Subgroups — #31010485

Order 17152.

598 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	#22570721	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863478	C₂	1	1	yes	maximal, characteristic
3	#22562369	C₂	1	2	yes	maximal, complemented
4	#22562365	C₂	1	2	yes	maximal, complemented
5	#22562355	C₂	1	2	yes	maximal, complemented
6	#22562345	C₂	1	2	yes	maximal, complemented
7	#22561995	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561964	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561926	C₂	1	1	yes	maximal, characteristic
10	#22561924	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
13	#16863949	—	2	1	no	—
14	C₂×#14524005	—	2	1	no	—
15	C₂×#14523983	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	1	yes	characteristic, complemented
17	#16863800	C₂²	1	1	yes	characteristic, complemented
18	#16863797	C₂²	1	1	yes	characteristic, complemented
19	#16863797	C₂²	1	1	yes	characteristic, complemented
20	#16863796	—	2	1	no	—
21	#16863793	—	2	1	no	—
22	#16863478	—	2	1	no	—
23	#16863478	—	2	1	no	—
24	#16863478	C₂²	1	2	yes	—
25	#16863478	C₂²	1	2	yes	—
26	#16863434	C₂²	1	1	yes	characteristic
27	#16863432	—	2	1	no	—
28	C₂×#14523963	—	2	1	no	—
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863223	C₂²	1	2	yes	complemented
31	#16863221	C₂²	1	2	yes	complemented
32	Dic₆₇⋊C₁₆	C₂²	1	2	yes	complemented
33	Dic₆₇⋊C₁₆	C₂²	1	2	yes	complemented
34	#16863198	C₂²	1	2	yes	—
35	#16863198	C₂²	1	2	yes	—
36	#16863196	C₂²	1	2	yes	—
37	#16863196	C₂²	1	2	yes	—
38	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
39	#427×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
41	#367×C₆₇	C₂²	1	2	yes	complemented
42	#366×C₆₇	C₂²	1	2	yes	complemented
43	C₂²×D₂₆₈	—	2	1	no	—
44	C₂²×C₄×D₆₇	—	4	1	no	—
45	C₂×D₅₃₆	—	2	1	no	—
46	C₂×D₅₃₆	—	2	2	no	—
47	C₂×D₅₃₆	Q₈	1	4	yes	complemented
48	C₂×D₅₃₆	C₂³	1	2	yes	—
49	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
50	#14524042	—	2	1	no	—
51	#14524042	—	2	1	no	—
52	#14524034	—	2	1	no	—
53	#14524034	—	2	1	no	—
54	#14524020	—	4	1	no	—
55	#14524020	—	4	1	no	—
56	C₄×Dic₁₃₄	—	4	1	no	—
57	C₄×Dic₁₃₄	—	4	1	no	—
58	#14524005	—	2	2	no	—
59	#14524005	—	2	2	no	—
60	#14524005	—	4	1	no	—
61	#14524005	—	4	1	no	—
62	#14524002	—	2	1	no	—
63	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
64	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
65	#14523983	—	2	2	no	—
66	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
67	#14523977	C₂³	1	1	yes	characteristic
68	#14523977	C₂³	1	1	yes	characteristic
69	#14523976	—	2	1	no	—
70	#14523976	—	2	1	no	—
71	#14523975	—	2	1	no	—
72	#14523974	—	2	1	no	—
73	#14523974	—	2	1	no	—
74	#14523963	—	2	2	no	—
75	#14523963	—	2	2	no	—
76	C₈×Dic₆₇	D₄	1	1	yes	characteristic
77	C₈×Dic₆₇	D₄	1	1	yes	characteristic
78	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
79	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#117×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	M₃₂×C₆₇	—	2	2	no	—
84	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
85	#107×C₆₇	—	2	1	no	—
86	#107×C₆₇	C₂³	1	1	yes	characteristic
87	#107×C₆₇	C₂³	1	1	yes	characteristic
88	#107×C₆₇	C₂³	1	2	yes	—
89	#106×C₆₇	—	2	1	no	—
90	C₂³×D₆₇	—	4	1	no	—
91	C₂×Dic₁₃₄	—	4	1	no	—
92	C₂×Dic₁₃₄	—	4	1	no	—
93	C₂×D₂₆₈	—	2	4	no	—
94	C₂×D₂₆₈	—	2	2	no	—
95	C₂×D₂₆₈	—	4	1	no	—
96	C₂×D₂₆₈	—	4	2	no	—
97	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
98	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
99	C₂×C₄×D₆₇	—	4	2	no	—
100	C₂×C₄×D₆₇	—	4	2	no	—
101	C₂×C₄×D₆₇	—	8	4	no	—
102	D₅₃₆	—	2	4	no	—
103	D₅₃₆	—	2	2	no	—
104	D₅₃₆	C₂×Q₈	1	4	yes	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₄⋊Dic₆₇	—	2	1	no	—
109	C₄⋊Dic₆₇	—	2	1	no	—
110	C₄⋊Dic₆₇	—	4	1	no	—
111	C₄⋊Dic₆₇	—	4	1	no	—
112	Dic₆₇⋊C₄	—	4	1	no	—
113	Dic₆₇⋊C₄	—	4	1	no	—
114	Dic₆₇⋊C₄	—	4	1	no	—
115	Dic₆₇⋊C₄	—	4	1	no	—
116	C₂²⋊Dic₆₇	—	4	1	no	—
117	(C₂×D₆₇)⋊C₄	—	4	2	no	—
118	(C₂×D₆₇)⋊C₄	—	4	2	no	—
119	C₄×Dic₆₇	—	4	1	no	—
120	C₄×Dic₆₇	D₈	1	1	yes	characteristic
121	C₄×Dic₆₇	D₈	1	1	yes	characteristic
122	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
124	C₆₇⋊C₁₆	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	4	1	no	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₆₇	#41	1	2	yes	—
130	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	1	no	—
135	C₂²⋊C₄×C₆₇	—	4	1	no	—
136	C₄²×C₆₇	—	4	1	no	—
137	C₁₆×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₆₇	—	4	2	no	—
140	C₂²×D₆₇	—	4	4	no	—
141	C₂²×D₆₇	—	8	2	no	—
142	C₂²×D₆₇	—	8	2	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	8	2	no	—
147	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#137	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	4	1	no	—
153	C₂×Dic₆₇	—	4	1	no	—
154	C₂×Dic₆₇	—	4	1	no	—
155	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
158	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
159	D₂₆₈	—	2	4	no	—
160	D₂₆₈	—	4	2	no	—
161	D₂₆₈	—	4	4	no	—
162	C₄×D₆₇	—	8	4	no	—
163	C₄×D₆₇	—	8	4	no	—
164	Dic₁₃₄	—	8	2	no	—
165	Dic₁₃₄	—	8	2	no	—
166	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
167	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
168	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	#474	1	2	yes	—
174	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₆₇	—	4	4	no	—
176	C₂×D₆₇	—	8	8	no	—
177	C₂×D₆₇	—	8	2	no	—
178	C₂×D₆₇	—	8	2	no	—
179	C₄×C₆₇	—	8	1	no	—
180	C₄×C₆₇	—	8	2	no	—
181	C₄×C₆₇	—	8	1	no	—
182	C₄×C₆₇	#474	1	2	yes	—
183	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
185	Dic₆₇	—	2	1	no	—
186	Dic₆₇	—	2	1	no	—
187	Dic₆₇	—	2	1	no	—
188	Dic₆₇	—	8	1	no	—
189	Dic₆₇	—	8	1	no	—
190	Dic₆₇	#509	1	2	yes	—
191	#33627	—	67	1	no	maximal, Sylow
192	C₂×C₆₇	—	2	2	no	—
193	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
194	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
195	C₂×C₆₇	#3414	1	1	yes	characteristic
196	D₆₇	—	8	4	no	—
197	#3160	—	67	1	no	—
198	C₂×C₄×D₈	—	67	1	no	—
199	#2228	—	67	2	no	—
200	#2228	—	67	2	no	—
201	#2226	—	67	2	no	—
202	#2224	—	67	2	no	—
203	#2159	—	67	1	no	—
204	#2159	D₆₇	1	1	yes	characteristic, complemented
205	#2150	—	67	1	no	—
206	#2140	—	67	1	no	—
207	#2140	—	67	1	no	—
208	C₆₇	#33627	1	1	yes	characteristic, complemented, Sylow
209	C₂²×D₈	—	67	1	no	—
210	#533	—	134	1	no	—
211	C₂×C₄×D₄	—	134	1	no	—
212	C₂×M₃₂	—	67	1	no	—
213	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
214	#500	—	67	1	no	—
215	#500	—	67	1	no	—
216	#499	—	67	1	no	—
217	#499	—	67	1	no	—
218	#499	—	134	1	no	—
219	#498	—	134	1	no	—
220	C₄×D₈	—	67	2	no	—
221	C₄×D₈	—	67	2	no	—
222	C₄×D₈	—	134	1	no	—
223	C₄×D₈	—	134	1	no	—
224	#427	—	67	1	no	—
225	#427	—	134	1	no	—
226	#427	C₂×D₆₇	1	1	yes	characteristic
227	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
228	C₂×#102	—	134	1	no	—
229	C₂×C₄×C₈	—	67	1	no	—
230	#367	—	67	2	no	—
231	#367	C₂×D₆₇	1	2	yes	complemented
232	#366	—	67	2	no	—
233	#366	C₂×D₆₇	1	2	yes	complemented
234	C₄⋊C₁₆	—	67	2	no	—
235	C₄⋊C₁₆	—	67	2	no	—
236	#357	—	67	2	no	—
237	#357	—	67	2	no	—
238	#357	—	67	2	no	—
239	#357	—	67	2	no	—
240	C₂²×D₄	—	134	1	no	—
241	C₂³×C₄	—	268	1	no	—
242	C₂×D₈	—	67	4	no	—
243	C₂×D₈	—	67	2	no	—
244	C₂×D₈	—	134	1	no	—
245	C₂×D₈	—	134	2	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	134	1	no	—
249	C₄⋊Q₈	—	134	1	no	—
250	#125	—	134	1	no	—
251	#125	—	134	1	no	—
252	C₂²⋊Q₈	—	268	1	no	—
253	C₂²⋊Q₈	—	268	1	no	—
254	C₄×Q₈	—	268	1	no	—
255	C₄×Q₈	—	268	1	no	—
256	C₄×D₄	—	268	1	no	—
257	C₄×D₄	—	268	1	no	—
258	C₄×D₄	—	134	2	no	—
259	C₄×D₄	—	134	2	no	—
260	#117	—	2	1	no	—
261	#117	—	134	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₂²⋊C₄	—	268	1	no	—
264	C₂×C₄²	—	67	1	no	—
265	M₃₂	—	2	2	no	—
266	M₃₂	—	134	2	no	—
267	C₂×C₁₆	—	67	2	no	—
268	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
269	#107	—	2	1	no	—
270	#107	—	67	1	no	—
271	#107	—	67	1	no	—
272	#107	—	134	1	no	—
273	#107	—	134	1	no	—
274	#107	C₂²×D₆₇	1	1	yes	characteristic
275	#107	C₂²×D₆₇	1	1	yes	characteristic
276	#107	C₂²×D₆₇	1	2	yes	—
277	#106	—	2	1	no	—
278	#106	—	134	1	no	—
279	#106	—	134	1	no	—
280	#106	—	134	1	no	—
281	#102	—	134	2	no	—
282	#102	—	134	2	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	67	1	no	—
285	C₄×C₈	—	67	1	no	—
286	C₄×C₈	—	67	1	no	—
287	C₂⁴	—	268	1	no	—
288	C₂×Q₈	—	268	1	no	—
289	C₂×Q₈	—	268	1	no	—
290	C₂×D₄	—	268	1	no	—
291	C₂×D₄	—	268	2	no	—
292	C₂×D₄	—	134	4	no	—
293	C₂×D₄	—	134	2	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	268	2	no	—
296	C₂²×C₄	—	268	2	no	—
297	C₂²×C₄	—	536	4	no	—
298	C₂²×C₄	—	67	1	no	—
299	C₂²×C₄	—	67	1	no	—
300	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
301	D₈	—	67	4	no	—
302	D₈	—	134	4	no	—
303	D₈	—	134	2	no	—
304	C₂×C₈	—	67	1	no	—
305	C₂×C₈	—	67	1	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₈	Q₈×D₆₇	1	2	yes	—
309	C₂×C₈	#11761628	1	2	yes	—
310	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	268	1	no	—
316	C₄⋊C₄	—	268	1	no	—
317	C₄⋊C₄	—	268	1	no	—
318	C₄⋊C₄	—	268	1	no	—
319	C₄⋊C₄	—	268	1	no	—
320	C₄⋊C₄	—	268	1	no	—
321	C₄⋊C₄	—	134	1	no	—
322	C₄⋊C₄	—	134	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	268	2	no	—
325	C₂²⋊C₄	—	268	2	no	—
326	C₂²⋊C₄	—	268	1	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	268	1	no	—
329	C₄²	—	67	1	no	—
330	C₄²	—	67	1	no	—
331	C₄²	—	67	1	no	—
332	C₄²	—	67	1	no	—
333	C₁₆	—	2	2	no	—
334	C₁₆	—	134	2	no	—
335	C₂³	—	268	2	no	—
336	C₂³	—	268	4	no	—
337	C₂³	—	536	2	no	—
338	C₂³	—	536	2	no	—
339	C₂³	D₈×D₆₇	1	1	yes	characteristic
340	Q₈	—	536	2	no	—
341	Q₈	—	536	2	no	—
342	D₄	—	268	4	no	—
343	D₄	—	268	2	no	—
344	D₄	—	134	4	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	268	1	no	—
350	C₂×C₄	—	268	1	no	—
351	C₂×C₄	—	268	1	no	—
352	C₂×C₄	—	536	4	no	—
353	C₂×C₄	—	536	4	no	—
354	C₂×C₄	—	67	1	no	—
355	C₂×C₄	—	67	1	no	—
356	C₂×C₄	—	67	1	no	—
357	C₂×C₄	—	67	1	no	—
358	C₂×C₄	—	134	2	no	—
359	C₂×C₄	#14524021	1	2	yes	—
360	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
361	C₂×C₄	#14524075	1	2	yes	—
362	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
363	C₈	—	134	1	no	—
364	C₈	—	134	1	no	—
365	C₈	#14524021	1	2	yes	—
366	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
367	C₈	C₂×#11761628	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	268	4	no	—
371	C₂²	—	536	8	no	—
372	C₂²	—	536	2	no	—
373	C₂²	—	536	2	no	—
374	C₂²	#16863675	1	2	yes	—
375	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	—	536	1	no	—
380	C₄	—	536	1	no	—
381	C₄	—	67	2	no	—
382	C₄	—	134	1	no	—
383	C₄	—	134	1	no	—
384	C₄	—	134	1	no	—
385	C₄	#16863675	1	2	yes	—
386	C₄	C₂×#14524021	1	1	yes	characteristic
387	C₄	C₂×#14524075	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	536	4	no	—
390	C₂	C₂×#16863675	1	1	yes	characteristic
391	C₂	#509×D₆₇	1	1	yes	characteristic
392	C₂	#22573344	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2159	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	#2159	D₆₇
hypocenter	C₆₇	#33627
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
1. #22570721	C₂	semidirect
3. #22562369	C₂	semidirect
4. #22562365	C₂	semidirect
5. #22562355	C₂	semidirect
6. #22562345	C₂	semidirect
7. #22561995	C₂	semidirect
8. #22561964	C₂	semidirect
11. C₆₇×#2159	C₂	semidirect
16. #16863800	C₂²	semidirect
17. #16863800	C₂²	semidirect
18. #16863797	C₂²	semidirect
19. #16863797	C₂²	semidirect
30. #16863223	C₂²	semidirect
31. #16863221	C₂²	semidirect
32. Dic₆₇⋊C₁₆	C₂²	semidirect
33. Dic₆₇⋊C₁₆	C₂²	semidirect
41. #367×C₆₇	C₂²	semidirect
42. #366×C₆₇	C₂²	semidirect
47. C₂×D₅₃₆	Q₈	semidirect
204. #2159	D₆₇	semidirect
208. C₆₇	#33627	semidirect
231. #367	C₂×D₆₇	semidirect
233. #366	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570721	2
C₂×#16863478	2
#22562369	2
#22562365	2
#22562355	2
#22562345	2
#22561995	2
#22561964	2
#22561926	2
#22561924	2
C₆₇×#2159	2
#33627	67