Subgroups — #30827608

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569158	C₂	1	2	yes	maximal
2	#22559106	C₂	1	2	yes	maximal, complemented
3	#22559060	C₂	1	4	yes	maximal, complemented
4	Q₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
6	#22558612	C₂	1	2	yes	maximal
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	4	no	—
10	#16863527	C₂²	1	4	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863423	—	2	2	no	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	#16863352	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	1	yes	characteristic
18	#16863149	C₂²	1	2	yes	—
19	#16863149	C₂²	1	1	yes	characteristic
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
22	#16863118	C₂²	1	2	yes	—
23	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#500×C₆₇	C₄	1	1	yes	characteristic
26	#494×C₆₇	C₄	1	1	yes	characteristic
27	#487×C₆₇	C₄	1	2	yes	—
28	#467×C₆₇	C₄	1	2	yes	—
29	#467×C₆₇	C₄	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#11761612	D₄	1	1	yes	characteristic
34	C₂×#11761612	D₄	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
43	#14523954	—	2	4	no	—
44	#14523954	D₄	1	2	yes	—
45	C₄×C₆₇⋊C₈	—	2	4	no	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
54	#125×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
59	#117×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	C₂×C₄	1	2	yes	—
62	#107×C₆₇	C₂×C₄	1	2	yes	—
63	#103×C₆₇	C₂×C₄	1	4	yes	—
64	#103×C₆₇	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	#11761612	—	2	2	no	—
70	#11761612	Q₁₆	1	4	yes	complemented
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	4	4	no	—
75	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
76	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	Q₁₆×C₆₇	—	2	4	no	—
83	Q₁₆×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	#41	1	1	yes	characteristic
101	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	—	2	4	no	—
105	Q₈×C₆₇	—	8	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	4	1	no	—
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₆₇	#135	1	2	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
119	C₈×C₆₇	—	2	1	no	—
120	C₈×C₆₇	—	2	1	no	—
121	C₈×C₆₇	—	2	1	no	—
122	C₈×C₆₇	#131	1	2	yes	—
123	C₆₇⋊C₈	—	2	2	no	—
124	C₆₇⋊C₈	—	4	2	no	—
125	C₆₇⋊C₈	—	4	2	no	—
126	C₆₇⋊C₈	—	4	2	no	—
127	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
138	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
139	C₄×C₆₇	#568	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
142	#18068	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	2	no	—
144	C₂×C₆₇	#2935	1	1	yes	characteristic
145	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₆₇	#2957	1	1	yes	characteristic
147	#3170	D₆₇	1	1	yes	characteristic
148	#2962	—	67	2	no	—
149	#1591	—	67	2	no	—
150	#1584	—	67	4	no	—
151	C₈×Q₁₆	—	67	2	no	—
152	C₂×#335	—	67	1	no	—
153	#1481	—	67	2	no	—
154	C₈×M₁₆	—	67	1	no	—
155	C₆₇	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	C₂×D₆₇	1	2	yes	—
157	#500	Dic₆₇	1	1	yes	characteristic
158	#494	Dic₆₇	1	1	yes	characteristic
159	#487	Dic₆₇	1	2	yes	—
160	#467	Dic₆₇	1	2	yes	—
161	#467	Dic₆₇	1	2	yes	—
162	#446	—	67	4	no	—
163	#446	—	134	4	no	—
164	C₈×Q₈	—	67	4	no	—
165	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
166	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
167	#431	—	67	1	no	—
168	#424	—	134	2	no	—
169	C₂×C₄⋊C₈	—	67	2	no	—
170	#405	—	134	2	no	—
171	C₄×M₁₆	—	67	1	no	—
172	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
173	#335	—	67	1	no	—
174	#335	—	67	2	no	—
175	#335	—	67	1	no	—
176	Q₈⋊C₈	—	67	4	no	—
177	Q₈⋊C₈	—	67	4	no	—
178	#322	—	67	2	no	—
179	C₈²	—	67	2	no	—
180	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
181	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
182	C₂×M₁₆	—	67	1	no	—
183	C₂×M₁₆	—	67	1	no	—
184	C₂²×C₈	—	134	2	no	—
185	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
188	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
189	#125	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
194	#117	—	2	2	no	—
195	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
196	#107	C₂×Dic₆₇	1	2	yes	—
197	#107	C₂×Dic₆₇	1	2	yes	—
198	C₄⋊C₈	—	67	2	no	—
199	C₄⋊C₈	—	67	2	no	—
200	C₄⋊C₈	—	134	4	no	—
201	C₄⋊C₈	—	134	4	no	—
202	C₄⋊C₈	—	134	2	no	—
203	C₄⋊C₈	—	134	1	no	—
204	#103	C₂×Dic₆₇	1	4	yes	—
205	#103	C₂×Dic₆₇	1	4	yes	—
206	C₂²⋊C₈	—	134	1	no	—
207	#97	—	67	2	no	—
208	#97	—	134	4	no	—
209	C₄×C₈	—	67	2	no	—
210	C₄×C₈	—	67	2	no	—
211	C₄×C₈	—	134	4	no	—
212	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
213	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	#11761627	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	67	4	no	—
224	M₁₆	—	134	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	268	4	no	—
228	C₂×C₈	—	67	2	no	—
229	C₂×C₈	—	67	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	—	134	2	no	—
232	C₂×C₈	—	134	2	no	—
233	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₄²	D₄×D₆₇	1	1	yes	characteristic
247	C₄²	#11761627	1	1	yes	characteristic
248	C₄²	C₂³×D₆₇	1	1	yes	characteristic
249	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
262	C₂×C₄	#14524066	1	2	yes	—
263	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	268	2	no	—
270	C₈	—	268	2	no	—
271	C₈	—	268	2	no	—
272	C₈	—	67	4	no	—
273	C₈	—	134	2	no	—
274	C₈	#14524050	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	Q₁₆×Dic₆₇	1	2	yes	—
278	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	Q₁₆×Dic₆₇	1	2	yes	—
285	C₄	C₂×#14524050	1	1	yes	characteristic
286	C₄	#16864163	1	1	yes	characteristic
287	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
288	C₄	C₂×#14524066	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#22568953	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×Dic₆₇	1	1	yes	characteristic
292	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#18068
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22559106	C₂	semidirect
3. #22559060	C₂	semidirect
4. Q₁₆×C₆₇⋊C₈	C₂	semidirect
70. #11761612	Q₁₆	semidirect
155. C₆₇	#18068	semidirect

Maximal subgroups

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

subgroup	index
#22569158	2
#22559106	2
#22559060	2
Q₁₆×C₆₇⋊C₈	2
C₂×#16863149	2
#22558612	2
C₈×#11761612	2
C₆₇×#3170	2
#18068	67