Subgroups — #30827519

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569295	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559106	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559083	C₂	1	2	yes	maximal, complemented
5	#22559078	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559015	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558971	C₂	1	1	yes	maximal, characteristic
9	#22558654	C₂	1	2	yes	maximal
10	C₈×#11761611	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₆₇	C₂²	1	2	yes	—
14	#16863527	C₂²	1	2	yes	—
15	#16863525	—	2	2	no	—
16	#16863524	—	2	2	no	—
17	C₈×Dic₁₃₄	C₂²	1	2	yes	—
18	#16863520	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic
21	#16863490	C₄	1	1	yes	characteristic
22	C₄×#11761618	C₂²	1	2	yes	—
23	#16863488	C₄	1	2	yes	—
24	#16863452	C₂²	1	2	yes	—
25	#16863451	—	2	2	no	—
26	#16863448	—	2	2	no	—
27	#16863447	C₂²	1	1	yes	characteristic
28	#107×D₆₇	C₄	1	1	yes	characteristic
29	#103×D₆₇	C₄	1	2	yes	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863117	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761618	C₂×C₄	1	2	yes	—
48	Q₁₆×D₆₇	—	2	4	no	—
49	C₂×#11761611	—	2	1	no	—
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
52	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₃₄	—	2	2	no	—
54	C₄×Dic₁₃₄	C₂³	1	2	yes	—
55	#14524008	—	2	2	no	—
56	#14524008	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×D₆₇	—	2	2	no	—
58	C₄⋊C₄×D₆₇	C₂×C₄	1	2	yes	—
59	C₄²×D₆₇	—	2	2	no	—
60	C₄²×D₆₇	C₂³	1	1	yes	characteristic
61	#14523977	C₂×C₄	1	1	yes	characteristic
62	#14523976	C₂×C₄	1	2	yes	—
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
68	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
69	#14523968	C₂×C₄	1	2	yes	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523966	C₂×C₄	1	2	yes	—
72	(C₂×D₆₇)⋊C₈	—	2	2	no	—
73	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₆₇)⋊C₈	D₄	1	1	yes	characteristic
75	#14523954	—	2	2	no	—
76	#14523953	—	2	2	no	—
77	#14523953	—	2	1	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₇	—	2	2	no	—
82	C₈×Dic₆₇	—	2	2	no	—
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
87	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	2	no	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	C₂×C₄	1	2	yes	—
91	#97×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
97	Q₈×D₆₇	—	2	4	no	—
98	Q₈×D₆₇	—	4	2	no	—
99	C₂×C₄×D₆₇	—	2	2	no	—
100	C₂×C₄×D₆₇	—	2	2	no	—
101	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
104	Dic₂₆₈	M₁₆	1	4	yes	complemented
105	#11761618	—	2	4	no	—
106	#11761611	—	2	4	no	—
107	C₂×C₆₇⋊C₈	—	2	1	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₈×D₆₇	—	2	2	no	—
114	C₈×D₆₇	—	4	1	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
117	Dic₆₇⋊C₄	—	2	2	no	—
118	Dic₆₇⋊C₄	—	2	2	no	—
119	Dic₆₇⋊C₄	—	2	2	no	—
120	C₄×Dic₆₇	—	2	2	no	—
121	C₄×Dic₆₇	—	2	2	no	—
122	C₄×Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₆₇	#41	1	1	yes	characteristic
126	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
127	Q₁₆×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	2	no	—
140	Q₈×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
151	C₄×D₆₇	—	2	2	no	—
152	C₄×D₆₇	—	2	1	no	—
153	C₄×D₆₇	—	2	1	no	—
154	C₄×D₆₇	—	2	1	no	—
155	C₄×D₆₇	—	2	1	no	—
156	C₄×D₆₇	—	4	4	no	—
157	C₄×D₆₇	—	4	4	no	—
158	Dic₁₃₄	—	2	2	no	—
159	Dic₁₃₄	—	4	4	no	—
160	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
161	C₈×C₆₇	—	2	2	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
167	C₆₇⋊C₈	—	2	2	no	—
168	C₆₇⋊C₈	—	2	1	no	—
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	2	1	no	—
171	C₆₇⋊C₈	—	2	1	no	—
172	C₆₇⋊C₈	—	4	2	no	—
173	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₆₇	—	2	1	no	—
175	C₂×D₆₇	—	2	1	no	—
176	C₂×D₆₇	#443	1	2	yes	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₆₇	#568	1	1	yes	characteristic
182	C₄×C₆₇	#578	1	1	yes	characteristic
183	C₄×C₆₇	#578	1	1	yes	characteristic
184	Dic₆₇	—	2	4	no	—
185	Dic₆₇	—	2	1	no	—
186	Dic₆₇	—	2	1	no	—
187	Dic₆₇	—	2	1	no	—
188	Dic₆₇	—	4	2	no	—
189	#18058	—	67	1	no	maximal, Sylow
190	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
191	C₂×C₆₇	#2950	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
193	D₆₇	—	2	2	no	—
194	#2984	—	67	2	no	—
195	C₂×C₄×Q₁₆	—	67	1	no	—
196	#1591	—	67	1	no	—
197	#1591	D₆₇	1	1	yes	characteristic, complemented
198	#1587	—	67	2	no	—
199	#1587	—	67	2	no	—
200	C₈×Q₁₆	—	67	1	no	—
201	C₈×Q₁₆	—	67	1	no	—
202	#1568	—	67	1	no	—
203	#1491	—	67	2	no	—
204	C₈×M₁₆	—	67	1	no	—
205	C₆₇	#18058	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₁₆	—	67	1	no	—
207	C₂×C₄×Q₈	—	67	2	no	—
208	#446	—	67	2	no	—
209	#446	—	134	2	no	—
210	#446	—	134	2	no	—
211	#446	C₂×D₆₇	1	2	yes	—
212	C₈×Q₈	—	67	2	no	—
213	C₈×Q₈	—	67	2	no	—
214	C₄×Q₁₆	—	67	1	no	—
215	C₄×Q₁₆	—	67	2	no	—
216	C₄×Q₁₆	—	67	1	no	—
217	C₄×Q₁₆	—	67	2	no	—
218	C₄×Q₁₆	—	67	1	no	—
219	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
220	#432	—	67	2	no	—
221	#432	—	134	2	no	—
222	#431	—	67	1	no	—
223	#431	—	134	2	no	—
224	C₂×#107	—	67	1	no	—
225	C₂×#103	—	67	2	no	—
226	C₄×M₁₆	—	67	1	no	—
227	C₂×C₄×C₈	—	67	1	no	—
228	#335	—	67	1	no	—
229	#335	C₂×D₆₇	1	1	yes	characteristic
230	#334	—	67	2	no	—
231	Q₈⋊C₈	—	67	2	no	—
232	Q₈⋊C₈	—	67	2	no	—
233	Q₈⋊C₈	—	67	2	no	—
234	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
235	#322	—	67	1	no	—
236	#322	—	67	1	no	—
237	C₈²	—	67	1	no	—
238	C₈²	C₂×D₆₇	1	1	yes	characteristic
239	C₂²×Q₈	—	67	2	no	—
240	C₂×Q₁₆	—	67	2	no	—
241	C₂×Q₁₆	—	67	1	no	—
242	C₂×Q₁₆	—	134	4	no	—
243	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
244	C₂×M₁₆	—	134	1	no	—
245	C₂²×C₈	—	67	1	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₄×Q₈	—	67	2	no	—
248	C₄×Q₈	—	67	2	no	—
249	C₄×Q₈	—	67	2	no	—
250	C₄×Q₈	—	134	2	no	—
251	C₄×Q₈	—	134	2	no	—
252	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
253	C₂×C₄⋊C₄	—	67	2	no	—
254	C₂×C₄⋊C₄	—	134	2	no	—
255	C₂×C₄²	—	67	1	no	—
256	C₂×C₄²	—	134	2	no	—
257	#107	—	67	1	no	—
258	#107	—	67	2	no	—
259	#107	C₄×D₆₇	1	1	yes	characteristic
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	67	1	no	—
262	C₄⋊C₈	—	67	1	no	—
263	C₄⋊C₈	—	67	2	no	—
264	C₄⋊C₈	—	134	2	no	—
265	C₄⋊C₈	—	134	2	no	—
266	C₄⋊C₈	—	134	2	no	—
267	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
268	#103	—	67	2	no	—
269	#103	—	67	2	no	—
270	#103	—	67	2	no	—
271	#103	C₄×D₆₇	1	2	yes	—
272	C₂²⋊C₈	—	67	1	no	—
273	C₂²⋊C₈	—	67	1	no	—
274	C₂²⋊C₈	—	134	2	no	—
275	#97	—	2	2	no	—
276	#97	—	134	2	no	—
277	#97	—	134	1	no	—
278	#97	—	134	2	no	—
279	C₄×C₈	—	67	1	no	—
280	C₄×C₈	—	67	1	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	134	2	no	—
285	C₄×C₈	—	134	2	no	—
286	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₂×Q₈	—	268	2	no	—
290	C₂×Q₈	—	67	2	no	—
291	C₂×Q₈	—	134	2	no	—
292	C₂×Q₈	—	134	4	no	—
293	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
294	C₂²×C₄	—	67	1	no	—
295	C₂²×C₄	—	67	1	no	—
296	C₂²×C₄	—	67	1	no	—
297	C₂²×C₄	—	134	2	no	—
298	C₂²×C₄	—	134	2	no	—
299	Q₁₆	—	2	2	no	—
300	Q₁₆	—	67	4	no	—
301	Q₁₆	—	134	4	no	—
302	M₁₆	—	134	4	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	268	1	no	—
306	C₂×C₈	—	67	1	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	67	1	no	—
309	C₂×C₈	—	67	1	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	1	no	—
312	C₂×C₈	—	134	2	no	—
313	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
315	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
316	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	67	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	—	134	2	no	—
321	C₄⋊C₄	—	134	2	no	—
322	C₄⋊C₄	—	134	2	no	—
323	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
324	C₄²	—	2	2	no	—
325	C₄²	—	67	1	no	—
326	C₄²	—	67	1	no	—
327	C₄²	—	67	1	no	—
328	C₄²	—	134	2	no	—
329	C₄²	—	134	2	no	—
330	C₄²	—	134	2	no	—
331	C₄²	C₂³×D₆₇	1	1	yes	characteristic
332	C₂³	—	67	1	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	268	4	no	—
336	Q₈	—	67	4	no	—
337	Q₈	—	134	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	268	4	no	—
341	C₂×C₄	—	268	4	no	—
342	C₂×C₄	—	67	1	no	—
343	C₂×C₄	—	67	1	no	—
344	C₂×C₄	—	67	1	no	—
345	C₂×C₄	—	134	2	no	—
346	C₂×C₄	—	134	2	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	1	no	—
349	C₂×C₄	—	134	1	no	—
350	C₂×C₄	—	134	1	no	—
351	C₂×C₄	—	134	1	no	—
352	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
354	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	2	no	—
360	C₈	—	268	2	no	—
361	C₈	—	134	2	no	—
362	C₈	—	134	1	no	—
363	C₈	—	134	1	no	—
364	C₈	—	134	1	no	—
365	C₈	—	134	1	no	—
366	C₈	M₁₆×D₆₇	1	2	yes	—
367	C₂²	—	67	2	no	—
368	C₂²	—	134	1	no	—
369	C₂²	—	134	1	no	—
370	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	4	2	no	—
373	C₄	—	4	2	no	—
374	C₄	—	268	2	no	—
375	C₄	—	134	1	no	—
376	C₄	—	134	1	no	—
377	C₄	—	134	4	no	—
378	C₄	—	134	1	no	—
379	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
380	C₄	#16864161	1	1	yes	characteristic
381	C₄	#16864209	1	1	yes	characteristic
382	C₄	#16864210	1	1	yes	characteristic
383	C₂	—	134	2	no	—
384	C₂	#22568945	1	1	yes	characteristic
385	C₂	#22569072	1	1	yes	characteristic
386	C₂	#443×D₆₇	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×#1591	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1591	D₆₇
hypocenter	C₆₇	#18058
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #22559106	C₂	semidirect
4. #22559083	C₂	semidirect
5. #22559078	C₂	semidirect
6. C₈×Dic₂₆₈	C₂	semidirect
7. #22559015	C₂	semidirect
11. C₆₇×#1591	C₂	semidirect
104. Dic₂₆₈	M₁₆	semidirect
197. #1591	D₆₇	semidirect
205. C₆₇	#18058	semidirect

Maximal subgroups

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

subgroup	index
#22569295	2
C₄×Q₁₆×D₆₇	2
#22559106	2
#22559083	2
#22559078	2
C₈×Dic₂₆₈	2
#22559015	2
#22558971	2
#22558654	2
C₈×#11761611	2
C₆₇×#1591	2
#18058	67