₄⋊C₈	—	134	2	no	—
199	#97	—	2	2	no	—
200	#97	—	134	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	—	67	8	no	—
203	C₄×C₈	—	67	1	no	—
204	C₄×C₈	—	67	2	no	—
205	C₄×C₈	—	67	1	no	—
206	C₄×C₈	—	134	4	no	—
207	C₄×C₈	—	134	2	no	—
208	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
209	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	134	2	no	—
212	C₂²×C₄	—	134	2	no	—
213	C₂²×C₄	#11761628	1	2	yes	—
214	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	134	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	67	2	no	—
220	C₂×C₈	—	67	2	no	—
221	C₂×C₈	—	67	4	no	—
222	C₂×C₈	—	67	4	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₆₇	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	67	4	no	—
231	C₄⋊C₄	—	134	4	no	—
232	C₄⋊C₄	—	134	2	no	—
233	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	134	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	67	4	no	—
238	C₄²	—	134	8	no	—
239	C₄²	—	134	2	no	—
240	C₄²	—	134	2	no	—
241	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
244	C₂³	C₂×#11761628	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	268	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	134	4	no	—
250	C₂×C₄	—	134	8	no	—
251	C₂×C₄	—	134	4	no	—
252	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
254	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
256	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
257	C₂×C₄	#41×D₆₇	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	268	2	no	—
261	C₈	—	67	4	no	—
262	C₈	—	67	4	no	—
263	C₈	#14524041	1	2	yes	—
264	C₈	#14524041	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
267	C₂²	#16863950	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	268	4	no	—
271	C₄	—	134	8	no	—
272	C₄	#16863511	1	4	yes	—
273	C₄	C₄×Q₈×D₆₇	1	2	yes	—
274	C₄	C₂×#14524041	1	1	yes	characteristic
275	C₄	#16863942	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	C₄×#14524041	1	1	yes	characteristic
278	C₂	C₂×#16863511	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#21732
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
11. #16863799	C₄	semidirect
12. #16863795	C₄	semidirect
13. #16863792	C₄	semidirect
54. C₈×Dic₆₇	Q₈	semidirect
149. C₆₇	#21732	semidirect

Maximal subgroups

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

subgroup	index
#22570719	2
#22569143	2
#22561187	2
#22561185	2
#22560954	2
#22560380	2
#22560089	2
C₄×C₈×Dic₆₇	2
C₆₇×#2960	2
#21732	67