Subgroups — #30827628

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569214	C₂	1	2	yes	maximal, complemented
2	#22559100	C₂	1	2	yes	maximal, complemented
3	#22559060	C₂	1	2	yes	maximal, complemented
4	#22559039	C₂	1	2	yes	maximal, complemented
5	SD₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558617	C₂	1	2	yes	maximal, complemented
8	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3168	C₂	1	1	yes	maximal, characteristic
10	#16863527	C₂²	1	2	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
12	#16863474	—	2	2	no	—
13	#16863474	C₂²	1	2	yes	—
14	#16863468	—	2	2	no	—
15	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
16	#16863457	—	2	2	no	—
17	#16863449	C₂²	1	2	yes	—
18	#16863449	C₂²	1	1	yes	characteristic
19	#16863352	—	2	2	no	—
20	C₄×#11761612	C₂²	1	1	yes	characteristic
21	#16863149	C₂²	1	2	yes	—
22	#16863147	C₂²	1	1	yes	characteristic
23	#16863147	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
25	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
26	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
28	#16863118	C₂²	1	2	yes	—
29	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
30	#529×C₆₇	C₂²	1	2	yes	—
31	#499×C₆₇	C₄	1	1	yes	characteristic
32	#495×C₆₇	C₄	1	1	yes	characteristic
33	#488×C₆₇	C₄	1	2	yes	—
34	#467×C₆₇	C₄	1	1	yes	characteristic
35	#466×C₆₇	C₄	1	1	yes	characteristic
36	#465×C₆₇	C₄	1	2	yes	—
37	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
39	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761612	—	2	2	no	—
42	C₂×#11761612	D₄	1	2	yes	—
43	C₂²×C₆₇⋊C₈	—	2	2	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
53	#14523954	—	2	2	no	—
54	#14523954	D₄	1	1	yes	characteristic
55	#14523954	D₄	1	1	yes	characteristic
56	C₄×C₆₇⋊C₈	—	2	2	no	—
57	C₄×C₆₇⋊C₈	—	2	2	no	—
58	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#126×C₆₇	—	2	2	no	—
66	#125×C₆₇	C₂×C₄	1	2	yes	—
67	#124×C₆₇	C₂×C₄	1	2	yes	—
68	#123×C₆₇	—	2	2	no	—
69	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
70	#121×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	C₂³	1	2	yes	—
74	#117×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
77	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
78	#106×C₆₇	C₂×C₄	1	2	yes	—
79	#103×C₆₇	C₂×C₄	1	2	yes	—
80	#103×C₆₇	C₂×C₄	1	2	yes	—
81	#102×C₆₇	C₂×C₄	1	2	yes	—
82	#102×C₆₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#11761612	—	2	4	no	—
88	#11761612	—	4	4	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	4	4	no	—
93	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
102	Q₁₆×C₆₇	—	4	1	no	—
103	SD₁₆×C₆₇	—	2	4	no	—
104	D₈×C₆₇	—	4	1	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	#41	1	1	yes	characteristic
126	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	2	no	—
130	Q₈×C₆₇	—	2	2	no	—
131	D₄×C₆₇	—	2	2	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	8	2	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₇	—	2	1	no	—
150	C₈×C₆₇	—	2	1	no	—
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	#131	1	2	yes	—
153	C₆₇⋊C₈	—	2	2	no	—
154	C₆₇⋊C₈	—	4	2	no	—
155	C₆₇⋊C₈	—	4	2	no	—
156	C₆₇⋊C₈	—	4	2	no	—
157	C₆₇⋊C₈	#135	1	4	yes	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	4	2	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	—	4	2	no	—
164	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
173	C₄×C₆₇	#568	1	1	yes	characteristic
174	C₄×C₆₇	C₂×#135	1	2	yes	—
175	#18070	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	4	1	no	—
177	C₂×C₆₇	—	8	2	no	—
178	C₂×C₆₇	#2935	1	1	yes	characteristic
179	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
180	C₂×C₆₇	#2957	1	1	yes	characteristic
181	#3168	D₆₇	1	1	yes	characteristic
182	#2969	—	67	2	no	—
183	#1590	—	67	2	no	—
184	#1584	—	67	2	no	—
185	#1581	—	67	2	no	—
186	C₈×SD₁₆	—	67	2	no	—
187	#1564	—	67	1	no	—
188	#1482	—	67	2	no	—
189	C₈×M₁₆	—	67	1	no	—
190	C₆₇	#18070	1	1	yes	characteristic, complemented, Sylow
191	#529	C₂×D₆₇	1	2	yes	—
192	#499	Dic₆₇	1	1	yes	characteristic
193	#495	Dic₆₇	1	1	yes	characteristic
194	#488	Dic₆₇	1	2	yes	—
195	#467	Dic₆₇	1	1	yes	characteristic
196	#466	Dic₆₇	1	1	yes	characteristic
197	#465	Dic₆₇	1	2	yes	—
198	#446	—	67	2	no	—
199	C₈×Q₈	—	67	2	no	—
200	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
201	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
202	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
203	#436	—	67	2	no	—
204	#436	—	134	2	no	—
205	#435	—	134	2	no	—
206	C₈×D₄	—	67	2	no	—
207	#433	—	134	2	no	—
208	#431	—	67	1	no	—
209	#431	—	67	2	no	—
210	#405	—	134	2	no	—
211	C₄×M₁₆	—	67	1	no	—
212	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
213	#335	—	67	2	no	—
214	#334	—	67	1	no	—
215	#334	—	67	1	no	—
216	Q₈⋊C₈	—	67	2	no	—
217	Q₈⋊C₈	—	67	2	no	—
218	D₄⋊C₈	—	67	2	no	—
219	D₄⋊C₈	—	67	2	no	—
220	#322	—	67	2	no	—
221	C₈²	—	67	2	no	—
222	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
223	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
224	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
225	C₂×M₁₆	—	67	2	no	—
226	C₂×M₁₆	—	134	2	no	—
227	C₂²×C₈	—	134	2	no	—
228	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
229	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
230	#126	—	2	2	no	—
231	#125	C₂×Dic₆₇	1	2	yes	—
232	#124	C₂×Dic₆₇	1	2	yes	—
233	#123	—	2	2	no	—
234	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
235	#121	C₂×Dic₆₇	1	2	yes	—
236	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	C₂²×D₆₇	1	2	yes	—
239	#117	—	2	2	no	—
240	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
241	#107	C₂×Dic₆₇	1	1	yes	characteristic
242	#107	C₂×Dic₆₇	1	1	yes	characteristic
243	#106	C₂×Dic₆₇	1	2	yes	—
244	C₄⋊C₈	—	67	2	no	—
245	C₄⋊C₈	—	67	2	no	—
246	C₄⋊C₈	—	134	2	no	—
247	C₄⋊C₈	—	134	2	no	—
248	C₄⋊C₈	—	134	1	no	—
249	#103	C₂×Dic₆₇	1	2	yes	—
250	#103	C₂×Dic₆₇	1	2	yes	—
251	#102	C₂×Dic₆₇	1	2	yes	—
252	#102	C₂×Dic₆₇	1	2	yes	—
253	C₂²⋊C₈	—	134	2	no	—
254	C₂²⋊C₈	—	134	2	no	—
255	C₂²⋊C₈	—	134	2	no	—
256	C₂²⋊C₈	—	134	1	no	—
257	#97	—	67	1	no	—
258	#97	—	67	1	no	—
259	#97	—	134	2	no	—
260	C₄×C₈	—	67	2	no	—
261	C₄×C₈	—	67	2	no	—
262	C₄×C₈	—	134	2	no	—
263	C₄×C₈	—	134	2	no	—
264	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
265	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
268	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
274	C₂²×C₄	#11761627	1	1	yes	characteristic
275	Q₁₆	—	4	1	no	—
276	SD₁₆	—	2	4	no	—
277	D₈	—	4	1	no	—
278	M₁₆	—	268	4	no	—
279	M₁₆	—	134	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	268	4	no	—
283	C₂×C₈	—	67	2	no	—
284	C₂×C₈	—	67	2	no	—
285	C₂×C₈	—	134	2	no	—
286	C₂×C₈	—	134	2	no	—
287	C₂×C₈	—	134	2	no	—
288	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
297	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
305	C₄²	D₄×D₆₇	1	1	yes	characteristic
306	C₄²	#11761627	1	1	yes	characteristic
307	C₄²	C₂³×D₆₇	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	Q₈	—	2	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	8	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
325	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
326	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	268	2	no	—
334	C₈	—	268	2	no	—
335	C₈	—	268	2	no	—
336	C₈	—	67	4	no	—
337	C₈	—	134	2	no	—
338	C₈	#14524050	1	2	yes	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	C₂×#14524050	1	1	yes	characteristic
354	C₄	#16864163	1	1	yes	characteristic
355	C₄	#135×D₆₇	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	8	2	no	—
358	C₂	#22568953	1	1	yes	characteristic
359	C₂	#135×Dic₆₇	1	1	yes	characteristic
360	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×#3168	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#466×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3168	D₆₇
hypocenter	C₆₇	#18070
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #22569214	C₂	semidirect
2. #22559100	C₂	semidirect
3. #22559060	C₂	semidirect
4. #22559039	C₂	semidirect
5. SD₁₆×C₆₇⋊C₈	C₂	semidirect
6. #22558957	C₂	semidirect
7. #22558617	C₂	semidirect
8. C₈×#11761612	C₂	semidirect
190. C₆₇	#18070	semidirect

Maximal subgroups

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

subgroup	index
#22569214	2
#22559100	2
#22559060	2
#22559039	2
SD₁₆×C₆₇⋊C₈	2
#22558957	2
#22558617	2
C₈×#11761612	2
C₆₇×#3168	2
#18070	67