Subgroups — #31012667

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572603	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22562433	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562396	C₂	1	2	yes	maximal, complemented
5	#22562356	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562308	C₂	1	2	yes	maximal, complemented
7	#22562249	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562136	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₆₇	C₂²	1	2	yes	—
13	#16864106	—	2	1	no	—
14	D₄×D₂₆₈	—	2	1	no	—
15	C₂×#14523986	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₆₇	C₂²	1	2	yes	complemented
17	#16863801	C₂²	1	1	yes	characteristic, complemented
18	#16863773	C₂²	1	1	yes	characteristic
19	#16863730	—	2	2	no	—
20	#16863691	—	2	1	no	—
21	#16863634	C₂²	1	2	yes	complemented
22	#16863633	C₂²	1	2	yes	—
23	#16863591	—	2	1	no	—
24	#16863531	—	2	2	no	—
25	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
26	#107×D₆₇	C₂²	1	2	yes	complemented
27	#16863223	C₂²	1	2	yes	complemented
28	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
29	#16863198	C₂²	1	2	yes	complemented
30	#16863198	C₂²	1	2	yes	complemented
31	#16863197	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	complemented
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
37	#367×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#358×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#357×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₂₆₈	—	4	1	no	—
42	C₂×D₄×D₆₇	—	2	2	no	—
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	C₂×#11761613	—	2	2	no	—
46	D₈×D₆₇	—	2	4	no	—
47	D₈×D₆₇	—	4	2	no	—
48	C₂×C₈×D₆₇	C₂³	1	2	yes	—
49	#14524043	—	2	1	no	—
50	#14524040	—	2	1	no	—
51	#14524035	—	4	1	no	—
52	#14524021	—	4	2	no	—
53	#14524019	—	2	2	no	—
54	#14524018	—	2	2	no	—
55	#14524017	—	4	1	no	—
56	#14524013	—	4	2	no	—
57	C₄×D₂₆₈	—	4	1	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	#14523986	—	4	2	no	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₆₇	—	4	2	no	—
63	#14523977	—	2	1	no	—
64	#14523977	C₂³	1	2	yes	—
65	#14523976	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	#14523967	—	2	2	no	—
68	#14523963	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	—	2	1	no	—
73	C₄×D₄×C₆₇	—	2	1	no	—
74	SD₃₂×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	#103×C₆₇	—	2	1	no	—
80	#102×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×D₆₇	—	4	2	no	—
83	C₂×Dic₁₃₄	—	4	1	no	—
84	C₂×D₂₆₈	—	2	2	no	—
85	C₂×D₂₆₈	—	4	2	no	—
86	C₂×D₂₆₈	—	4	1	no	—
87	C₂×D₂₆₈	—	4	1	no	—
88	C₂×D₂₆₈	—	8	4	no	—
89	C₂×D₂₆₈	D₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₆₇	—	4	2	no	—
91	D₄×D₆₇	—	4	4	no	—
92	D₄×D₆₇	—	8	2	no	—
93	C₂×C₄×D₆₇	—	4	2	no	—
94	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
95	D₅₃₆	—	2	2	no	—
96	D₅₃₆	—	2	2	no	—
97	D₅₃₆	—	8	1	no	—
98	#11761613	—	4	4	no	—
99	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
100	C₈×D₆₇	—	2	2	no	—
101	C₈×D₆₇	—	2	2	no	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	—	4	1	no	—
104	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
105	Dic₆₇⋊C₄	—	4	2	no	—
106	Dic₆₇⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	(C₂×D₆₇)⋊C₄	—	4	2	no	—
109	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
110	C₆₇⋊C₁₆	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	4	1	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	—	2	1	no	—
116	D₈×C₆₇	—	2	1	no	—
117	D₈×C₆₇	#41	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₂²⋊C₄×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₆₇	—	2	1	no	—
127	C₁₆×C₆₇	—	2	1	no	—
128	C₂³×C₆₇	—	4	1	no	—
129	C₂²×D₆₇	—	4	2	no	—
130	C₂²×D₆₇	—	8	4	no	—
131	C₂²×D₆₇	—	8	4	no	—
132	C₂²×D₆₇	—	16	2	no	—
133	C₂²×D₆₇	C₂×D₈	1	2	yes	—
134	Q₈×C₆₇	—	4	1	no	—
135	Q₈×C₆₇	—	4	1	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	8	2	no	—
142	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₆₇	—	8	4	no	—
146	C₂×Dic₆₇	—	4	2	no	—
147	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
148	D₂₆₈	—	4	2	no	—
149	D₂₆₈	—	4	2	no	—
150	D₂₆₈	—	8	4	no	—
151	D₂₆₈	—	8	2	no	—
152	D₂₆₈	—	16	1	no	—
153	D₂₆₈	SD₃₂	1	4	yes	complemented
154	C₄×D₆₇	—	2	2	no	—
155	C₄×D₆₇	—	2	2	no	—
156	C₄×D₆₇	—	16	2	no	—
157	Dic₁₃₄	—	16	1	no	—
158	C₈×C₆₇	—	2	1	no	—
159	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
160	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
161	C₆₇⋊C₈	—	2	2	no	—
162	C₂²×C₆₇	—	4	2	no	—
163	C₂²×C₆₇	—	8	1	no	—
164	C₂²×C₆₇	—	8	1	no	—
165	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
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	8	no	—
172	C₂×D₆₇	C₂×SD₃₂	1	4	yes	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	8	1	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	—	8	1	no	—
177	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
178	C₄×C₆₇	#546	1	1	yes	characteristic
179	C₄×C₆₇	#577	1	1	yes	characteristic
180	Dic₆₇	—	2	2	no	—
181	Dic₆₇	—	8	2	no	—
182	#33862	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	—	8	2	no	—
184	C₂×C₆₇	#3294	1	1	yes	characteristic
185	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
186	C₂×C₆₇	#3415	1	1	yes	characteristic
187	D₆₇	—	2	4	no	—
188	D₆₇	—	16	2	no	—
189	#3338	—	67	1	no	—
190	D₄×D₈	—	67	1	no	—
191	#2240	—	67	1	no	—
192	#2233	—	67	2	no	—
193	#2226	—	67	1	no	—
194	#2217	—	67	2	no	—
195	#2207	—	67	1	no	—
196	#2187	—	67	2	no	—
197	C₄×SD₃₂	D₆₇	1	1	yes	characteristic, complemented
198	D₄×C₁₆	—	67	1	no	—
199	C₂×#357	—	67	2	no	—
200	C₆₇	#33862	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₈	—	67	2	no	—
202	#554	—	134	1	no	—
203	D₄²	—	134	1	no	—
204	C₂×SD₃₂	—	67	2	no	—
205	C₂×SD₃₂	C₂×D₆₇	1	1	yes	characteristic
206	C₂²×C₁₆	—	67	2	no	—
207	#500	—	67	1	no	—
208	#493	—	67	1	no	—
209	#484	—	134	2	no	—
210	#477	—	134	1	no	—
211	#467	—	67	2	no	—
212	#466	—	67	2	no	—
213	#459	—	134	1	no	—
214	#447	—	134	2	no	—
215	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
216	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
217	C₈×D₄	—	67	1	no	—
218	C₂×#107	—	67	2	no	—
219	#367	—	67	2	no	—
220	#367	C₂×D₆₇	1	1	yes	characteristic, complemented
221	C₄⋊C₁₆	—	67	1	no	—
222	#358	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	#357	C₂×D₆₇	1	1	yes	characteristic
228	C₂²⋊C₁₆	—	67	2	no	—
229	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
230	C₂²×D₄	—	268	1	no	—
231	C₂²×D₄	—	134	2	no	—
232	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
233	C₂×D₈	—	268	2	no	—
234	C₂×D₈	—	67	2	no	—
235	C₂×D₈	—	134	2	no	—
236	C₂×D₈	—	134	4	no	—
237	C₂×D₈	—	134	1	no	—
238	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
239	C₂²×C₈	—	67	2	no	—
240	C₄⋊Q₈	—	134	1	no	—
241	#127	—	134	1	no	—
242	#125	—	268	1	no	—
243	C₂²⋊Q₈	—	268	2	no	—
244	C₂²⋊Q₈	—	134	2	no	—
245	#121	—	268	1	no	—
246	#121	—	134	2	no	—
247	#120	—	268	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	268	1	no	—
251	C₄×D₄	—	67	1	no	—
252	C₂×C₄⋊C₄	—	134	2	no	—
253	SD₃₂	—	2	2	no	—
254	SD₃₂	—	268	2	no	—
255	C₂×C₁₆	—	268	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	C₄⋊C₈	—	67	1	no	—
265	#103	—	2	1	no	—
266	#103	—	134	2	no	—
267	#102	—	2	1	no	—
268	#102	—	134	2	no	—
269	C₂²⋊C₈	—	67	2	no	—
270	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	C₂⁴	—	268	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	268	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	268	2	no	—
276	C₂×D₄	—	268	2	no	—
277	C₂×D₄	—	268	4	no	—
278	C₂×D₄	—	268	1	no	—
279	C₂×D₄	—	268	1	no	—
280	C₂×D₄	—	536	4	no	—
281	C₂×D₄	—	536	2	no	—
282	C₂×D₄	—	67	1	no	—
283	C₂×D₄	—	134	2	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	268	2	no	—
286	C₂²×C₄	—	67	2	no	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	1	no	—
289	D₈	—	2	1	no	—
290	D₈	—	268	4	no	—
291	D₈	—	536	1	no	—
292	D₈	—	134	2	no	—
293	D₈	—	134	2	no	—
294	D₈	#11761628	1	2	yes	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	268	2	no	—
304	C₄⋊C₄	—	268	2	no	—
305	C₄⋊C₄	—	268	1	no	—
306	C₄⋊C₄	—	67	1	no	—
307	C₄⋊C₄	—	134	2	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	268	2	no	—
310	C₂²⋊C₄	—	268	2	no	—
311	C₂²⋊C₄	—	67	2	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₆₇	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	134	2	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	268	2	no	—
319	C₂³	—	536	4	no	—
320	C₂³	—	536	4	no	—
321	C₂³	—	1072	2	no	—
322	C₂³	—	67	2	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	1072	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	268	2	no	—
329	D₄	—	268	2	no	—
330	D₄	—	536	4	no	—
331	D₄	—	536	4	no	—
332	D₄	—	536	2	no	—
333	D₄	—	1072	1	no	—
334	D₄	—	67	4	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	268	2	no	—
340	C₂×C₄	—	1072	2	no	—
341	C₂×C₄	—	67	2	no	—
342	C₂×C₄	—	134	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
345	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	134	2	no	—
349	C₈	C₂×#11761628	1	1	yes	characteristic
350	C₈	#14524089	1	1	yes	characteristic
351	C₂²	—	4	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	536	2	no	—
355	C₂²	—	536	2	no	—
356	C₂²	—	536	2	no	—
357	C₂²	—	1072	8	no	—
358	C₂²	—	67	4	no	—
359	C₂²	—	134	2	no	—
360	C₂²	—	134	2	no	—
361	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	536	2	no	—
367	C₄	—	134	2	no	—
368	C₄	SD₃₂×D₆₇	1	2	yes	—
369	C₄	#16864042	1	1	yes	characteristic
370	C₄	#16864203	1	1	yes	characteristic
371	C₂	—	8	2	no	—
372	C₂	—	1072	2	no	—
373	C₂	—	134	4	no	—
374	C₂	#22572024	1	1	yes	characteristic
375	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
376	C₂	#22573347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×SD₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	D₈×D₂₆₈	C₂
hypercenter	C₄×SD₃₂	D₆₇
hypocenter	C₆₇	#33862
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22572603	C₂	semidirect
3. #22562433	C₂	semidirect
4. #22562396	C₂	semidirect
5. #22562356	C₂	semidirect
6. #22562308	C₂	semidirect
7. #22562249	C₂	semidirect
8. #22562136	C₂	semidirect
9. C₁₆×D₂₆₈	C₂	semidirect
10. #357×D₆₇	C₂	semidirect
11. C₄×SD₃₂×C₆₇	C₂	semidirect
15. C₂×#14523986	C₂²	semidirect
16. C₂×C₁₆×D₆₇	C₂²	semidirect
17. #16863801	C₂²	semidirect
21. #16863634	C₂²	semidirect
26. #107×D₆₇	C₂²	semidirect
27. #16863223	C₂²	semidirect
28. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
29. #16863198	C₂²	semidirect
30. #16863198	C₂²	semidirect
32. #16863196	C₂²	semidirect
33. (C₂×D₆₇)⋊C₁₆	C₂²	semidirect
35. C₄×Q₁₆×C₆₇	C₂²	semidirect
37. #367×C₆₇	C₂²	semidirect
38. #358×C₆₇	C₂²	semidirect
40. C₄×C₁₆×C₆₇	C₂²	semidirect
153. D₂₆₈	SD₃₂	semidirect
197. C₄×SD₃₂	D₆₇	semidirect
200. C₆₇	#33862	semidirect
215. C₄×Q₁₆	C₂×D₆₇	semidirect
220. #367	C₂×D₆₇	semidirect
222. #358	C₂×D₆₇	semidirect
229. 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
#22572603	2
D₈×D₂₆₈	2
#22562433	2
#22562396	2
#22562356	2
#22562308	2
#22562249	2
#22562136	2
C₁₆×D₂₆₈	2
#357×D₆₇	2
C₄×SD₃₂×C₆₇	2
#33862	67