Subgroups — #30847348

Order 17152.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572557	C₂	1	1	yes	maximal, characteristic, complemented
2	#22572516	C₂	1	2	yes	maximal, complemented
3	#22571931	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559806	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559757	C₂	1	2	yes	maximal, complemented
6	#22559598	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559574	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559496	C₂	1	2	yes	maximal, complemented
9	#22559495	C₂	1	2	yes	maximal, complemented
10	#22559095	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	—	2	1	no	—
13	#16864051	C₂²	1	2	yes	—
14	#16864050	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863794	C₂²	1	2	yes	—
17	#16863776	C₂²	1	1	yes	characteristic
18	#16863747	C₂²	1	2	yes	—
19	#16863733	—	2	2	no	—
20	#16863710	—	2	2	no	—
21	#16863677	C₂²	1	2	yes	—
22	#16863676	C₂²	1	2	yes	—
23	#16863641	—	2	2	no	—
24	#16863637	—	2	1	no	—
25	#16863632	—	2	1	no	—
26	#16863602	C₂²	1	2	yes	—
27	#16863601	C₂²	1	2	yes	—
28	#16863560	—	2	2	no	—
29	#16863538	—	2	2	no	—
30	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
31	#16863475	C₂²	1	2	yes	—
32	C₂×#14523977	—	2	1	no	—
33	#16863394	—	2	2	no	—
34	#16863149	C₂²	1	1	yes	characteristic
35	#16863143	C₂²	1	2	yes	—
36	#16863141	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761628	—	2	2	no	—
46	C₂×#11761627	—	2	2	no	—
47	#14524065	—	4	4	no	—
48	C₂×Dic₂₆₈	—	2	1	no	—
49	C₂×Dic₂₆₈	—	2	1	no	—
50	C₂×#11761617	—	2	2	no	—
51	C₂×#11761616	—	2	2	no	—
52	C₂×D₅₃₆	D₄	1	1	yes	characteristic
53	C₂×#11761611	—	2	2	no	—
54	#14524043	C₂³	1	2	yes	—
55	#14524043	C₂³	1	2	yes	—
56	#14524027	—	4	2	no	—
57	#14524025	—	4	2	no	—
58	#14524023	—	2	2	no	—
59	#14524022	—	2	2	no	—
60	#14524017	—	2	2	no	—
61	#14524014	—	2	2	no	—
62	C₄×D₂₆₈	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊C₄×D₆₇	—	2	2	no	—
65	#14523977	—	2	2	no	—
66	#14523977	—	2	1	no	—
67	#14523977	—	2	1	no	—
68	#14523977	D₄	1	1	yes	characteristic
69	#14523976	—	2	2	no	—
70	#14523975	—	4	2	no	—
71	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
72	#14523968	—	2	2	no	—
73	#14523968	—	2	2	no	—
74	#14523968	—	2	2	no	—
75	#14523966	—	2	2	no	—
76	#14523965	—	2	2	no	—
77	#14523965	D₄	1	2	yes	—
78	#14523964	—	2	2	no	—
79	(C₂×D₆₇)⋊C₈	—	2	2	no	—
80	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	—	2	1	no	—
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	#102×C₆₇	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	1	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	#11761628	—	4	4	no	—
96	#11761628	—	8	2	no	—
97	#11761627	—	4	4	no	—
98	#11761627	—	8	2	no	—
99	C₂×Dic₁₃₄	—	2	2	no	—
100	C₂×Dic₁₃₄	—	2	2	no	—
101	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
102	C₂×C₂²⋊D₆₇	—	4	2	no	—
103	C₂×C₂²⋊D₆₇	—	4	2	no	—
104	C₂²×Dic₆₇	—	4	2	no	—
105	C₂×C₄×D₆₇	—	2	2	no	—
106	C₂×C₄×D₆₇	—	4	2	no	—
107	Dic₂₆₈	—	4	2	no	—
108	Dic₂₆₈	—	8	1	no	—
109	#11761617	—	4	4	no	—
110	#11761616	—	8	2	no	—
111	D₅₃₆	—	2	2	no	—
112	#11761611	—	8	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₄⋊Dic₆₇	—	2	2	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	—	2	2	no	—
117	C₄⋊Dic₆₇	—	2	2	no	—
118	C₄⋊Dic₆₇	—	4	2	no	—
119	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
120	Dic₆₇⋊C₄	—	4	2	no	—
121	C₂²⋊Dic₆₇	—	4	2	no	—
122	(C₂×D₆₇)⋊C₄	—	2	2	no	—
123	(C₂×D₆₇)⋊C₄	—	4	2	no	—
124	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	D₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	4	2	no	—
133	C₂×C₈×C₆₇	—	4	2	no	—
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₆₇	—	2	1	no	—
141	C₂²⋊C₄×C₆₇	—	2	1	no	—
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂²×D₆₇	—	2	2	no	—
146	D₄×C₆₇	—	2	1	no	—
147	D₄×C₆₇	—	2	1	no	—
148	D₄×C₆₇	—	2	1	no	—
149	D₄×C₆₇	#135	1	2	yes	complemented
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²⋊D₆₇	—	8	4	no	—
160	C₂²⋊D₆₇	—	8	4	no	—
161	C₂×Dic₆₇	—	2	2	no	—
162	C₂×Dic₆₇	—	4	2	no	—
163	C₂×Dic₆₇	—	4	2	no	—
164	C₂×Dic₆₇	—	8	4	no	—
165	D₂₆₈	—	2	2	no	—
166	D₂₆₈	—	2	2	no	—
167	C₄×D₆₇	—	4	2	no	—
168	C₄×D₆₇	—	4	2	no	—
169	C₄×D₆₇	—	8	4	no	—
170	Dic₁₃₄	—	4	2	no	—
171	Dic₁₃₄	—	4	2	no	—
172	Dic₁₃₄	—	4	2	no	—
173	Dic₁₃₄	—	4	2	no	—
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	—	2	1	no	—
176	C₈×C₆₇	—	2	1	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	—	4	1	no	—
179	C₈×C₆₇	—	4	1	no	—
180	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
181	C₆₇⋊C₈	—	4	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	1	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	D₄²	1	1	yes	characteristic
189	C₂×D₆₇	—	4	2	no	—
190	C₂×D₆₇	—	4	2	no	—
191	C₂×D₆₇	—	4	2	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	4	1	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
196	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₆₇	#576	1	1	yes	characteristic
198	C₄×C₆₇	#577	1	1	yes	characteristic
199	Dic₆₇	—	4	2	no	—
200	Dic₆₇	—	8	2	no	—
201	Dic₆₇	—	8	2	no	—
202	#19902	—	67	1	no	maximal, Sylow
203	C₂×C₆₇	—	4	2	no	—
204	C₂×C₆₇	—	8	1	no	—
205	C₂×C₆₇	#3272	1	1	yes	characteristic
206	C₂×C₆₇	#3280	1	1	yes	characteristic
207	C₂×C₆₇	#3293	1	1	yes	characteristic
208	D₆₇	—	8	2	no	—
209	#3334	—	67	1	no	—
210	#3330	—	67	2	no	—
211	#3283	—	67	1	no	—
212	#1707	—	67	1	no	—
213	#1699	—	67	2	no	—
214	#1673	—	67	1	no	—
215	#1669	—	67	1	no	—
216	#1654	—	67	2	no	—
217	#1654	—	67	2	no	—
218	#1589	—	67	1	no	—
219	C₈×D₈	D₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#19902	1	1	yes	characteristic, complemented, Sylow
221	C₂×#135	—	134	1	no	—
222	#547	—	67	2	no	—
223	#547	—	67	2	no	—
224	#500	—	67	1	no	—
225	#498	—	67	2	no	—
226	#494	—	67	1	no	—
227	#487	—	67	2	no	—
228	#484	—	134	2	no	—
229	#480	—	134	2	no	—
230	#474	—	67	2	no	—
231	#474	—	67	2	no	—
232	#468	—	134	2	no	—
233	#467	—	134	1	no	—
234	#466	—	134	1	no	—
235	#461	—	67	2	no	—
236	#461	—	67	2	no	—
237	#452	—	134	2	no	—
238	#449	—	134	2	no	—
239	C₄×D₈	—	67	1	no	—
240	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
241	#436	—	67	2	no	—
242	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
243	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
244	C₂×#107	—	134	1	no	—
245	#417	—	134	2	no	—
246	#335	—	67	1	no	—
247	#335	C₂×D₆₇	1	1	yes	characteristic
248	#332	—	67	2	no	—
249	#332	—	67	2	no	—
250	D₄⋊C₈	—	67	2	no	—
251	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
252	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
253	C₈²	C₂×D₆₇	1	1	yes	characteristic
254	C₂×#41	—	134	2	no	—
255	C₂×#41	—	134	2	no	—
256	#135	—	268	4	no	—
257	C₂×Q₁₆	—	134	1	no	—
258	C₂×Q₁₆	—	134	1	no	—
259	C₂×SD₁₆	—	134	2	no	—
260	C₂×SD₁₆	—	134	2	no	—
261	C₂×D₈	—	67	1	no	—
262	C₂×D₈	D₂₆₈	1	1	yes	characteristic
263	C₂×M₁₆	—	134	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	—	67	2	no	—
267	C₄⋊Q₈	—	67	2	no	—
268	#123	—	268	2	no	—
269	#123	—	268	2	no	—
270	C₂²⋊Q₈	—	134	2	no	—
271	C₂²⋊Q₈	—	134	2	no	—
272	#121	—	134	2	no	—
273	#121	—	134	2	no	—
274	C₄×D₄	—	67	2	no	—
275	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
276	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
277	C₂×C₄⋊C₄	—	134	2	no	—
278	C₂×C₄⋊C₄	—	134	2	no	—
279	#107	—	67	1	no	—
280	#107	—	134	2	no	—
281	#107	—	134	1	no	—
282	#107	—	134	1	no	—
283	#107	—	134	2	no	—
284	#107	D₂₆₈	1	1	yes	characteristic
285	#106	—	268	2	no	—
286	C₄⋊C₈	—	67	2	no	—
287	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
289	#103	—	134	2	no	—
290	#103	—	134	2	no	—
291	#103	—	134	2	no	—
292	#103	—	134	2	no	—
293	#102	—	67	2	no	—
294	#102	—	134	2	no	—
295	#102	—	134	2	no	—
296	#102	D₂₆₈	1	1	yes	characteristic
297	#102	D₂₆₈	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	134	2	no	—
301	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
302	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
304	#41	—	268	4	no	—
305	#41	—	268	4	no	—
306	#41	—	536	2	no	—
307	#41	—	536	2	no	—
308	C₂×Q₈	—	134	2	no	—
309	C₂×Q₈	—	134	2	no	—
310	C₂×D₄	—	268	2	no	—
311	C₂×D₄	—	268	2	no	—
312	C₂×D₄	—	67	2	no	—
313	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
314	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	268	2	no	—
318	C₂²×C₄	—	268	2	no	—
319	C₂²×C₄	—	134	2	no	—
320	Q₁₆	—	268	2	no	—
321	Q₁₆	—	536	1	no	—
322	SD₁₆	—	268	4	no	—
323	SD₁₆	—	536	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	134	2	no	—
326	M₁₆	—	536	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	134	2	no	—
333	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
334	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
335	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
336	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
337	C₄⋊C₄	—	268	2	no	—
338	C₄⋊C₄	—	268	2	no	—
339	C₄⋊C₄	—	67	2	no	—
340	C₄⋊C₄	—	134	2	no	—
341	C₄⋊C₄	—	134	2	no	—
342	C₄⋊C₄	—	134	2	no	—
343	C₄⋊C₄	—	134	2	no	—
344	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	268	2	no	—
349	C₂²⋊C₄	—	268	2	no	—
350	C₂²⋊C₄	—	134	2	no	—
351	C₄²	C₂³×D₆₇	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	134	2	no	—
355	Q₈	—	268	2	no	—
356	Q₈	—	268	2	no	—
357	Q₈	—	268	2	no	—
358	Q₈	—	268	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	536	4	no	—
363	D₄	—	536	4	no	—
364	D₄	—	134	2	no	—
365	D₄	—	134	2	no	—
366	D₄	#14524065	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	268	2	no	—
374	C₂×C₄	—	268	2	no	—
375	C₂×C₄	—	268	2	no	—
376	C₂×C₄	—	268	2	no	—
377	C₂×C₄	—	536	4	no	—
378	C₂×C₄	—	536	4	no	—
379	C₂×C₄	—	134	2	no	—
380	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
382	C₂×C₄	#14524089	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	268	2	no	—
390	C₈	#14524067	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	268	2	no	—
398	C₂²	—	268	2	no	—
399	C₂²	—	268	2	no	—
400	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	268	2	no	—
405	C₄	—	536	2	no	—
406	C₄	—	536	2	no	—
407	C₄	C₂×#14524065	1	1	yes	characteristic
408	C₄	C₂×#14524067	1	1	yes	characteristic
409	C₄	#16864201	1	1	yes	characteristic
410	C₄	#16864203	1	1	yes	characteristic
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	536	2	no	—
414	C₂	#22571810	1	1	yes	characteristic
415	C₂	#22571896	1	1	yes	characteristic
416	C₂	#22572014	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×D₈×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
hypercenter	C₈×D₈	D₆₇
hypocenter	C₆₇	#19902
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #22572557	C₂	semidirect
2. #22572516	C₂	semidirect
3. #22571931	C₂	semidirect
4. #22559806	C₂	semidirect
5. #22559757	C₂	semidirect
6. #22559598	C₂	semidirect
7. #22559574	C₂	semidirect
8. #22559496	C₂	semidirect
9. #22559495	C₂	semidirect
10. #22559095	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
149. D₄×C₆₇	#135	semidirect
219. C₈×D₈	D₆₇	semidirect
220. C₆₇	#19902	semidirect
366. D₄	#14524065	semidirect

Maximal subgroups

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

subgroup	index
#22572557	2
#22572516	2
#22571931	2
#22559806	2
#22559757	2
#22559598	2
#22559574	2
#22559496	2
#22559495	2
#22559095	2
C₈×D₈×C₆₇	2
#19902	67