Subgroups — #30827642

Order 17152.

578 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570792	C₂	1	1	yes	maximal, characteristic
2	#22569151	C₂	1	2	yes	maximal
3	#22559119	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559118	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559083	C₂	1	2	yes	maximal, complemented
6	#22559080	C₂	1	2	yes	maximal, complemented
7	#22559027	C₂	1	1	yes	maximal, characteristic, complemented
8	#334×D₆₇	C₂	1	1	yes	maximal, characteristic
9	#22558610	C₂	1	2	yes	maximal
10	#22558495	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863950	C₂²	1	2	yes	—
13	#16863800	C₄	1	1	yes	characteristic
14	#16863774	C₄	1	1	yes	characteristic
15	#16863745	C₄	1	2	yes	—
16	#16863638	C₄	1	1	yes	characteristic
17	#16863636	C₄	1	2	yes	—
18	#16863634	C₄	1	1	yes	characteristic
19	#16863527	C₂²	1	2	yes	—
20	#16863526	—	2	2	no	—
21	#16863526	C₂²	1	2	yes	—
22	#16863525	—	2	2	no	—
23	#16863520	C₂²	1	2	yes	—
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	C₄×#11761618	C₂²	1	2	yes	—
26	#16863447	C₂²	1	1	yes	characteristic
27	#16863421	—	2	2	no	—
28	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
29	#16863350	—	2	2	no	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863147	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	#16863117	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761618	C₂×C₄	1	2	yes	—
47	C₂×#11761611	D₄	1	2	yes	—
48	C₂×C₈×D₆₇	—	2	2	no	—
49	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	#14524042	C₂×C₄	1	2	yes	—
52	#14524041	C₂×C₄	1	2	yes	—
53	#14524033	—	2	2	no	—
54	#14524023	C₂×C₄	1	2	yes	—
55	#14524021	—	2	2	no	—
56	#14524019	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₃₄	—	2	2	no	—
58	C₄×Dic₁₃₄	C₂³	1	2	yes	—
59	#14524000	—	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	—	2	1	no	—
68	Dic₆₇⋊C₈	—	2	2	no	—
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	1	no	—
73	#14523954	—	2	2	no	—
74	#14523953	—	2	2	no	—
75	#14523953	D₄	1	2	yes	—
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₆₇	—	2	2	no	—
80	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
84	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	2	no	—
86	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₆₇	—	2	2	no	—
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₃₄	—	2	2	no	—
93	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
97	Dic₂₆₈	—	2	2	no	—
98	#11761618	—	4	2	no	—
99	#11761611	—	2	4	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
105	C₈×D₆₇	—	2	2	no	—
106	C₈×D₆₇	—	4	1	no	—
107	C₈×D₆₇	—	4	4	no	—
108	C₄⋊Dic₆₇	—	2	2	no	—
109	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
110	Dic₆₇⋊C₄	—	2	2	no	—
111	Dic₆₇⋊C₄	—	2	2	no	—
112	Dic₆₇⋊C₄	—	2	2	no	—
113	Dic₆₇⋊C₄	—	2	2	no	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	(C₂×D₆₇)⋊C₄	—	2	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	C₄×Dic₆₇	—	2	2	no	—
118	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₆₇	#41	1	1	yes	characteristic
121	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
122	Q₁₆×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
133	Q₈×C₆₇	—	2	2	no	—
134	Q₈×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₆₇	—	2	2	no	—
141	C₂×Dic₆₇	—	2	2	no	—
142	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
145	C₄×D₆₇	—	2	1	no	—
146	C₄×D₆₇	—	2	1	no	—
147	C₄×D₆₇	—	4	1	no	—
148	C₄×D₆₇	#137	1	4	yes	—
149	Dic₁₃₄	—	2	2	no	—
150	Dic₁₃₄	—	2	2	no	—
151	Dic₁₃₄	—	8	2	no	—
152	C₈×C₆₇	—	2	1	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	4	1	no	—
155	C₈×C₆₇	—	4	2	no	—
156	C₈×C₆₇	#131	1	2	yes	—
157	C₆₇⋊C₈	—	2	1	no	—
158	C₆₇⋊C₈	—	2	1	no	—
159	C₆₇⋊C₈	—	2	2	no	—
160	C₆₇⋊C₈	—	4	1	no	—
161	C₆₇⋊C₈	—	4	2	no	—
162	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×D₆₇	—	2	1	no	—
164	C₂×D₆₇	—	2	1	no	—
165	C₂×D₆₇	#441	1	2	yes	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
170	C₄×C₆₇	#568	1	1	yes	characteristic
171	C₄×C₆₇	C₂×#137	1	2	yes	—
172	Dic₆₇	—	2	1	no	—
173	Dic₆₇	—	2	1	no	—
174	Dic₆₇	—	4	2	no	—
175	Dic₆₇	—	4	2	no	—
176	Dic₆₇	#441	1	2	yes	—
177	#18072	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	#2935	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
180	C₂×C₆₇	#2957	1	1	yes	characteristic
181	D₆₇	—	2	2	no	—
182	#3170	—	67	1	no	—
183	#2962	—	67	2	no	—
184	#1593	—	67	1	no	—
185	#1593	—	67	1	no	—
186	#1587	—	67	2	no	—
187	#1587	—	67	2	no	—
188	#1579	—	67	1	no	—
189	#1579	D₆₇	1	1	yes	characteristic, complemented
190	C₂×#334	—	67	1	no	—
191	#1481	—	67	2	no	—
192	#1451	—	67	1	no	—
193	C₆₇	#18072	1	1	yes	characteristic, complemented, Sylow
194	#533	—	67	2	no	—
195	#500	—	67	1	no	—
196	#494	—	67	1	no	—
197	#487	—	67	2	no	—
198	#467	—	67	1	no	—
199	#467	—	67	1	no	—
200	#467	—	67	2	no	—
201	#446	—	67	2	no	—
202	#446	—	67	2	no	—
203	#446	—	134	2	no	—
204	#446	—	134	2	no	—
205	C₈×Q₈	—	67	2	no	—
206	C₈×Q₈	C₂×D₆₇	1	2	yes	—
207	C₄×Q₁₆	—	67	2	no	—
208	C₄×Q₁₆	—	67	1	no	—
209	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
210	#431	—	67	1	no	—
211	#424	—	134	2	no	—
212	C₂×C₄⋊C₈	—	67	2	no	—
213	#405	—	134	2	no	—
214	C₄×M₁₆	—	67	1	no	—
215	C₂×C₄×C₈	—	67	1	no	—
216	#334	—	67	2	no	—
217	#334	—	67	1	no	—
218	#334	C₂×D₆₇	1	1	yes	characteristic
219	Q₈⋊C₈	—	67	2	no	—
220	Q₈⋊C₈	—	67	2	no	—
221	Q₈⋊C₈	—	67	2	no	—
222	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
223	#322	—	67	1	no	—
224	#322	—	67	1	no	—
225	#322	—	67	1	no	—
226	#322	C₂×D₆₇	1	1	yes	characteristic
227	C₂×Q₁₆	—	67	2	no	—
228	C₂×Q₁₆	—	67	1	no	—
229	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
230	C₂×M₁₆	—	67	2	no	—
231	C₂²×C₈	—	67	1	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	—	134	2	no	—
234	C₄⋊Q₈	—	67	2	no	—
235	C₄⋊Q₈	—	67	2	no	—
236	#125	—	134	2	no	—
237	C₂²⋊Q₈	—	67	2	no	—
238	C₂²⋊Q₈	—	67	2	no	—
239	C₂²⋊Q₈	—	134	2	no	—
240	C₄×Q₈	—	67	2	no	—
241	C₄×Q₈	—	134	2	no	—
242	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
243	#117	—	134	2	no	—
244	C₂×C₄²	—	67	1	no	—
245	#107	—	67	1	no	—
246	#107	—	67	2	no	—
247	#107	C₄×D₆₇	1	1	yes	characteristic
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	67	2	no	—
250	C₄⋊C₈	—	134	2	no	—
251	C₄⋊C₈	—	134	2	no	—
252	C₄⋊C₈	—	134	1	no	—
253	C₄⋊C₈	—	134	2	no	—
254	C₄⋊C₈	—	134	2	no	—
255	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
256	#103	—	67	2	no	—
257	#103	—	67	2	no	—
258	#103	—	67	2	no	—
259	#103	C₄×D₆₇	1	2	yes	—
260	C₂²⋊C₈	—	134	1	no	—
261	#97	—	67	2	no	—
262	#97	—	134	2	no	—
263	#97	—	134	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	67	1	no	—
266	C₄×C₈	—	67	1	no	—
267	C₄×C₈	—	67	1	no	—
268	C₄×C₈	—	67	1	no	—
269	C₄×C₈	—	67	1	no	—
270	C₄×C₈	—	134	2	no	—
271	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
274	C₂×Q₈	—	67	2	no	—
275	C₂×Q₈	—	134	2	no	—
276	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
277	C₂²×C₄	—	67	1	no	—
278	C₂²×C₄	—	67	1	no	—
279	C₂²×C₄	—	67	1	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	268	2	no	—
282	Q₁₆	—	134	2	no	—
283	M₁₆	—	134	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	268	1	no	—
287	C₂×C₈	—	268	4	no	—
288	C₂×C₈	—	67	1	no	—
289	C₂×C₈	—	67	1	no	—
290	C₂×C₈	—	67	2	no	—
291	C₂×C₈	—	134	2	no	—
292	C₂×C₈	—	134	1	no	—
293	C₂×C₈	—	134	2	no	—
294	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₂×C₈	D₄×D₆₇	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	67	2	no	—
299	C₄⋊C₄	—	134	2	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₄⋊C₄	—	134	2	no	—
302	C₄⋊C₄	—	134	2	no	—
303	C₄⋊C₄	—	134	2	no	—
304	C₄⋊C₄	—	134	2	no	—
305	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
306	C₂²⋊C₄	—	134	2	no	—
307	C₂²⋊C₄	—	134	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	67	1	no	—
310	C₄²	—	67	1	no	—
311	C₄²	—	67	1	no	—
312	C₄²	—	134	2	no	—
313	C₄²	C₂³×D₆₇	1	1	yes	characteristic
314	C₂³	—	67	1	no	—
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	—	536	2	no	—
318	Q₈	—	134	2	no	—
319	Q₈	—	134	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	268	1	no	—
323	C₂×C₄	—	67	1	no	—
324	C₂×C₄	—	67	4	no	—
325	C₂×C₄	—	67	1	no	—
326	C₂×C₄	—	67	1	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	1	no	—
330	C₂×C₄	—	134	1	no	—
331	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
333	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	268	1	no	—
339	C₈	—	268	2	no	—
340	C₈	—	134	1	no	—
341	C₈	—	134	1	no	—
342	C₈	—	134	2	no	—
343	C₈	#14524049	1	2	yes	—
344	C₂²	—	67	2	no	—
345	C₂²	—	134	1	no	—
346	C₂²	—	134	1	no	—
347	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	268	2	no	—
352	C₄	—	268	2	no	—
353	C₄	—	67	2	no	—
354	C₄	—	134	1	no	—
355	C₄	—	134	1	no	—
356	C₄	C₂×#14524049	1	1	yes	characteristic
357	C₄	#16864161	1	1	yes	characteristic
358	C₄	#137×D₆₇	1	2	yes	—
359	C₂	—	134	2	no	—
360	C₂	#22568948	1	1	yes	characteristic
361	C₂	#441×D₆₇	1	1	yes	characteristic
362	C₂	#22569123	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×#1579	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1579	D₆₇
hypocenter	C₆₇	#18072
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #22559119	C₂	semidirect
4. #22559118	C₂	semidirect
5. #22559083	C₂	semidirect
6. #22559080	C₂	semidirect
7. #22559027	C₂	semidirect
11. C₆₇×#1579	C₂	semidirect
189. #1579	D₆₇	semidirect
193. C₆₇	#18072	semidirect

Maximal subgroups

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

subgroup	index
#22570792	2
#22569151	2
#22559119	2
#22559118	2
#22559083	2
#22559080	2
#22559027	2
#334×D₆₇	2
#22558610	2
#22558495	2
C₆₇×#1579	2
#18072	67