Subgroups — #30839847

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569673	C₂	1	2	yes	maximal, complemented
3	#22559770	C₂	1	4	yes	maximal, complemented
4	#22559722	C₂	1	2	yes	maximal, complemented
5	#22559688	C₂	1	2	yes	maximal, complemented
6	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558884	C₂	1	2	yes	maximal, complemented
8	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524069	—	2	2	no	—
10	#16863946	C₂²	1	2	yes	—
11	#16863801	C₂²	1	1	yes	characteristic, complemented
12	#16863773	C₂²	1	1	yes	characteristic, complemented
13	#16863743	C₂²	1	2	yes	complemented
14	#16863632	—	2	1	no	—
15	#16863632	—	2	1	no	—
16	#16863611	—	2	4	no	—
17	#16863601	C₂²	1	4	yes	—
18	#16863591	C₂²	1	4	yes	—
19	C₄×D₅₃₆	—	2	1	no	—
20	C₄×D₅₃₆	—	2	1	no	—
21	#16863420	C₂²	1	2	yes	—
22	#16863395	—	2	2	no	—
23	#16863147	C₂²	1	2	yes	complemented
24	#16863147	C₂²	1	2	yes	complemented
25	#16863141	C₂²	1	4	yes	complemented
26	#16863139	C₂²	1	4	yes	complemented
27	C₂²×M₁₆×C₆₇	—	2	1	no	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	#436×C₆₇	—	2	1	no	—
30	#436×C₆₇	—	2	1	no	—
31	#436×C₆₇	C₂²	1	2	yes	—
32	#436×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761628	—	2	2	no	—
37	#14524069	—	4	8	no	—
38	C₂×#11761616	—	2	4	no	—
39	C₂×D₅₃₆	—	2	1	no	—
40	C₂×D₅₃₆	—	2	1	no	—
41	C₂×#11761613	—	2	4	no	—
42	C₂×#11761612	—	2	2	no	—
43	#14524043	C₂³	1	2	yes	—
44	#14524040	C₂³	1	2	yes	—
45	#14524031	—	2	2	no	—
46	#14524017	—	2	2	no	—
47	#14524017	—	2	2	no	—
48	C₄×D₂₆₈	—	2	2	no	—
49	C₄×D₂₆₈	—	2	2	no	—
50	#14523977	—	2	1	no	—
51	#14523977	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
54	#14523966	—	2	4	no	—
55	#14523965	—	2	2	no	—
56	#14523965	—	2	2	no	—
57	#14523963	—	2	4	no	—
58	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	4	4	no	—
63	C₂²×C₈×C₆₇	—	2	1	no	—
64	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	2	1	no	—
68	C₄×D₄×C₆₇	D₄	1	2	yes	—
69	C₄×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₆₇	—	2	1	no	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	C₂²⋊C₈×C₆₇	—	2	2	no	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#11761628	—	4	4	no	—
82	#11761628	—	4	4	no	—
83	C₂×Dic₁₃₄	—	2	2	no	—
84	C₂×D₂₆₈	—	2	2	no	—
85	C₂×D₂₆₈	—	2	2	no	—
86	C₂×D₂₆₈	—	2	2	no	—
87	C₂×C₂²⋊D₆₇	—	4	2	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	#11761616	—	4	8	no	—
90	D₅₃₆	—	4	2	no	—
91	D₅₃₆	—	4	2	no	—
92	#11761613	—	4	8	no	—
93	#11761612	—	4	4	no	—
94	C₂×C₆₇⋊C₈	—	2	4	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	(C₂×D₆₇)⋊C₄	—	4	2	no	—
98	(C₂×D₆₇)⋊C₄	—	4	2	no	—
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
111	M₁₆×C₆₇	—	4	4	no	—
112	M₁₆×C₆₇	—	4	4	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	4	1	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	#41	1	1	yes	characteristic
128	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	#121	1	1	yes	characteristic
134	C₂²×D₆₇	—	4	2	no	—
135	D₄×C₆₇	—	2	4	no	—
136	D₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	4	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₆₇	—	8	4	no	—
153	C₂×Dic₆₇	—	4	2	no	—
154	D₂₆₈	—	4	2	no	—
155	D₂₆₈	—	4	2	no	—
156	D₂₆₈	—	4	2	no	—
157	D₂₆₈	—	4	2	no	—
158	D₂₆₈	—	8	2	no	—
159	C₄×D₆₇	—	8	4	no	—
160	Dic₁₃₄	—	4	2	no	—
161	Dic₁₃₄	—	4	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	AGL(1,ℤ/8)	1	2	yes	complemented
167	C₆₇⋊C₈	—	4	4	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	8	4	no	—
174	C₂²×C₆₇	#469	1	2	yes	—
175	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
176	C₂×D₆₇	—	8	2	no	—
177	C₂×D₆₇	—	8	2	no	—
178	C₂×D₆₇	—	8	2	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	#469	1	2	yes	—
184	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
186	C₄×C₆₇	#575	1	1	yes	characteristic
187	Dic₆₇	—	8	2	no	—
188	#19280	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	—	2	2	no	—
190	C₂×C₆₇	—	8	2	no	—
191	C₂×C₆₇	#3036	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
193	C₂×C₆₇	#3067	1	1	yes	characteristic
194	D₆₇	—	16	2	no	—
195	#3169	—	67	1	no	—
196	#3036	—	67	2	no	—
197	C₂×#436	D₆₇	1	1	yes	characteristic, complemented
198	#1701	—	67	4	no	—
199	#1693	—	67	2	no	—
200	#1687	—	67	2	no	—
201	#1566	—	67	1	no	—
202	#1547	—	67	2	no	—
203	C₆₇	#19280	1	1	yes	characteristic, complemented, Sylow
204	C₂×AGL(1,ℤ/8)	—	134	2	no	—
205	C₂²×M₁₆	—	2	1	no	—
206	#532	—	67	2	no	—
207	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
208	#500	—	67	1	no	—
209	#493	—	67	1	no	—
210	#486	—	67	2	no	—
211	#466	—	134	1	no	—
212	#466	—	134	1	no	—
213	#463	—	134	4	no	—
214	#461	—	67	4	no	—
215	#459	—	67	4	no	—
216	C₄×D₈	—	134	1	no	—
217	C₄×D₈	—	134	1	no	—
218	#436	—	2	1	no	—
219	#436	—	2	1	no	—
220	#436	C₂×D₆₇	1	2	yes	—
221	#436	C₂×D₆₇	1	2	yes	—
222	#423	—	67	2	no	—
223	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
224	#417	—	134	2	no	—
225	C₂×C₂²⋊C₈	—	2	1	no	—
226	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
227	#334	—	67	2	no	—
228	#334	—	67	2	no	—
229	#332	—	67	4	no	—
230	#331	—	67	4	no	—
231	C₂×#41	—	134	2	no	—
232	C₂²×D₄	D₂₆₈	1	1	yes	characteristic
233	C₂³×C₄	—	2	1	no	—
234	AGL(1,ℤ/8)	—	268	8	no	—
235	C₂×SD₁₆	—	134	4	no	—
236	C₂×D₈	—	134	4	no	—
237	C₂×D₈	—	134	1	no	—
238	C₂×D₈	—	134	1	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	4	4	no	—
242	C₂×M₁₆	—	134	2	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
246	C₄⋊Q₈	—	67	2	no	—
247	#127	—	67	2	no	—
248	#124	—	134	2	no	—
249	#121	—	134	2	no	—
250	#121	—	134	2	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	134	2	no	—
254	C₄×D₄	—	134	2	no	—
255	C₄×D₄	D₂₆₈	1	2	yes	—
256	C₄×D₄	C₂²×D₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
260	#107	—	134	1	no	—
261	#107	—	134	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	67	4	no	—
264	C₄⋊C₈	—	134	2	no	—
265	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
266	#103	—	134	4	no	—
267	#102	—	134	2	no	—
268	#102	—	134	2	no	—
269	#102	—	134	4	no	—
270	C₂²⋊C₈	—	2	2	no	—
271	C₂²⋊C₈	—	2	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	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₂⁴	—	2	1	no	—
277	#41	—	268	4	no	—
278	#41	—	268	4	no	—
279	C₂×Q₈	—	134	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	268	2	no	—
283	C₂×D₄	—	134	2	no	—
284	C₂×D₄	—	134	2	no	—
285	C₂×D₄	—	134	2	no	—
286	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	268	2	no	—
293	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
295	C₂²×C₄	#11761628	1	1	yes	characteristic
296	SD₁₆	—	268	8	no	—
297	D₈	—	268	2	no	—
298	D₈	—	268	8	no	—
299	D₈	—	268	2	no	—
300	M₁₆	—	4	4	no	—
301	M₁₆	—	4	4	no	—
302	M₁₆	—	268	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	134	4	no	—
308	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	134	2	no	—
314	C₄⋊C₄	—	134	2	no	—
315	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	268	2	no	—
319	C₂²⋊C₄	—	268	2	no	—
320	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
321	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
322	C₄²	#11761628	1	1	yes	characteristic
323	C₄²	C₂³×D₆₇	1	1	yes	characteristic
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	268	2	no	—
329	C₂³	#14524017	1	1	yes	characteristic
330	Q₈	—	268	2	no	—
331	Q₈	—	268	2	no	—
332	D₄	—	2	4	no	—
333	D₄	—	4	2	no	—
334	D₄	—	268	2	no	—
335	D₄	—	268	2	no	—
336	D₄	—	268	2	no	—
337	D₄	—	268	2	no	—
338	D₄	—	536	2	no	—
339	D₄	—	536	4	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	268	2	no	—
349	C₂×C₄	—	536	4	no	—
350	C₂×C₄	#14524017	1	1	yes	characteristic
351	C₂×C₄	#14524017	1	1	yes	characteristic
352	C₂×C₄	#14524017	1	1	yes	characteristic
353	C₂×C₄	#14524071	1	4	yes	—
354	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
356	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	268	4	no	—
362	C₈	#14524069	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	8	4	no	—
369	C₂²	—	536	2	no	—
370	C₂²	—	536	2	no	—
371	C₂²	—	536	2	no	—
372	C₂²	#16863649	1	2	yes	—
373	C₂²	C₂×#14524017	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	536	2	no	—
379	C₄	#16863649	1	2	yes	—
380	C₄	C₂×#14524069	1	1	yes	characteristic
381	C₄	C₂×#14524071	1	2	yes	—
382	C₄	#16864194	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	8	2	no	—
385	C₂	—	1072	2	no	—
386	C₂	#22569673	1	1	yes	characteristic
387	C₂	C₂×#16863649	1	1	yes	characteristic
388	C₂	#22569929	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#436×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#19280
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22570788	C₂	semidirect
2. #22569673	C₂	semidirect
3. #22559770	C₂	semidirect
4. #22559722	C₂	semidirect
5. #22559688	C₂	semidirect
6. #22558965	C₂	semidirect
7. #22558884	C₂	semidirect
8. C₂×#436×C₆₇	C₂	semidirect
11. #16863801	C₂²	semidirect
12. #16863773	C₂²	semidirect
13. #16863743	C₂²	semidirect
23. #16863147	C₂²	semidirect
24. #16863147	C₂²	semidirect
25. #16863141	C₂²	semidirect
26. #16863139	C₂²	semidirect
166. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
197. C₂×#436	D₆₇	semidirect
203. C₆₇	#19280	semidirect
362. C₈	#14524069	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22569673	2
#22559770	2
#22559722	2
#22559688	2
#22558965	2
#22558884	2
C₂×#436×C₆₇	2
#19280	67