Subgroups — #30864680

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	#22569146	C₂	1	2	yes	maximal
3	#22561189	C₂	1	2	yes	maximal
4	#22560960	C₂	1	2	yes	maximal
5	#22560955	C₂	1	2	yes	maximal
6	#22560096	C₂	1	2	yes	maximal
7	#22560091	C₂	1	2	yes	maximal
8	#22559895	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1848	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	#16863525	—	2	8	no	—
16	#16863434	C₂²	1	2	yes	—
17	#16863434	C₂²	1	2	yes	—
18	#16863424	C₂²	1	4	yes	—
19	#16863354	C₂²	1	4	yes	—
20	C₂×#14523954	C₂²	1	1	yes	characteristic
21	C₂×#14523953	C₂²	1	1	yes	characteristic
22	C₂×#14523953	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863242	C₂²	1	1	yes	characteristic
26	#16863157	C₂²	1	4	yes	—
27	#16863157	C₂²	1	4	yes	—
28	C₂×#108×C₆₇	C₂²	1	2	yes	—
29	C₂×#108×C₆₇	C₂²	1	2	yes	—
30	#423×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
34	#14524043	C₂×C₄	1	4	yes	—
35	#14524035	C₂×C₄	1	4	yes	—
36	#14524022	—	2	4	no	—
37	C₄×Dic₁₃₄	—	2	4	no	—
38	#14524002	C₂³	1	4	yes	—
39	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
41	#14523977	C₂×C₄	1	4	yes	—
42	#14523977	C₂×C₄	1	4	yes	—
43	#14523975	C₂×C₄	1	4	yes	—
44	#14523975	C₂×C₄	1	4	yes	—
45	Dic₆₇⋊C₈	—	2	8	no	—
46	#14523954	D₄	1	4	yes	—
47	#14523953	—	2	4	no	—
48	#14523953	—	2	2	no	—
49	#14523953	Q₈	1	4	yes	complemented
50	C₈×Dic₆₇	—	2	4	no	—
51	C₈×Dic₆₇	—	2	2	no	—
52	C₈×Dic₆₇	Q₈	1	4	yes	complemented
53	C₆₇⋊#95	—	2	2	no	—
54	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#108×C₆₇	—	2	4	no	—
59	#108×C₆₇	—	2	4	no	—
60	C₄⋊C₈×C₆₇	—	2	4	no	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₁₃₄	—	2	4	no	—
66	C₂²×Dic₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	2	1	no	—
68	C₂×C₆₇⋊C₈	—	2	1	no	—
69	C₂×C₆₇⋊C₈	—	2	1	no	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
71	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
72	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
73	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
76	Dic₆₇⋊C₄	—	2	8	no	—
77	C₂²⋊Dic₆₇	—	2	4	no	—
78	C₄×Dic₆₇	—	2	4	no	—
79	C₄×Dic₆₇	—	2	1	no	—
80	C₄×Dic₆₇	—	2	1	no	—
81	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
82	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₆₇	—	2	4	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	#41	1	2	yes	—
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	2	yes	—
99	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₆₇	—	2	8	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	4	2	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	Dic₁₃₄	—	8	4	no	—
115	C₈×C₆₇	—	4	4	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	C₆₇⋊C₈	—	2	2	no	—
121	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	—
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	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₄×C₆₇	#532	1	1	yes	characteristic
132	Dic₆₇	—	4	2	no	—
133	Dic₆₇	—	4	2	no	—
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	D₆₇	1	1	yes	characteristic
146	#1773	—	67	2	no	—
147	#1772	—	67	2	no	—
148	#1726	—	67	1	no	—
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	C₂×D₆₇	1	2	yes	—
157	C₂×#108	C₂×D₆₇	1	2	yes	—
158	#427	—	67	2	no	—
159	#427	—	67	2	no	—
160	#424	—	67	4	no	—
161	#423	C₂×D₆₇	1	2	yes	—
162	#405	—	67	4	no	—
163	C₂×#97	—	67	1	no	—
164	C₂×#97	—	67	1	no	—
165	C₂×#97	—	67	1	no	—
166	C₂×C₄×C₈	—	67	1	no	—
167	C₂×C₄×C₈	—	67	1	no	—
168	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
169	#376	—	67	1	no	—
170	#339	—	67	4	no	—
171	#339	—	67	4	no	—
172	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
173	C₂²×C₈	—	67	1	no	—
174	C₂²×C₈	—	67	1	no	—
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	—	67	1	no	—
183	C₂×C₄²	—	67	1	no	—
184	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
185	#108	—	2	4	no	—
186	#108	—	2	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	—	2	4	no	—
192	C₄⋊C₈	—	134	8	no	—
193	#97	—	67	4	no	—
194	#97	—	67	4	no	—
195	#97	—	134	4	no	—
196	#97	—	134	2	no	—
197	C₄×C₈	—	67	4	no	—
198	C₄×C₈	—	134	4	no	—
199	C₄×C₈	—	134	2	no	—
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	—	134	2	no	—
205	C₂×Q₈	—	134	4	no	—
206	C₂²×C₄	—	134	1	no	—
207	C₂²×C₄	—	134	1	no	—
208	C₂²×C₄	#11761629	1	2	yes	—
209	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	67	2	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	67	2	no	—
216	C₂×C₈	—	67	2	no	—
217	C₂×C₈	—	134	1	no	—
218	C₂×C₈	—	134	1	no	—
219	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
223	C₂×C₈	Q₈×D₆₇	1	2	yes	—
224	C₂×C₈	Q₈×D₆₇	1	2	yes	—
225	C₂×C₈	#11761627	1	2	yes	—
226	C₂×C₈	#11761627	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	—	67	2	no	—
232	C₄²	—	67	2	no	—
233	C₄²	—	134	4	no	—
234	C₄²	—	134	1	no	—
235	C₄²	—	134	1	no	—
236	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₄²	D₄×D₆₇	1	2	yes	—
239	C₂³	C₂×#11761629	1	1	yes	characteristic
240	Q₈	—	536	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	268	2	no	—
243	C₂×C₄	—	268	2	no	—
244	C₂×C₄	—	134	8	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	#14524005	1	2	yes	—
248	C₂×C₄	#14524005	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	67	4	no	—
257	C₈	—	134	2	no	—
258	C₈	#14524008	1	2	yes	—
259	C₈	#14524008	1	2	yes	—
260	C₈	#14524008	1	2	yes	—
261	C₈	#14524008	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×#14524005	1	1	yes	characteristic
264	C₂²	#16863951	1	1	yes	characteristic
265	C₂²	#16863951	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	268	2	no	—
269	C₄	—	268	2	no	—
270	C₄	—	268	8	no	—
271	C₄	C₂×#14524008	1	1	yes	characteristic
272	C₄	C₂×#14524008	1	1	yes	characteristic
273	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
274	C₄	#16863944	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	#22562776	1	1	yes	characteristic
278	C₂	#22569125	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#21733
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

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
49. #14523953	Q₈	semidirect
52. C₈×Dic₆₇	Q₈	semidirect
149. C₆₇	#21733	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22569146	2
#22561189	2
#22560960	2
#22560955	2
#22560096	2
#22560091	2
#22559895	2
C₆₇×#1848	2
#21733	67