Subgroups — #30826387

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 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	#22569140	C₂	1	2	yes	maximal
3	#22559119	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559116	C₂	1	2	yes	maximal, complemented
5	#22559028	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559027	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559025	C₂	1	2	yes	maximal, complemented
8	#22558963	C₂	1	1	yes	maximal, characteristic
9	#22558662	C₂	1	2	yes	maximal
10	#322×D₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1593	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	#16863526	C₂²	1	2	yes	—
20	#16863524	—	2	2	no	—
21	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
22	#16863520	—	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	#16863421	—	2	2	no	—
27	#16863418	C₂²	1	2	yes	—
28	#16863353	—	2	2	no	—
29	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₇	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	#16863118	C₂²	1	1	yes	characteristic
39	#16863117	C₂²	1	1	yes	characteristic
40	#446×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	—	2	2	no	—
48	C₂×C₈×D₆₇	—	2	1	no	—
49	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₇	D₄	1	2	yes	—
52	#14524042	C₂×C₄	1	2	yes	—
53	#14524041	C₂×C₄	1	2	yes	—
54	#14524033	—	2	2	no	—
55	#14524023	C₂×C₄	1	2	yes	—
56	#14524021	—	2	2	no	—
57	#14524019	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₃₄	—	2	2	no	—
59	C₄×Dic₁₃₄	C₂³	1	2	yes	—
60	#14524000	—	2	2	no	—
61	C₄²×D₆₇	C₂³	1	1	yes	characteristic
62	#14523977	C₂×C₄	1	1	yes	characteristic
63	#14523976	C₂×C₄	1	2	yes	—
64	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
65	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	2	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	#14523953	—	2	2	no	—
73	C₄×C₆₇⋊C₈	—	2	2	no	—
74	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₆₇	—	2	2	no	—
78	C₈×Dic₆₇	—	2	1	no	—
79	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₇	D₄	1	2	yes	—
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	#97×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	—	2	4	no	—
99	#11761611	—	4	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	4	no	—
106	C₈×D₆₇	—	2	2	no	—
107	C₈×D₆₇	—	2	2	no	—
108	C₈×D₆₇	—	4	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
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	Dic₆₇⋊C₄	—	2	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	2	2	no	—
118	C₄×Dic₆₇	—	2	2	no	—
119	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₆₇	#41	1	1	yes	characteristic
122	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	2	no	—
135	Q₈×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₆₇	—	2	2	no	—
142	C₂×Dic₆₇	—	2	2	no	—
143	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
146	C₄×D₆₇	—	2	1	no	—
147	C₄×D₆₇	—	2	1	no	—
148	C₄×D₆₇	—	2	2	no	—
149	C₄×D₆₇	#137	1	4	yes	—
150	Dic₁₃₄	—	2	2	no	—
151	Dic₁₃₄	—	2	2	no	—
152	Dic₁₃₄	—	4	4	no	—
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	—	2	2	no	—
155	C₈×C₆₇	—	4	1	no	—
156	C₈×C₆₇	—	4	2	no	—
157	C₈×C₆₇	#131	1	2	yes	—
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	C₆₇⋊C₈	#131	1	2	yes	—
163	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₆₇	—	2	1	no	—
165	C₂×D₆₇	—	2	1	no	—
166	C₂×D₆₇	#441	1	2	yes	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
171	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
172	C₄×C₆₇	C₂×#137	1	2	yes	—
173	Dic₆₇	—	2	1	no	—
174	Dic₆₇	—	4	2	no	—
175	Dic₆₇	—	4	2	no	—
176	Dic₆₇	#441	1	2	yes	—
177	Dic₆₇	#444	1	2	yes	—
178	#17961	—	67	1	no	maximal, Sylow
179	C₂×C₆₇	#2931	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
181	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
182	D₆₇	—	2	2	no	—
183	#3170	—	67	1	no	—
184	#2960	—	67	2	no	—
185	#1593	—	67	2	no	—
186	#1593	—	67	1	no	—
187	#1593	D₆₇	1	1	yes	characteristic, complemented
188	#1579	—	67	1	no	—
189	#1579	—	67	1	no	—
190	#1579	—	67	2	no	—
191	#1566	—	67	1	no	—
192	#1493	—	67	2	no	—
193	C₂×#322	—	67	1	no	—
194	C₆₇	#17961	1	1	yes	characteristic, complemented, Sylow
195	#533	—	67	2	no	—
196	#500	—	67	1	no	—
197	#494	—	67	1	no	—
198	#487	—	67	2	no	—
199	#467	—	67	1	no	—
200	#467	—	67	1	no	—
201	#467	—	67	2	no	—
202	#446	—	67	2	no	—
203	#446	—	134	2	no	—
204	#446	C₂×D₆₇	1	2	yes	—
205	C₈×Q₈	—	67	2	no	—
206	C₈×Q₈	—	67	2	no	—
207	C₈×Q₈	—	134	2	no	—
208	C₄×Q₁₆	—	67	2	no	—
209	C₄×Q₁₆	—	67	1	no	—
210	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
211	#424	—	134	2	no	—
212	#423	—	67	2	no	—
213	#405	—	134	2	no	—
214	C₂×C₄×C₈	—	67	1	no	—
215	C₂×C₄×C₈	—	67	1	no	—
216	C₂×C₄×C₈	—	67	1	no	—
217	#334	—	67	2	no	—
218	#334	—	67	1	no	—
219	#334	C₂×D₆₇	1	1	yes	characteristic
220	Q₈⋊C₈	—	67	2	no	—
221	Q₈⋊C₈	—	67	2	no	—
222	Q₈⋊C₈	—	67	2	no	—
223	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
224	#322	—	67	1	no	—
225	#322	—	67	1	no	—
226	#322	—	67	1	no	—
227	#322	C₂×D₆₇	1	1	yes	characteristic
228	C₂×Q₁₆	—	67	2	no	—
229	C₂×Q₁₆	—	67	1	no	—
230	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
231	C₂×M₁₆	—	134	2	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	—	67	1	no	—
234	C₂²×C₈	—	67	2	no	—
235	C₂²×C₈	—	134	1	no	—
236	C₄⋊Q₈	—	67	2	no	—
237	C₄⋊Q₈	—	67	2	no	—
238	#125	—	134	2	no	—
239	C₂²⋊Q₈	—	67	2	no	—
240	C₂²⋊Q₈	—	67	2	no	—
241	C₂²⋊Q₈	—	134	2	no	—
242	C₄×Q₈	—	67	2	no	—
243	C₄×Q₈	—	134	2	no	—
244	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
245	#117	—	134	2	no	—
246	C₂×C₄²	—	67	1	no	—
247	#107	—	67	1	no	—
248	#107	—	67	2	no	—
249	#107	C₄×D₆₇	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	67	2	no	—
252	C₄⋊C₈	—	134	2	no	—
253	C₄⋊C₈	—	134	2	no	—
254	C₄⋊C₈	—	134	2	no	—
255	C₄⋊C₈	—	134	2	no	—
256	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
257	#103	—	67	2	no	—
258	#103	—	67	2	no	—
259	#103	—	67	2	no	—
260	#103	C₄×D₆₇	1	2	yes	—
261	#97	—	2	2	no	—
262	#97	—	134	2	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	1	no	—
265	C₄×C₈	—	67	2	no	—
266	C₄×C₈	—	67	1	no	—
267	C₄×C₈	—	67	1	no	—
268	C₄×C₈	—	67	1	no	—
269	C₄×C₈	—	134	2	no	—
270	C₄×C₈	—	134	1	no	—
271	C₄×C₈	—	134	2	no	—
272	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	C₂×Q₈	—	67	2	no	—
276	C₂×Q₈	—	134	2	no	—
277	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
278	C₂²×C₄	—	67	1	no	—
279	C₂²×C₄	—	67	1	no	—
280	C₂²×C₄	—	67	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	134	4	no	—
283	Q₁₆	—	134	2	no	—
284	M₁₆	—	268	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	268	2	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	4	no	—
293	C₂×C₈	—	134	2	no	—
294	C₂×C₈	—	134	1	no	—
295	C₂×C₈	—	134	2	no	—
296	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
298	C₂×C₈	D₄×D₆₇	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	67	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	—	134	2	no	—
306	C₄⋊C₄	—	134	2	no	—
307	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
308	C₂²⋊C₄	—	134	2	no	—
309	C₂²⋊C₄	—	134	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	67	1	no	—
312	C₄²	—	67	1	no	—
313	C₄²	—	67	1	no	—
314	C₄²	—	134	2	no	—
315	C₄²	C₂³×D₆₇	1	1	yes	characteristic
316	C₂³	—	67	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	268	4	no	—
320	Q₈	—	134	2	no	—
321	Q₈	—	134	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	67	1	no	—
325	C₂×C₄	—	67	1	no	—
326	C₂×C₄	—	67	4	no	—
327	C₂×C₄	—	67	1	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	—	134	1	no	—
332	C₂×C₄	—	134	1	no	—
333	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
335	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	268	1	no	—
341	C₈	—	268	2	no	—
342	C₈	—	67	2	no	—
343	C₈	—	134	1	no	—
344	C₈	—	134	2	no	—
345	C₈	#14524051	1	2	yes	—
346	C₂²	—	67	2	no	—
347	C₂²	—	134	1	no	—
348	C₂²	—	134	1	no	—
349	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	268	2	no	—
354	C₄	—	268	2	no	—
355	C₄	—	67	2	no	—
356	C₄	—	67	2	no	—
357	C₄	—	134	1	no	—
358	C₄	#131×D₆₇	1	1	yes	characteristic
359	C₄	C₂×#14524051	1	1	yes	characteristic
360	C₄	#137×D₆₇	1	2	yes	—
361	C₂	—	134	2	no	—
362	C₂	#22568918	1	1	yes	characteristic
363	C₂	#441×D₆₇	1	1	yes	characteristic
364	C₂	#444×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×#1593	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1593	D₆₇
hypocenter	C₆₇	#17961
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #22559119	C₂	semidirect
4. #22559116	C₂	semidirect
5. #22559028	C₂	semidirect
6. #22559027	C₂	semidirect
7. #22559025	C₂	semidirect
11. C₆₇×#1593	C₂	semidirect
187. #1593	D₆₇	semidirect
194. C₆₇	#17961	semidirect

Maximal subgroups

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

subgroup	index
#22570792	2
#22569140	2
#22559119	2
#22559116	2
#22559028	2
#22559027	2
#22559025	2
#22558963	2
#22558662	2
#322×D₆₇	2
C₆₇×#1593	2
#17961	67