Subgroups — #30847385

Order 17152.

604 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	Q₈×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22572632	C₂	1	1	yes	maximal, characteristic
3	#22572505	C₂	1	2	yes	maximal, complemented
4	#22559819	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559785	C₂	1	2	yes	maximal, complemented
6	#22559613	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559598	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559439	C₂	1	2	yes	maximal, complemented
9	#22559376	C₂	1	2	yes	maximal, complemented
10	#22559095	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864111	C₂²	1	2	yes	—
13	Q₈×D₂₆₈	C₂²	1	2	yes	—
14	#16863801	—	2	1	no	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863801	C₂²	1	1	yes	characteristic
17	#16863747	C₂²	1	2	yes	—
18	#16863747	C₂²	1	2	yes	—
19	#16863740	—	2	2	no	—
20	#16863721	—	2	2	no	—
21	#16863691	C₂²	1	2	yes	—
22	#16863687	C₂²	1	2	yes	—
23	#16863677	—	2	2	no	—
24	#16863677	C₂²	1	2	yes	—
25	#16863641	—	2	2	no	—
26	#16863632	—	2	1	no	—
27	#16863601	C₂²	1	2	yes	—
28	#16863533	—	2	2	no	—
29	C₄×D₅₃₆	—	2	1	no	—
30	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
31	#16863475	C₂²	1	2	yes	—
32	#16863434	—	2	1	no	—
33	#16863403	—	2	2	no	—
34	#16863149	C₂²	1	1	yes	characteristic
35	#16863144	C₂²	1	2	yes	—
36	#16863141	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	#334×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	—	2	2	no	—
46	C₂×#11761616	—	2	2	no	—
47	C₂×D₅₃₆	D₄	1	1	yes	characteristic
48	C₂×#11761611	—	2	2	no	—
49	#14524043	—	2	2	no	—
50	#14524043	C₂³	1	2	yes	—
51	#14524043	C₂³	1	2	yes	—
52	#14524029	—	4	2	no	—
53	#14524023	—	2	2	no	—
54	#14524023	—	4	2	no	—
55	#14524017	—	2	2	no	—
56	#14524014	—	2	2	no	—
57	C₄×D₂₆₈	—	2	2	no	—
58	C₄×D₂₆₈	C₂³	1	2	yes	—
59	#14524002	—	2	2	no	—
60	#14524001	—	2	2	no	—
61	#14523977	—	2	2	no	—
62	#14523977	—	2	1	no	—
63	#14523977	—	2	1	no	—
64	#14523977	—	2	1	no	—
65	#14523977	—	2	1	no	—
66	#14523977	D₄	1	1	yes	characteristic
67	#14523976	—	2	2	no	—
68	#14523975	—	2	2	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
70	#14523968	—	2	2	no	—
71	#14523968	—	2	2	no	—
72	#14523968	—	4	2	no	—
73	#14523967	—	2	2	no	—
74	#14523966	—	2	2	no	—
75	#14523965	—	2	2	no	—
76	#14523965	—	2	2	no	—
77	#14523965	D₄	1	2	yes	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	—	2	1	no	—
81	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
83	#106×C₆₇	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	—	2	1	no	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#103×C₆₇	D₄	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₄×C₈×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₃₄	—	2	2	no	—
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	—	4	2	no	—
97	Q₈×D₆₇	—	4	4	no	—
98	Q₈×D₆₇	—	8	2	no	—
99	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
100	C₂×C₂²⋊D₆₇	—	4	2	no	—
101	C₂×C₄×D₆₇	—	2	2	no	—
102	C₂×C₄×D₆₇	—	4	2	no	—
103	#11761616	—	8	2	no	—
104	D₅₃₆	—	2	2	no	—
105	#11761611	—	8	2	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	C₄⋊Dic₆₇	—	2	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	—	2	2	no	—
111	C₄⋊Dic₆₇	—	4	2	no	—
112	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
113	C₂²⋊Dic₆₇	—	4	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	(C₂×D₆₇)⋊C₄	—	4	2	no	—
116	(C₂×D₆₇)⋊C₄	—	4	2	no	—
117	C₄×Dic₆₇	—	4	2	no	—
118	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	SD₁₆×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	4	2	no	—
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₆₇	—	2	1	no	—
134	C₄²×C₆₇	—	2	1	no	—
135	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₆₇	—	2	1	no	—
137	C₂²×D₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	2	1	no	—
139	Q₈×C₆₇	#135	1	2	yes	—
140	D₄×C₆₇	—	2	1	no	—
141	D₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₆₇	—	16	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	4	2	no	—
153	C₂×Dic₆₇	—	4	2	no	—
154	D₂₆₈	—	2	2	no	—
155	D₂₆₈	—	2	2	no	—
156	C₄×D₆₇	—	4	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	C₄×D₆₇	—	8	4	no	—
159	Dic₁₃₄	—	4	2	no	—
160	Dic₁₃₄	—	4	2	no	—
161	Dic₁₃₄	—	8	4	no	—
162	Dic₁₃₄	—	8	2	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	4	1	no	—
167	C₈×C₆₇	—	4	1	no	—
168	C₈×C₆₇	—	4	1	no	—
169	C₈×C₆₇	#137	1	2	yes	—
170	C₆₇⋊C₈	—	4	2	no	—
171	C₂²×C₆₇	—	4	1	no	—
172	C₂²×C₆₇	—	4	1	no	—
173	C₂²×C₆₇	—	4	1	no	—
174	C₂²×C₆₇	D₄²	1	1	yes	characteristic
175	C₂×D₆₇	—	4	2	no	—
176	C₂×D₆₇	—	4	2	no	—
177	C₂×D₆₇	—	4	2	no	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	—	4	1	no	—
182	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
183	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
184	C₄×C₆₇	#576	1	1	yes	characteristic
185	C₄×C₆₇	#578	1	1	yes	characteristic
186	Dic₆₇	—	4	2	no	—
187	Dic₆₇	—	8	2	no	—
188	Dic₆₇	—	8	2	no	—
189	#19905	—	67	1	no	maximal, Sylow
190	C₂×C₆₇	—	8	1	no	—
191	C₂×C₆₇	#3276	1	1	yes	characteristic
192	C₂×C₆₇	#3280	1	1	yes	characteristic
193	C₂×C₆₇	#3290	1	1	yes	characteristic
194	D₆₇	—	8	2	no	—
195	Q₈×D₈	—	67	1	no	—
196	#3341	—	67	1	no	—
197	#3329	—	67	2	no	—
198	#1710	—	67	1	no	—
199	#1703	—	67	2	no	—
200	#1676	—	67	1	no	—
201	#1673	—	67	1	no	—
202	#1646	—	67	2	no	—
203	#1637	—	67	2	no	—
204	#1589	—	67	1	no	—
205	C₈×SD₁₆	D₆₇	1	1	yes	characteristic, complemented
206	C₆₇	#19905	1	1	yes	characteristic, complemented, Sylow
207	#555	—	67	2	no	—
208	D₄×Q₈	—	67	2	no	—
209	#500	—	67	1	no	—
210	#500	—	67	1	no	—
211	#500	—	134	1	no	—
212	#487	—	67	2	no	—
213	#487	—	67	2	no	—
214	#485	—	134	2	no	—
215	#482	—	134	2	no	—
216	#477	—	67	2	no	—
217	#476	—	67	2	no	—
218	#474	—	67	2	no	—
219	#474	—	134	2	no	—
220	#468	—	134	2	no	—
221	#466	—	134	1	no	—
222	#461	—	67	2	no	—
223	#448	—	134	2	no	—
224	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
225	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
226	C₄×D₈	—	67	1	no	—
227	C₄×D₈	—	134	1	no	—
228	#436	—	67	2	no	—
229	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
230	#427	—	134	1	no	—
231	#419	—	134	2	no	—
232	#335	—	67	1	no	—
233	#334	C₂×D₆₇	1	1	yes	characteristic
234	#333	—	67	2	no	—
235	#332	—	67	2	no	—
236	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
237	D₄⋊C₈	—	67	2	no	—
238	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
239	C₈²	C₂×D₆₇	1	1	yes	characteristic
240	C₂²×Q₈	—	134	2	no	—
241	C₂×SD₁₆	—	134	2	no	—
242	C₂×SD₁₆	D₂₆₈	1	1	yes	characteristic
243	C₂×D₈	—	67	1	no	—
244	C₂×M₁₆	—	134	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	67	2	no	—
247	C₄⋊Q₈	—	67	2	no	—
248	C₄⋊Q₈	—	134	2	no	—
249	#124	—	268	2	no	—
250	C₂²⋊Q₈	—	268	2	no	—
251	C₂²⋊Q₈	—	134	2	no	—
252	#121	—	134	2	no	—
253	#121	—	134	2	no	—
254	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
255	C₄×D₄	—	67	2	no	—
256	C₄×D₄	—	134	2	no	—
257	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
258	#117	—	134	2	no	—
259	#117	—	134	2	no	—
260	#107	—	67	1	no	—
261	#107	—	134	2	no	—
262	#107	—	134	1	no	—
263	#107	—	134	1	no	—
264	#107	—	134	1	no	—
265	#107	—	134	1	no	—
266	#107	—	134	2	no	—
267	#106	—	134	2	no	—
268	#106	D₂₆₈	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	67	2	no	—
271	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
273	#103	—	268	2	no	—
274	#103	—	134	2	no	—
275	#103	—	134	2	no	—
276	#103	—	134	2	no	—
277	#103	—	134	2	no	—
278	#103	D₂₆₈	1	1	yes	characteristic
279	#102	—	67	2	no	—
280	#102	—	134	2	no	—
281	#102	—	134	2	no	—
282	#102	D₂₆₈	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	134	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₂×Q₈	—	268	2	no	—
290	C₂×Q₈	—	268	4	no	—
291	C₂×Q₈	—	536	2	no	—
292	C₂×Q₈	—	134	2	no	—
293	C₂×Q₈	—	134	2	no	—
294	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
295	C₂×D₄	—	268	2	no	—
296	C₂×D₄	—	67	2	no	—
297	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	268	2	no	—
300	C₂²×C₄	—	134	2	no	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	536	2	no	—
303	D₈	—	134	2	no	—
304	M₁₆	—	536	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
311	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
312	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
313	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	268	2	no	—
316	C₄⋊C₄	—	67	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	—	134	2	no	—
321	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
322	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	268	2	no	—
325	C₂²⋊C₄	—	268	2	no	—
326	C₂²⋊C₄	—	268	2	no	—
327	C₂²⋊C₄	—	134	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	268	2	no	—
330	C₄²	C₂³×D₆₇	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	134	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	268	2	no	—
335	Q₈	—	268	2	no	—
336	Q₈	—	536	4	no	—
337	Q₈	—	536	2	no	—
338	Q₈	#14524065	1	2	yes	—
339	D₄	—	2	1	no	—
340	D₄	—	2	1	no	—
341	D₄	—	1072	2	no	—
342	D₄	—	134	2	no	—
343	D₄	—	134	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	268	2	no	—
350	C₂×C₄	—	268	2	no	—
351	C₂×C₄	—	268	2	no	—
352	C₂×C₄	—	268	2	no	—
353	C₂×C₄	—	536	4	no	—
354	C₂×C₄	—	134	2	no	—
355	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
357	C₂×C₄	#14524089	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	268	2	no	—
365	C₈	#14524072	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	268	2	no	—
370	C₂²	—	268	2	no	—
371	C₂²	—	268	2	no	—
372	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	268	2	no	—
378	C₄	—	536	2	no	—
379	C₄	—	536	2	no	—
380	C₄	C₂×#14524065	1	1	yes	characteristic
381	C₄	C₂×#14524072	1	1	yes	characteristic
382	C₄	#16864201	1	1	yes	characteristic
383	C₄	#16864210	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	536	2	no	—
386	C₂	#22571856	1	1	yes	characteristic
387	C₂	#22571896	1	1	yes	characteristic
388	C₂	#22571996	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×SD₁₆×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
prime radical	#22572632	C₂
hypercenter	C₈×SD₁₆	D₆₇
hypocenter	C₆₇	#19905
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₅₃₆	C₂	semidirect
3. #22572505	C₂	semidirect
4. #22559819	C₂	semidirect
5. #22559785	C₂	semidirect
6. #22559613	C₂	semidirect
7. #22559598	C₂	semidirect
8. #22559439	C₂	semidirect
9. #22559376	C₂	semidirect
10. #22559095	C₂	semidirect
11. C₈×SD₁₆×C₆₇	C₂	semidirect
205. C₈×SD₁₆	D₆₇	semidirect
206. C₆₇	#19905	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₅₃₆	2
#22572632	2
#22572505	2
#22559819	2
#22559785	2
#22559613	2
#22559598	2
#22559439	2
#22559376	2
#22559095	2
C₈×SD₁₆×C₆₇	2
#19905	67