Subgroups — #30864603

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562958	C₂	1	2	yes	maximal
2	#22561183	C₂	1	2	yes	maximal
3	#22560953	C₂	1	2	yes	maximal
4	#22560938	C₂	1	2	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#22560109	C₂	1	2	yes	maximal
7	#106×Dic₆₇	C₂	1	2	yes	maximal
8	#22559895	C₂	1	1	yes	maximal, characteristic
9	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523975	C₂²	1	2	yes	—
12	C₂×#14523954	C₂²	1	1	yes	characteristic
13	C₂×#14523953	C₂²	1	1	yes	characteristic
14	C₂×#14523953	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863303	—	2	4	no	—
18	#16863295	C₂²	1	4	yes	—
19	#16863282	—	2	4	no	—
20	#16863262	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	#16863242	C₂²	1	1	yes	characteristic
23	#16863158	C₂²	1	4	yes	—
24	#16863158	C₂²	1	4	yes	—
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	#498×C₆₇	C₄	1	8	yes	complemented
27	C₂×#106×C₆₇	C₂²	1	2	yes	—
28	C₂×#106×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₆₇	—	2	4	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₇	—	2	4	no	—
35	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	#14523977	—	2	4	no	—
38	#14523975	—	2	4	no	—
39	#14523954	—	2	2	no	—
40	#14523953	—	2	2	no	—
41	#14523953	Q₈	1	4	yes	complemented
42	C₈×Dic₆₇	—	2	2	no	—
43	C₈×Dic₆₇	Q₈	1	4	yes	complemented
44	C₆₇⋊#95	—	2	4	no	—
45	C₆₇⋊#95	—	2	4	no	—
46	C₆₇⋊#95	—	2	2	no	—
47	C₆₇⋊#95	—	2	2	no	—
48	C₆₇⋊#95	D₄	1	4	yes	—
49	C₂²×Q₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#106×C₆₇	C₂×C₄	1	8	yes	—
58	#106×C₆₇	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₂²×Dic₆₇	—	2	4	no	—
61	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
62	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	—	2	1	no	—
66	C₂×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	8	4	no	—
72	C₄×Dic₆₇	—	2	1	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₄×Dic₆₇	—	8	4	no	—
75	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
76	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
77	C₂×Q₈×C₆₇	—	2	8	no	—
78	C₂×Q₈×C₆₇	—	8	4	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₂×C₈×C₆₇	#41	1	2	yes	—
88	C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₆₇	#41	1	2	yes	—
93	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₆₇	—	8	8	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	8	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₂×Dic₆₇	—	2	4	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	4	no	—
109	C₂×Dic₆₇	—	4	4	no	—
110	C₂×Dic₆₇	—	4	4	no	—
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
113	C₈×C₆₇	C₄×Q₈	1	4	yes	—
114	C₈×C₆₇	C₄×Q₈	1	4	yes	—
115	C₆₇⋊C₈	—	2	2	no	—
116	C₆₇⋊C₈	—	2	2	no	—
117	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₂²×C₆₇	#532	1	1	yes	characteristic
120	C₂²×C₆₇	#577	1	4	yes	—
121	C₄×C₆₇	—	2	4	no	—
122	C₄×C₆₇	—	4	8	no	—
123	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₄×C₆₇	#533	1	1	yes	characteristic
126	C₄×C₆₇	#533	1	1	yes	characteristic
127	Dic₆₇	—	2	4	no	—
128	Dic₆₇	—	4	2	no	—
129	Dic₆₇	—	8	4	no	—
130	#21724	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₆₇	#2949	1	2	yes	—
133	C₂×C₆₇	#3163	1	2	yes	—
134	C₂×C₆₇	#3173	1	2	yes	—
135	C₂×#498	D₆₇	1	1	yes	characteristic
136	#2331	—	67	2	no	—
137	#1993	—	67	2	no	—
138	#1944	—	67	2	no	—
139	#1941	—	67	2	no	—
140	#1846	—	67	1	no	—
141	#1776	—	67	2	no	—
142	C₄×#106	—	67	2	no	—
143	#1726	—	67	1	no	—
144	C₆₇	#21724	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
146	#498	Dic₆₇	1	8	yes	complemented
147	C₂×#107	—	67	2	no	—
148	C₂×#106	—	67	2	no	—
149	C₂×#106	C₂×D₆₇	1	2	yes	—
150	C₂×#106	C₂×D₆₇	1	2	yes	—
151	C₂×#97	—	67	1	no	—
152	C₂×#97	—	67	1	no	—
153	C₂×#97	—	67	1	no	—
154	C₂×C₄×C₈	—	67	1	no	—
155	C₂×C₄×C₈	—	67	1	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#391	—	134	4	no	—
158	#389	—	67	4	no	—
159	#387	—	134	4	no	—
160	#382	—	67	2	no	—
161	C₄×C₄⋊C₄	—	67	4	no	—
162	#376	—	67	1	no	—
163	#340	—	67	4	no	—
164	#340	—	67	4	no	—
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×C₈	—	67	1	no	—
167	C₂²×C₈	—	67	1	no	—
168	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	67	4	no	—
174	C₂×C₄⋊C₄	—	134	4	no	—
175	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
176	C₂×C₄²	—	67	1	no	—
177	C₂×C₄²	—	67	1	no	—
178	C₂×C₄²	—	134	4	no	—
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	#107	—	134	4	no	—
181	#106	—	134	4	no	—
182	#106	C₂×Dic₆₇	1	8	yes	—
183	#106	C₂×Dic₆₇	1	8	yes	—
184	#97	—	67	4	no	—
185	#97	—	134	2	no	—
186	#97	—	134	2	no	—
187	C₄×C₈	—	67	4	no	—
188	C₄×C₈	—	134	2	no	—
189	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
190	#95	—	67	4	no	—
191	#95	—	134	4	no	—
192	#95	—	134	4	no	—
193	#95	—	134	2	no	—
194	#95	—	134	2	no	—
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	67	2	no	—
199	C₂²×C₄	—	67	2	no	—
200	C₂²×C₄	—	134	4	no	—
201	C₂²×C₄	D₄×D₆₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
203	C₂×C₈	—	67	2	no	—
204	C₂×C₈	—	67	2	no	—
205	C₂×C₈	—	134	1	no	—
206	C₂×C₈	—	134	1	no	—
207	C₂×C₈	—	134	1	no	—
208	C₂×C₈	—	134	1	no	—
209	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
210	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
211	C₂×C₈	Q₈×D₆₇	1	2	yes	—
212	C₂×C₈	Q₈×D₆₇	1	2	yes	—
213	C₂×C₈	#11761629	1	2	yes	—
214	C₂×C₈	#11761629	1	2	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	536	4	no	—
218	C₄⋊C₄	—	134	4	no	—
219	C₄⋊C₄	—	134	4	no	—
220	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
221	C₄²	—	536	4	no	—
222	C₄²	—	67	2	no	—
223	C₄²	—	67	2	no	—
224	C₄²	—	134	1	no	—
225	C₄²	—	134	1	no	—
226	C₄²	C₂²×Dic₆₇	1	2	yes	—
227	C₄²	#11761627	1	2	yes	—
228	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
229	Q₈	—	8	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	268	4	no	—
235	C₂×C₄	—	268	4	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	67	4	no	—
238	C₂×C₄	—	134	4	no	—
239	C₂×C₄	—	134	2	no	—
240	C₂×C₄	—	134	4	no	—
241	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
242	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
246	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
247	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
248	C₈	—	134	2	no	—
249	C₈	—	134	2	no	—
250	C₈	Q₈×Dic₆₇	1	4	yes	—
251	C₈	Q₈×Dic₆₇	1	4	yes	—
252	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
253	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
254	C₂²	#16863947	1	1	yes	characteristic
255	C₂²	#16864202	1	4	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	4	8	no	—
258	C₄	—	268	2	no	—
259	C₄	—	536	4	no	—
260	C₄	—	134	4	no	—
261	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
262	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
263	C₄	#16863951	1	1	yes	characteristic
264	C₄	#16863951	1	1	yes	characteristic
265	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
266	C₂	#22569069	1	2	yes	—
267	C₂	#22570734	1	2	yes	—
268	C₂	#22570828	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#498×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#498	D₆₇
hypocenter	C₆₇	#21724
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
26. #498×C₆₇	C₄	semidirect
41. #14523953	Q₈	semidirect
43. C₈×Dic₆₇	Q₈	semidirect
144. C₆₇	#21724	semidirect
146. #498	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562958	2
#22561183	2
#22560953	2
#22560938	2
#22560373	2
#22560109	2
#106×Dic₆₇	2
#22559895	2
C₂×#498×C₆₇	2
#21724	67