Subgroups — #31010678

Order 17152.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569858	C₂	1	1	yes	maximal, characteristic
2	#22568806	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562403	C₂	1	2	yes	maximal, complemented
4	#22562402	C₂	1	2	yes	maximal, complemented
5	#22562396	C₂	1	2	yes	maximal, complemented
6	#22562391	C₂	1	2	yes	maximal, complemented
7	#22561998	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561924	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16863198	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561822	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	#16863965	—	2	1	no	—
15	#16863876	—	2	1	no	—
16	C₂×#14523983	C₂²	1	1	yes	characteristic
17	#16863650	—	2	2	no	—
18	#16863633	C₂²	1	2	yes	—
19	#16863633	C₂²	1	2	yes	—
20	#16863439	—	2	1	no	—
21	#16863435	—	2	2	no	—
22	#16863433	C₂²	1	2	yes	complemented
23	C₂×#14523977	C₂²	1	1	yes	characteristic
24	#107×D₆₇	C₂²	1	2	yes	complemented
25	#16863400	—	2	1	no	—
26	#16863223	C₂²	1	2	yes	complemented
27	#16863221	C₂²	1	2	yes	complemented
28	#16863198	C₂²	1	2	yes	—
29	#16863198	C₂²	1	2	yes	—
30	#16863196	C₂²	1	2	yes	—
31	#16863196	C₂²	1	2	yes	—
32	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	complemented
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	#427×C₆₇	C₂²	1	1	yes	characteristic
36	#427×C₆₇	C₂²	1	1	yes	characteristic
37	#367×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#367×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#366×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#366×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂×#11761628	D₄	1	1	yes	characteristic
42	C₂²×D₂₆₈	—	2	1	no	—
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	—	2	2	no	—
45	C₂×D₅₃₆	—	2	2	no	—
46	C₂×D₅₃₆	C₂³	1	2	yes	—
47	#14524049	—	2	2	no	—
48	#14524049	—	2	2	no	—
49	C₂×#11761612	D₄	1	1	yes	characteristic
50	C₂×#11761611	D₄	1	2	yes	—
51	C₂×C₈×D₆₇	C₂³	1	2	yes	—
52	#14524040	—	4	1	no	—
53	#14524031	—	4	1	no	—
54	#14524018	—	2	2	no	—
55	#14524018	—	2	2	no	—
56	#14524013	—	4	2	no	—
57	C₄×Dic₁₃₄	—	4	1	no	—
58	C₄×D₂₆₈	—	4	1	no	—
59	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
60	#14524001	—	2	2	no	—
61	C₂×C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊C₄×D₆₇	—	2	2	no	—
63	#14523983	—	2	2	no	—
64	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
65	#14523977	—	2	1	no	—
66	#14523977	C₂³	1	2	yes	—
67	#14523976	—	2	2	no	—
68	#14523974	—	2	2	no	—
69	#14523963	—	2	2	no	—
70	#14523963	—	2	2	no	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	#117×C₆₇	—	2	1	no	—
73	#117×C₆₇	—	2	1	no	—
74	C₂×C₁₆×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂³	1	1	yes	characteristic
80	#107×C₆₇	C₂³	1	1	yes	characteristic
81	#106×C₆₇	—	2	1	no	—
82	#106×C₆₇	—	2	1	no	—
83	C₂³×D₆₇	—	4	1	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
87	C₂×D₂₆₈	—	2	2	no	—
88	C₂×D₂₆₈	—	4	2	no	—
89	C₂×D₂₆₈	—	4	2	no	—
90	C₂×D₂₆₈	—	4	1	no	—
91	C₂×D₂₆₈	—	4	2	no	—
92	C₂×D₂₆₈	—	4	2	no	—
93	C₂×D₂₆₈	D₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₆₇	D₈	1	2	yes	—
95	C₂²×Dic₆₇	—	4	1	no	—
96	C₂×C₄×D₆₇	—	4	2	no	—
97	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
98	D₅₃₆	—	2	2	no	—
99	D₅₃₆	—	2	2	no	—
100	D₅₃₆	—	4	2	no	—
101	#11761612	—	2	2	no	—
102	#11761611	—	2	4	no	—
103	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
104	C₈×D₆₇	—	2	2	no	—
105	C₈×D₆₇	—	2	2	no	—
106	C₄⋊Dic₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	—	4	1	no	—
108	Dic₆₇⋊C₄	—	4	2	no	—
109	(C₂×D₆₇)⋊C₄	—	4	2	no	—
110	(C₂×D₆₇)⋊C₄	—	4	2	no	—
111	(C₂×D₆₇)⋊C₄	—	4	2	no	—
112	C₄×Dic₆₇	—	4	2	no	—
113	C₆₇⋊C₁₆	—	2	2	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	#41	1	2	yes	—
117	C₂×C₈×C₆₇	#41	1	2	yes	—
118	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₂²⋊C₄×C₆₇	—	4	1	no	—
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	—	4	1	no	—
127	C₁₆×C₆₇	—	2	1	no	—
128	C₁₆×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₆₇	—	4	2	no	—
131	C₂²×D₆₇	—	8	2	no	—
132	C₂²×D₆₇	—	16	1	no	—
133	C₂²×D₆₇	—	16	1	no	—
134	C₂²×D₆₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	#123	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#137	1	2	yes	—
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₆₇	—	2	4	no	—
144	C₂×Dic₆₇	—	4	2	no	—
145	C₂×Dic₆₇	—	16	1	no	—
146	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
147	D₂₆₈	—	2	2	no	—
148	D₂₆₈	—	4	2	no	—
149	D₂₆₈	—	4	2	no	—
150	D₂₆₈	—	8	2	no	—
151	D₂₆₈	—	16	2	no	—
152	D₂₆₈	—	16	2	no	—
153	C₄×D₆₇	—	2	2	no	—
154	C₄×D₆₇	—	2	2	no	—
155	C₄×D₆₇	—	16	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	C₈×C₆₇	#123	1	2	yes	—
158	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
159	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
160	C₆₇⋊C₈	—	2	2	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	#480	1	2	yes	—
164	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
165	C₂×D₆₇	—	2	2	no	—
166	C₂×D₆₇	—	2	2	no	—
167	C₂×D₆₇	—	2	2	no	—
168	C₂×D₆₇	—	8	2	no	—
169	C₂×D₆₇	—	8	2	no	—
170	C₂×D₆₇	—	8	2	no	—
171	C₂×D₆₇	—	16	4	no	—
172	C₄×C₆₇	—	8	1	no	—
173	C₄×C₆₇	—	8	1	no	—
174	C₄×C₆₇	—	8	1	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	#480	1	2	yes	—
177	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
178	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
179	Dic₆₇	—	2	2	no	—
180	Dic₆₇	—	8	2	no	—
181	#33650	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	2	no	—
183	C₂×C₆₇	C₂×#480	1	1	yes	characteristic
184	C₂×C₆₇	#3415	1	1	yes	characteristic
185	C₂×C₆₇	#3416	1	1	yes	characteristic
186	D₆₇	—	4	2	no	—
187	D₆₇	—	16	2	no	—
188	#3058	—	67	1	no	—
189	#2913	—	67	1	no	—
190	#2234	—	67	2	no	—
191	#2234	—	67	2	no	—
192	#2233	—	67	2	no	—
193	#2232	—	67	2	no	—
194	#2159	—	67	1	no	—
195	#2158	D₆₇	1	1	yes	characteristic, complemented
196	#2140	—	67	1	no	—
197	C₂×#357	—	67	1	no	—
198	#2113	—	67	1	no	—
199	C₆₇	#33650	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	—	67	1	no	—
201	C₂×#131	—	67	1	no	—
202	#535	—	134	1	no	—
203	#520	—	134	1	no	—
204	C₂×M₃₂	—	67	1	no	—
205	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
206	#469	—	134	2	no	—
207	#466	—	67	2	no	—
208	#466	—	67	2	no	—
209	#428	—	134	2	no	—
210	#428	—	134	1	no	—
211	#427	—	67	2	no	—
212	#427	C₂×D₆₇	1	1	yes	characteristic
213	#427	C₂×D₆₇	1	1	yes	characteristic
214	C₂×#107	—	67	1	no	—
215	C₂×#107	—	67	2	no	—
216	#418	—	134	1	no	—
217	#367	—	67	2	no	—
218	#367	C₂×D₆₇	1	1	yes	characteristic, complemented
219	#367	C₂×D₆₇	1	1	yes	characteristic, complemented
220	#366	—	67	2	no	—
221	#366	C₂×D₆₇	1	1	yes	characteristic, complemented
222	#366	C₂×D₆₇	1	1	yes	characteristic, complemented
223	#357	—	67	2	no	—
224	#357	—	67	2	no	—
225	#357	—	67	2	no	—
226	#357	—	67	2	no	—
227	C₂²⋊C₁₆	—	67	2	no	—
228	C₂²⋊C₁₆	—	67	2	no	—
229	C₂×#41	—	67	1	no	—
230	C₂²×D₄	—	134	1	no	—
231	C₂×D₈	—	67	2	no	—
232	C₂×D₈	—	134	1	no	—
233	C₂×D₈	—	134	2	no	—
234	C₂×D₈	—	134	2	no	—
235	#131	—	134	2	no	—
236	#131	—	134	2	no	—
237	C₂×M₁₆	—	67	1	no	—
238	C₂×M₁₆	—	67	2	no	—
239	C₂²×C₈	—	67	2	no	—
240	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	#127	—	268	1	no	—
242	#124	—	268	1	no	—
243	#121	—	134	2	no	—
244	#121	—	134	2	no	—
245	#120	—	268	2	no	—
246	C₄×Q₈	—	268	1	no	—
247	C₄×D₄	—	268	1	no	—
248	C₄×D₄	—	268	2	no	—
249	#117	—	2	1	no	—
250	#117	—	2	1	no	—
251	#117	—	134	2	no	—
252	C₂×C₄⋊C₄	—	134	2	no	—
253	C₂×C₄⋊C₄	—	134	1	no	—
254	M₃₂	—	134	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	—	67	2	no	—
257	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
259	#107	—	67	2	no	—
260	#107	—	134	2	no	—
261	#107	—	134	1	no	—
262	#107	C₂²×D₆₇	1	1	yes	characteristic
263	#107	C₂²×D₆₇	1	1	yes	characteristic
264	#107	C₂²×D₆₇	1	1	yes	characteristic
265	#107	C₂²×D₆₇	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	2	1	no	—
268	#106	—	134	2	no	—
269	#102	—	134	2	no	—
270	#102	—	134	2	no	—
271	C₂⁴	—	268	1	no	—
272	#41	—	134	2	no	—
273	#41	—	134	2	no	—
274	C₂×Q₈	—	67	1	no	—
275	C₂×D₄	—	268	2	no	—
276	C₂×D₄	—	268	2	no	—
277	C₂×D₄	—	268	2	no	—
278	C₂×D₄	—	268	1	no	—
279	C₂×D₄	—	268	2	no	—
280	C₂×D₄	—	67	2	no	—
281	C₂×D₄	—	67	1	no	—
282	C₂×D₄	—	134	2	no	—
283	C₂²×C₄	—	268	2	no	—
284	C₂²×C₄	—	268	1	no	—
285	C₂²×C₄	—	67	2	no	—
286	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
287	D₈	—	268	2	no	—
288	D₈	—	134	2	no	—
289	D₈	—	134	2	no	—
290	M₁₆	—	134	2	no	—
291	M₁₆	—	134	4	no	—
292	C₂×C₈	—	67	2	no	—
293	C₂×C₈	—	134	2	no	—
294	C₂×C₈	—	134	2	no	—
295	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
296	C₂×C₈	#11761628	1	2	yes	—
297	C₂×C₈	#11761629	1	2	yes	—
298	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	268	2	no	—
304	C₄⋊C₄	—	268	1	no	—
305	C₄⋊C₄	—	134	2	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	268	2	no	—
309	C₂²⋊C₄	—	268	2	no	—
310	C₂²⋊C₄	—	268	2	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	1	no	—
313	C₄²	—	268	2	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	134	2	no	—
317	C₂³	—	268	2	no	—
318	C₂³	—	536	2	no	—
319	C₂³	—	1072	1	no	—
320	C₂³	—	1072	1	no	—
321	C₂³	—	67	2	no	—
322	C₂³	D₈×D₆₇	1	1	yes	characteristic
323	Q₈	—	134	2	no	—
324	D₄	—	268	2	no	—
325	D₄	—	268	2	no	—
326	D₄	—	536	2	no	—
327	D₄	—	1072	2	no	—
328	D₄	—	1072	2	no	—
329	D₄	—	134	4	no	—
330	D₄	—	134	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	268	2	no	—
336	C₂×C₄	—	1072	2	no	—
337	C₂×C₄	—	1072	1	no	—
338	C₂×C₄	—	67	2	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	#14524028	1	2	yes	—
342	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
343	C₂×C₄	#14524075	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
345	C₈	—	134	2	no	—
346	C₈	#14524028	1	2	yes	—
347	C₈	C₂×#11761628	1	1	yes	characteristic
348	C₈	C₂×#11761629	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	536	2	no	—
352	C₂²	—	536	2	no	—
353	C₂²	—	536	2	no	—
354	C₂²	—	1072	4	no	—
355	C₂²	—	134	2	no	—
356	C₂²	—	134	2	no	—
357	C₂²	—	134	2	no	—
358	C₂²	#16863711	1	2	yes	—
359	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	536	2	no	—
365	C₄	—	134	2	no	—
366	C₄	#16863711	1	2	yes	—
367	C₄	C₂×#14524028	1	1	yes	characteristic
368	C₄	C₂×#14524075	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	268	2	no	—
371	C₂	—	1072	2	no	—
372	C₂	C₂×#16863711	1	1	yes	characteristic
373	C₂	#22573347	1	1	yes	characteristic
374	C₂	#22573352	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×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22569858	C₂
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33650
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #22568806	C₂	semidirect
3. #22562403	C₂	semidirect
4. #22562402	C₂	semidirect
5. #22562396	C₂	semidirect
6. #22562391	C₂	semidirect
7. #22561998	C₂	semidirect
8. #22561924	C₂	semidirect
9. C₂×#16863198	C₂	semidirect
10. #22561822	C₂	semidirect
11. C₆₇×#2158	C₂	semidirect
22. #16863433	C₂²	semidirect
24. #107×D₆₇	C₂²	semidirect
26. #16863223	C₂²	semidirect
27. #16863221	C₂²	semidirect
32. (C₂×D₆₇)⋊C₁₆	C₂²	semidirect
33. (C₂×D₆₇)⋊C₁₆	C₂²	semidirect
37. #367×C₆₇	C₂²	semidirect
38. #367×C₆₇	C₂²	semidirect
39. #366×C₆₇	C₂²	semidirect
40. #366×C₆₇	C₂²	semidirect
195. #2158	D₆₇	semidirect
199. C₆₇	#33650	semidirect
218. #367	C₂×D₆₇	semidirect
219. #367	C₂×D₆₇	semidirect
221. #366	C₂×D₆₇	semidirect
222. #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
#22569858	2
#22568806	2
#22562403	2
#22562402	2
#22562396	2
#22562391	2
#22561998	2
#22561924	2
C₂×#16863198	2
#22561822	2
C₆₇×#2158	2
#33650	67