Subgroups — #31009074

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573310	C₂	1	2	yes	maximal, complemented
2	#22569107	C₂	1	1	yes	maximal, characteristic
3	#22562051	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562050	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562049	C₂	1	2	yes	maximal, complemented
6	#22562047	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562043	C₂	1	2	yes	maximal, complemented
8	#22561973	C₂	1	1	yes	maximal, characteristic, complemented
9	#350×D₆₇	C₂	1	2	yes	maximal, complemented
10	#22561804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864193	C₄	1	1	yes	characteristic
13	#131×D₆₇	C₂²	1	2	yes	—
14	#16863814	—	2	2	no	—
15	#16863814	C₂²	1	2	yes	complemented
16	#16863813	C₂²	1	2	yes	complemented
17	#16863812	—	2	2	no	—
18	#16863812	C₂²	1	2	yes	complemented
19	C₂×#14523982	—	2	2	no	—
20	C₂×#14523982	C₂²	1	1	yes	characteristic
21	M₃₂×D₆₇	—	2	2	no	—
22	M₃₂×D₆₇	C₂²	1	2	yes	—
23	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
24	#16863518	C₄	1	1	yes	characteristic
25	#16863517	C₄	1	1	yes	characteristic
26	#16863515	C₄	1	2	yes	—
27	#16863514	C₂²	1	1	yes	characteristic
28	#16863511	C₂²	1	2	yes	—
29	#16863443	C₄	1	1	yes	characteristic
30	#16863411	C₄	1	2	yes	—
31	#16863350	C₂²	1	1	yes	characteristic
32	#16863217	C₂²	1	1	yes	characteristic, complemented
33	#16863216	C₂²	1	2	yes	complemented
34	#16863181	C₂²	1	2	yes	complemented
35	#16863180	C₂²	1	2	yes	complemented
36	#16863179	C₂²	1	2	yes	complemented
37	#16863172	C₂²	1	1	yes	characteristic, complemented
38	#16863171	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
40	#504×C₆₇	C₂²	1	2	yes	complemented
41	#443×C₆₇	C₂²	1	1	yes	characteristic
42	#364×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#350×C₆₇	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#41×D₆₇	C₂×C₄	1	2	yes	—
46	#14524075	C₈	1	2	yes	—
47	#14524072	C₈	1	2	yes	—
48	#14524070	C₈	1	2	yes	—
49	#14524067	C₈	1	2	yes	—
50	#14524065	C₂×C₄	1	1	yes	characteristic
51	#14524063	C₂×C₄	1	2	yes	—
52	#14524051	—	2	2	no	—
53	#14524051	C₂×C₄	1	2	yes	—
54	#14524050	C₂×C₄	1	2	yes	—
55	#14524049	—	2	2	no	—
56	#14524049	C₂³	1	2	yes	—
57	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₆₇	—	2	2	no	—
59	M₁₆×D₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₆₇	—	2	2	no	—
61	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
62	#14524000	C₂×C₄	1	1	yes	characteristic
63	#14523983	—	2	2	no	—
64	#14523983	C₂³	1	2	yes	—
65	#14523982	—	2	2	no	—
66	#14523982	—	2	2	no	—
67	#14523982	—	2	2	no	—
68	#14523982	D₄	1	2	yes	—
69	#14523982	D₄	1	2	yes	—
70	C₂×C₆₇⋊C₁₆	—	2	2	no	—
71	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₁₆×D₆₇	—	2	2	no	—
74	C₁₆×D₆₇	—	2	2	no	—
75	C₁₆×D₆₇	—	2	2	no	—
76	C₁₆×D₆₇	—	2	2	no	—
77	#14523979	C₂×C₄	1	1	yes	characteristic
78	#14523978	C₂×C₄	1	2	yes	—
79	#14523971	C₂×C₄	1	2	yes	—
80	#14523970	C₂×C₄	1	2	yes	—
81	#14523969	C₂×C₄	1	2	yes	—
82	#14523954	C₂×C₄	1	1	yes	characteristic
83	#14523953	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
85	#135×C₆₇	C₂×C₄	1	1	yes	characteristic
86	#131×C₆₇	C₂³	1	2	yes	—
87	M₃₂×C₆₇	—	2	2	no	—
88	M₃₂×C₆₇	C₂³	1	2	yes	—
89	C₂×C₁₆×C₆₇	—	2	2	no	—
90	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
92	#108×C₆₇	C₂×C₄	1	1	yes	characteristic
93	#104×C₆₇	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#11761629	C₂×C₈	1	2	yes	—
96	#11761628	—	2	2	no	—
97	#11761628	C₂²×C₄	1	2	yes	—
98	#11761627	—	2	2	no	—
99	#11761627	C₂×C₈	1	2	yes	—
100	Q₈×D₆₇	C₂×C₈	1	2	yes	—
101	D₄×D₆₇	—	2	2	no	—
102	D₄×D₆₇	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₆₇	—	2	2	no	—
104	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
105	Dic₂₆₈	C₂×C₈	1	1	yes	characteristic
106	#11761618	C₂×C₈	1	2	yes	—
107	#11761617	C₂×C₈	1	2	yes	—
108	#11761616	C₂×C₈	1	2	yes	—
109	#11761615	C₂×C₈	1	2	yes	—
110	D₅₃₆	C₂×C₈	1	1	yes	characteristic
111	#11761613	C₂×C₈	1	2	yes	—
112	#11761612	—	2	2	no	—
113	#11761612	C₂²×C₄	1	2	yes	—
114	#11761611	—	2	2	no	—
115	#11761611	—	2	2	no	—
116	#11761611	—	2	2	no	—
117	#11761611	C₂×C₈	1	2	yes	—
118	#11761611	C₂×C₈	1	2	yes	—
119	C₂×C₆₇⋊C₈	—	2	2	no	—
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₆₇	—	2	2	no	—
123	C₈×D₆₇	—	2	2	no	—
124	C₈×D₆₇	—	2	2	no	—
125	C₈×D₆₇	C₂×D₄	1	2	yes	—
126	C₈×D₆₇	#41	1	2	yes	—
127	Dic₆₇⋊C₄	—	2	1	no	—
128	(C₂×D₆₇)⋊C₄	—	2	1	no	—
129	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₆₇⋊C₁₆	—	2	2	no	—
131	C₆₇⋊C₁₆	—	2	1	no	—
132	C₆₇⋊C₁₆	C₂×D₄	1	2	yes	—
133	#41×C₆₇	C₂²×C₄	1	2	yes	—
134	Q₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
135	SD₁₆×C₆₇	C₂×C₈	1	2	yes	—
136	D₈×C₆₇	C₂×C₈	1	1	yes	characteristic
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₁₆×C₆₇	—	2	2	no	—
144	C₁₆×C₆₇	—	2	1	no	—
145	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×D₆₇	—	2	2	no	—
148	C₂²×D₆₇	#131	1	1	yes	characteristic
149	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
150	D₄×C₆₇	—	2	2	no	—
151	D₄×C₆₇	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₆₇	—	2	2	no	—
156	C₂²⋊D₆₇	—	2	2	no	—
157	C₂²⋊D₆₇	—	2	2	no	—
158	C₂×Dic₆₇	—	2	2	no	—
159	C₂×Dic₆₇	—	2	1	no	—
160	C₂×Dic₆₇	#131	1	1	yes	characteristic
161	D₂₆₈	—	2	2	no	—
162	D₂₆₈	C₂²×C₈	1	2	yes	—
163	C₄×D₆₇	—	2	2	no	—
164	C₄×D₆₇	—	2	2	no	—
165	C₄×D₆₇	—	2	2	no	—
166	C₄×D₆₇	C₄×D₄	1	2	yes	—
167	C₄×D₆₇	C₂²×C₈	1	2	yes	—
168	Dic₁₃₄	—	2	2	no	—
169	Dic₁₃₄	C₂²×C₈	1	2	yes	—
170	C₈×C₆₇	—	2	2	no	—
171	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
175	C₆₇⋊C₈	—	2	2	no	—
176	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
177	C₆₇⋊C₈	C₂²×C₈	1	2	yes	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
180	C₂×D₆₇	—	2	2	no	—
181	C₂×D₆₇	—	4	1	no	—
182	C₂×D₆₇	—	8	2	no	—
183	C₂×D₆₇	C₈×D₄	1	2	yes	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	4	1	no	—
186	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
188	Dic₆₇	—	2	2	no	—
189	Dic₆₇	—	4	1	no	—
190	Dic₆₇	C₈×D₄	1	2	yes	—
191	#33442	—	67	1	no	maximal, Sylow
192	C₂×C₆₇	—	2	1	no	—
193	C₂×C₆₇	—	4	2	no	—
194	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
195	D₆₇	—	2	2	no	—
196	D₆₇	—	4	2	no	—
197	#3407	—	67	2	no	—
198	#2955	—	67	1	no	—
199	#2171	—	67	2	no	—
200	#2171	—	67	1	no	—
201	#2171	—	67	1	no	—
202	#2170	—	67	2	no	—
203	#2170	—	67	1	no	—
204	#2170	D₆₇	1	1	yes	characteristic, complemented
205	#2153	—	67	1	no	—
206	C₂×#350	—	67	2	no	—
207	#2108	—	67	1	no	—
208	C₆₇	#33442	1	1	yes	characteristic, complemented, Sylow
209	#575	—	67	1	no	—
210	C₂×#131	—	67	2	no	—
211	#504	—	67	2	no	—
212	#504	—	67	2	no	—
213	#504	—	67	2	no	—
214	#504	—	134	2	no	—
215	#504	—	134	2	no	—
216	#504	C₂×D₆₇	1	2	yes	complemented
217	C₂×M₃₂	—	67	1	no	—
218	C₂×M₃₂	—	67	2	no	—
219	C₂×M₃₂	—	134	2	no	—
220	C₂×M₃₂	—	134	2	no	—
221	C₂²×C₁₆	—	67	1	no	—
222	#444	—	67	1	no	—
223	#444	—	67	1	no	—
224	#444	—	67	2	no	—
225	#443	—	67	2	no	—
226	#443	—	67	1	no	—
227	#443	C₂×D₆₇	1	1	yes	characteristic
228	#430	—	67	1	no	—
229	#421	—	67	2	no	—
230	#405	—	67	1	no	—
231	#364	—	67	2	no	—
232	#364	—	67	1	no	—
233	#364	C₂×D₆₇	1	1	yes	characteristic, complemented
234	#350	—	67	2	no	—
235	#350	—	67	2	no	—
236	#350	—	67	2	no	—
237	#350	C₂×D₆₇	1	2	yes	complemented
238	#346	—	67	1	no	—
239	#346	—	67	1	no	—
240	C₄×C₁₆	—	67	1	no	—
241	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
242	C₂×#41	—	67	2	no	—
243	#137	—	67	2	no	—
244	#137	—	67	2	no	—
245	AGL(1,ℤ/8)	—	67	2	no	—
246	AGL(1,ℤ/8)	—	67	2	no	—
247	#135	—	67	2	no	—
248	#135	—	67	1	no	—
249	#135	C₄×D₆₇	1	1	yes	characteristic
250	#131	—	67	2	no	—
251	#131	—	67	2	no	—
252	#131	—	67	2	no	—
253	#131	—	134	2	no	—
254	#131	—	134	2	no	—
255	#131	C₂²×D₆₇	1	2	yes	—
256	C₂×M₁₆	—	67	1	no	—
257	C₂×M₁₆	—	67	2	no	—
258	C₂×M₁₆	—	134	2	no	—
259	C₂²×C₈	—	67	1	no	—
260	C₂²×C₈	—	134	2	no	—
261	#117	—	67	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	67	2	no	—
264	M₃₂	—	67	2	no	—
265	M₃₂	—	67	2	no	—
266	M₃₂	—	134	2	no	—
267	M₃₂	—	134	2	no	—
268	M₃₂	—	134	2	no	—
269	M₃₂	—	134	2	no	—
270	M₃₂	C₂²×D₆₇	1	2	yes	—
271	C₂×C₁₆	—	2	2	no	—
272	C₂×C₁₆	—	67	1	no	—
273	C₂×C₁₆	—	67	1	no	—
274	C₂×C₁₆	—	134	2	no	—
275	C₂×C₁₆	—	134	2	no	—
276	C₂×C₁₆	—	134	2	no	—
277	C₂×C₁₆	—	134	2	no	—
278	C₂×C₁₆	—	134	2	no	—
279	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
281	#108	—	67	2	no	—
282	#108	—	67	1	no	—
283	#108	C₄×D₆₇	1	1	yes	characteristic
284	#104	—	67	2	no	—
285	#104	—	67	2	no	—
286	#104	—	67	2	no	—
287	#104	C₄×D₆₇	1	2	yes	—
288	#97	—	67	1	no	—
289	#97	—	67	1	no	—
290	C₄×C₈	—	67	1	no	—
291	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
292	#41	—	67	2	no	—
293	#41	—	67	2	no	—
294	#41	—	67	2	no	—
295	#41	—	134	2	no	—
296	#41	—	134	2	no	—
297	#41	C₂×C₄×D₆₇	1	2	yes	—
298	C₂×Q₈	—	67	2	no	—
299	C₂×D₄	—	67	2	no	—
300	C₂×D₄	—	134	2	no	—
301	C₂²×C₄	—	67	1	no	—
302	C₂²×C₄	—	134	2	no	—
303	Q₁₆	—	67	2	no	—
304	Q₁₆	—	67	1	no	—
305	Q₁₆	C₈×D₆₇	1	1	yes	characteristic
306	SD₁₆	—	67	2	no	—
307	SD₁₆	—	67	2	no	—
308	SD₁₆	—	67	2	no	—
309	SD₁₆	C₈×D₆₇	1	2	yes	—
310	D₈	—	67	2	no	—
311	D₈	—	67	1	no	—
312	D₈	C₈×D₆₇	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	67	2	no	—
315	M₁₆	—	67	2	no	—
316	M₁₆	—	67	2	no	—
317	M₁₆	—	134	2	no	—
318	M₁₆	—	134	2	no	—
319	M₁₆	—	134	2	no	—
320	M₁₆	—	134	2	no	—
321	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	67	1	no	—
324	C₂×C₈	—	67	1	no	—
325	C₂×C₈	—	67	2	no	—
326	C₂×C₈	—	67	2	no	—
327	C₂×C₈	—	134	2	no	—
328	C₂×C₈	—	134	2	no	—
329	C₂×C₈	—	134	2	no	—
330	C₂×C₈	—	134	2	no	—
331	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
333	C₄⋊C₄	—	134	1	no	—
334	C₂²⋊C₄	—	134	1	no	—
335	C₄²	—	67	1	no	—
336	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	67	2	no	—
340	C₁₆	—	134	2	no	—
341	C₁₆	—	134	1	no	—
342	C₁₆	D₄×D₆₇	1	1	yes	characteristic
343	C₁₆	D₄×D₆₇	1	1	yes	characteristic
344	C₂³	—	67	1	no	—
345	C₂³	—	134	2	no	—
346	Q₈	—	67	2	no	—
347	Q₈	—	134	2	no	—
348	Q₈	C₂×C₈×D₆₇	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	67	2	no	—
351	D₄	—	134	2	no	—
352	D₄	—	134	2	no	—
353	D₄	—	134	2	no	—
354	D₄	—	134	2	no	—
355	D₄	C₂×C₈×D₆₇	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	67	1	no	—
359	C₂×C₄	—	67	2	no	—
360	C₂×C₄	—	67	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	—	134	2	no	—
363	C₂×C₄	—	134	1	no	—
364	C₂×C₄	—	134	2	no	—
365	C₂×C₄	—	134	2	no	—
366	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	67	2	no	—
369	C₈	—	67	2	no	—
370	C₈	—	134	2	no	—
371	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
372	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
373	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
374	C₈	#41×D₆₇	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	268	1	no	—
377	C₂²	—	536	2	no	—
378	C₂²	—	67	2	no	—
379	C₂²	—	134	2	no	—
380	C₂²	#131×D₆₇	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	268	1	no	—
384	C₄	—	67	2	no	—
385	C₄	—	134	2	no	—
386	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	268	2	no	—
391	C₂	—	134	2	no	—
392	C₂	C₈×D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2170	C₂
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	#16864193	C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2170	D₆₇
hypocenter	C₆₇	#33442
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #22573310	C₂	semidirect
3. #22562051	C₂	semidirect
4. #22562050	C₂	semidirect
5. #22562049	C₂	semidirect
6. #22562047	C₂	semidirect
7. #22562043	C₂	semidirect
8. #22561973	C₂	semidirect
9. #350×D₆₇	C₂	semidirect
10. #22561804	C₂	semidirect
11. C₆₇×#2170	C₂	semidirect
15. #16863814	C₂²	semidirect
16. #16863813	C₂²	semidirect
18. #16863812	C₂²	semidirect
32. #16863217	C₂²	semidirect
33. #16863216	C₂²	semidirect
34. #16863181	C₂²	semidirect
35. #16863180	C₂²	semidirect
36. #16863179	C₂²	semidirect
37. #16863172	C₂²	semidirect
38. #16863171	C₂²	semidirect
39. C₁₆×Dic₆₇	C₂²	semidirect
40. #504×C₆₇	C₂²	semidirect
42. #364×C₆₇	C₂²	semidirect
43. #350×C₆₇	C₂²	semidirect
44. C₄×C₁₆×C₆₇	C₂²	semidirect
204. #2170	D₆₇	semidirect
208. C₆₇	#33442	semidirect
216. #504	C₂×D₆₇	semidirect
233. #364	C₂×D₆₇	semidirect
237. #350	C₂×D₆₇	semidirect
241. C₄×C₁₆	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573310	2
#22569107	2
#22562051	2
#22562050	2
#22562049	2
#22562047	2
#22562043	2
#22561973	2
#350×D₆₇	2
#22561804	2
C₆₇×#2170	2
#33442	67