Subgroups — #30749661

Order 17152.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22560189	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863149	C₂	1	2	yes	maximal, complemented
4	#22558044	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
13	#16863151	—	2	1	no	—
14	#16863149	—	2	4	no	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	#336×C₆₇	C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	—	2	4	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₂²×C₆₇⋊C₈	—	4	2	no	—
25	C₂²×C₆₇⋊C₈	—	4	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
31	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²⋊C₈×C₆₇	—	2	2	no	—
44	C₂²⋊C₈×C₆₇	C₈	1	4	yes	complemented
45	C₄×C₈×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	4	8	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	1	no	—
52	C₂×C₆₇⋊C₈	—	4	1	no	—
53	C₂×C₆₇⋊C₈	—	4	1	no	—
54	C₂×C₆₇⋊C₈	—	4	1	no	—
55	C₂×C₆₇⋊C₈	—	4	1	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	1	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	4	1	no	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
72	C₂²×C₄×C₆₇	D₈	1	2	yes	—
73	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	complemented
83	C₄²×C₆₇	—	2	2	no	—
84	C₄²×C₆₇	—	2	1	no	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	C₂×D₄	1	2	yes	—
88	C₄²×C₆₇	#41	1	2	yes	—
89	C₂³×C₆₇	—	2	1	no	—
90	C₂³×C₆₇	—	2	1	no	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	1	no	—
93	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
94	C₂³×C₆₇	#107	1	2	yes	—
95	C₂³×C₆₇	#108	1	2	yes	—
96	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₆₇	#107	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#131	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	—	4	2	no	—
128	C₆₇⋊C₈	—	4	4	no	—
129	C₆₇⋊C₈	—	8	1	no	—
130	C₆₇⋊C₈	—	8	1	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	#335	1	4	yes	complemented
138	C₂²×C₆₇	#385	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₆₇	#424	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
142	C₂²×C₆₇	#428	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
144	C₂²×C₆₇	#430	1	1	yes	characteristic
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	4	1	no	—
149	C₄×C₆₇	C₈×D₄	1	2	yes	—
150	C₄×C₆₇	#435	1	2	yes	—
151	C₄×C₆₇	#447	1	1	yes	characteristic
152	C₄×C₆₇	#451	1	1	yes	characteristic
153	C₄×C₆₇	#480	1	1	yes	characteristic
154	C₄×C₆₇	#484	1	1	yes	characteristic
155	#8829	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	—	2	4	no	—
157	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
158	C₂×C₆₇	#1565	1	1	yes	characteristic
159	C₂×C₆₇	#1571	1	1	yes	characteristic
160	C₂×C₆₇	#1573	1	1	yes	characteristic
161	C₂×C₆₇	#1850	1	1	yes	characteristic
162	C₂×C₆₇	#1852	1	1	yes	characteristic
163	C₂×C₆₇	#1855	1	1	yes	characteristic
164	C₂²×C₄⋊C₈	—	67	1	no	—
165	#1797	—	67	1	no	—
166	C₄×C₂²⋊C₈	D₆₇	1	1	yes	characteristic
167	C₂×#335	—	67	2	no	—
168	#1277	—	67	2	no	—
169	C₆₇	#8829	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₈	—	134	1	no	—
171	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	67	2	no	—
173	C₂×C₄⋊C₈	—	67	1	no	—
174	C₂×C₄⋊C₈	—	67	1	no	—
175	C₂×C₄⋊C₈	—	134	4	no	—
176	C₂×C₄⋊C₈	—	134	1	no	—
177	C₂×C₂²⋊C₈	—	134	1	no	—
178	C₂×C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
181	#336	—	134	1	no	—
182	#336	Dic₆₇	1	2	yes	—
183	#335	—	134	4	no	—
184	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
185	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₈	—	268	2	no	—
188	C₂²×C₈	—	268	2	no	—
189	C₂²×C₈	—	134	1	no	—
190	C₂²×C₈	—	134	1	no	—
191	C₂²×C₈	—	134	4	no	—
192	C₂²×C₈	—	134	2	no	—
193	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
194	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
198	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
201	C₄⋊C₈	—	67	4	no	—
202	C₄⋊C₈	—	134	4	no	—
203	C₄⋊C₈	—	134	2	no	—
204	C₄⋊C₈	—	134	2	no	—
205	C₄⋊C₈	—	134	2	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	—	268	2	no	—
208	C₂²⋊C₈	C₆₇⋊C₈	1	4	yes	complemented
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	2	no	—
211	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
225	C₂²×C₄	#11761612	1	2	yes	—
226	C₂²×C₄	D₅₃₆	1	2	yes	—
227	C₂²×C₄	Dic₂₆₈	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	268	8	no	—
237	C₂×C₈	—	268	2	no	—
238	C₂×C₈	—	268	2	no	—
239	C₂×C₈	—	268	1	no	—
240	C₂×C₈	—	268	1	no	—
241	C₂×C₈	—	268	1	no	—
242	C₂×C₈	—	268	1	no	—
243	C₂×C₈	—	268	1	no	—
244	C₂×C₈	—	268	1	no	—
245	C₂×C₈	—	134	4	no	—
246	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	complemented
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	D₄×D₆₇	1	2	yes	—
252	C₄²	#11761627	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
258	C₂³	#14523977	1	2	yes	—
259	C₂³	#14523979	1	2	yes	—
260	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
278	C₂×C₄	#14523977	1	2	yes	—
279	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
280	C₂×C₄	#14524013	1	1	yes	characteristic
281	C₂×C₄	#14524020	1	2	yes	—
282	C₂×C₄	#14524027	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
285	C₂×C₄	#14524050	1	2	yes	—
286	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
287	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
288	C₂×C₄	#14524071	1	1	yes	characteristic
289	C₂×C₄	#14524076	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	268	4	no	—
293	C₈	—	536	1	no	—
294	C₈	—	536	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#16863149	1	4	yes	complemented
302	C₂²	#16863271	1	1	yes	characteristic
303	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
304	C₂²	#16863423	1	1	yes	characteristic
305	C₂²	C₂×#14523977	1	1	yes	characteristic
306	C₂²	#16863438	1	1	yes	characteristic
307	C₂²	C₂×#14523979	1	1	yes	characteristic
308	C₂²	#16863445	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
314	C₄	#16863468	1	2	yes	—
315	C₄	#16863532	1	1	yes	characteristic
316	C₄	#16863554	1	1	yes	characteristic
317	C₄	#16863710	1	1	yes	characteristic
318	C₄	#16863732	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×#16863149	1	1	yes	characteristic
321	C₂	#22558961	1	1	yes	characteristic
322	C₂	#22558984	1	1	yes	characteristic
323	C₂	#22558994	1	1	yes	characteristic
324	C₂	#22560389	1	1	yes	characteristic
325	C₂	#22560399	1	1	yes	characteristic
326	C₂	#22560416	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₂²⋊C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₆₇
hypocenter	C₆₇	#8829
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16863149	C₂	semidirect
4. #22558044	C₂	semidirect
44. C₂²⋊C₈×C₆₇	C₈	semidirect
82. C₂×C₈×C₆₇	C₂×C₈	semidirect
137. C₂²×C₆₇	#335	semidirect
169. C₆₇	#8829	semidirect
208. C₂²⋊C₈	C₆₇⋊C₈	semidirect
246. C₂×C₈	C₂×C₆₇⋊C₈	semidirect
301. C₂²	#16863149	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
#22560189	2
C₂×#16863149	2
#22558044	2
C₄×C₂²⋊C₈×C₆₇	2
#8829	67