Subgroups — #30847411

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573198	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₅₃₆	C₂	1	1	yes	maximal, characteristic
3	#22572494	C₂	1	2	yes	maximal, complemented
4	#22559806	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559757	C₂	1	2	yes	maximal, complemented
6	#22559603	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559579	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559481	C₂	1	2	yes	maximal, complemented
9	#22559475	C₂	1	2	yes	maximal, complemented
10	#22559095	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864119	C₂²	1	2	yes	—
13	Q₈×D₂₆₈	C₂²	1	2	yes	—
14	#16863801	—	2	1	no	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863794	C₂²	1	2	yes	—
17	#16863780	—	2	1	no	—
18	#16863776	C₂²	1	1	yes	characteristic
19	#16863747	C₂²	1	2	yes	—
20	#16863739	—	2	2	no	—
21	#16863681	C₂²	1	2	yes	—
22	#16863677	—	2	2	no	—
23	#16863677	C₂²	1	2	yes	—
24	#16863653	—	2	2	no	—
25	#16863619	—	2	2	no	—
26	#16863606	C₂²	1	2	yes	—
27	#16863602	C₂²	1	2	yes	—
28	#16863550	—	2	2	no	—
29	C₄×Dic₂₆₈	—	2	1	no	—
30	C₄×D₅₃₆	—	2	1	no	—
31	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
32	#16863475	C₂²	1	2	yes	—
33	#16863403	—	2	2	no	—
34	#16863149	C₂²	1	1	yes	characteristic
35	#16863143	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	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	—	2	2	no	—
46	C₂×Dic₂₆₈	—	2	1	no	—
47	C₂×Dic₂₆₈	—	2	1	no	—
48	C₂×#11761618	—	2	2	no	—
49	C₂×#11761617	—	2	2	no	—
50	C₂×D₅₃₆	D₄	1	1	yes	characteristic
51	C₂×#11761611	—	2	2	no	—
52	#14524043	—	2	2	no	—
53	#14524043	C₂³	1	2	yes	—
54	#14524043	C₂³	1	2	yes	—
55	#14524038	—	4	2	no	—
56	#14524031	—	2	2	no	—
57	#14524023	—	2	2	no	—
58	#14524023	—	4	2	no	—
59	#14524019	—	2	2	no	—
60	C₄×Dic₁₃₄	—	2	2	no	—
61	C₄×D₂₆₈	—	2	2	no	—
62	C₄×D₂₆₈	C₂³	1	2	yes	—
63	#14524001	—	2	2	no	—
64	#14523977	—	2	2	no	—
65	#14523977	—	2	1	no	—
66	#14523977	—	2	1	no	—
67	#14523977	D₄	1	1	yes	characteristic
68	#14523975	—	4	2	no	—
69	#14523974	—	2	2	no	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	#14523968	—	2	2	no	—
72	#14523968	—	2	2	no	—
73	#14523968	—	2	2	no	—
74	#14523967	—	2	2	no	—
75	#14523966	—	2	2	no	—
76	#14523965	—	2	2	no	—
77	#14523965	D₄	1	2	yes	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	#107×C₆₇	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	—	2	1	no	—
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	#103×C₆₇	D₄	1	1	yes	characteristic
89	C₄×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×C₄×D₆₇	—	2	2	no	—
101	C₂×C₄×D₆₇	—	4	2	no	—
102	Dic₂₆₈	—	4	2	no	—
103	Dic₂₆₈	—	8	1	no	—
104	#11761618	—	8	2	no	—
105	#11761617	—	8	2	no	—
106	D₅₃₆	—	2	2	no	—
107	#11761611	—	8	2	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	C₄⋊Dic₆₇	—	2	2	no	—
111	C₄⋊Dic₆₇	—	2	2	no	—
112	C₄⋊Dic₆₇	—	2	2	no	—
113	C₄⋊Dic₆₇	—	4	2	no	—
114	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
115	Dic₆₇⋊C₄	—	4	2	no	—
116	(C₂×D₆₇)⋊C₄	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	4	2	no	—
118	(C₂×D₆₇)⋊C₄	—	4	2	no	—
119	(C₂×D₆₇)⋊C₄	—	4	2	no	—
120	C₄×Dic₆₇	—	4	2	no	—
121	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
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	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂²×D₆₇	—	2	2	no	—
139	Q₈×C₆₇	—	2	1	no	—
140	Q₈×C₆₇	—	2	1	no	—
141	Q₈×C₆₇	—	2	1	no	—
142	Q₈×C₆₇	#135	1	2	yes	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	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	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	4	2	no	—
152	C₂×Dic₆₇	—	4	2	no	—
153	D₂₆₈	—	2	2	no	—
154	D₂₆₈	—	2	2	no	—
155	C₄×D₆₇	—	4	2	no	—
156	C₄×D₆₇	—	4	2	no	—
157	C₄×D₆₇	—	8	4	no	—
158	Dic₁₃₄	—	4	2	no	—
159	Dic₁₃₄	—	4	2	no	—
160	Dic₁₃₄	—	4	2	no	—
161	Dic₁₃₄	—	4	2	no	—
162	Dic₁₃₄	—	8	4	no	—
163	C₈×C₆₇	—	2	2	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	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	D₄²	1	1	yes	characteristic
172	C₂×D₆₇	—	4	2	no	—
173	C₂×D₆₇	—	4	2	no	—
174	C₂×D₆₇	—	4	2	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	4	1	no	—
178	C₄×C₆₇	—	4	1	no	—
179	C₄×C₆₇	—	4	1	no	—
180	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
181	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
182	C₄×C₆₇	#576	1	1	yes	characteristic
183	C₄×C₆₇	#577	1	1	yes	characteristic
184	Dic₆₇	—	4	2	no	—
185	Dic₆₇	—	8	2	no	—
186	Dic₆₇	—	8	2	no	—
187	#19907	—	67	1	no	maximal, Sylow
188	C₂×C₆₇	#3278	1	1	yes	characteristic
189	C₂×C₆₇	#3280	1	1	yes	characteristic
190	C₂×C₆₇	#3293	1	1	yes	characteristic
191	D₆₇	—	8	2	no	—
192	#3393	—	67	1	no	—
193	Q₈×D₈	—	67	1	no	—
194	#3328	—	67	2	no	—
195	#1707	—	67	1	no	—
196	#1699	—	67	2	no	—
197	#1674	—	67	1	no	—
198	#1670	—	67	1	no	—
199	#1652	—	67	2	no	—
200	#1651	—	67	2	no	—
201	#1589	—	67	1	no	—
202	C₈×Q₁₆	D₆₇	1	1	yes	characteristic, complemented
203	C₆₇	#19907	1	1	yes	characteristic, complemented, Sylow
204	#556	—	67	2	no	—
205	D₄×Q₈	—	67	2	no	—
206	#500	—	67	1	no	—
207	#500	—	134	1	no	—
208	#498	—	67	2	no	—
209	#495	—	134	1	no	—
210	#494	—	67	1	no	—
211	#487	—	67	2	no	—
212	#485	—	134	2	no	—
213	#475	—	67	2	no	—
214	#474	—	67	2	no	—
215	#474	—	134	2	no	—
216	#470	—	134	2	no	—
217	#464	—	134	2	no	—
218	#462	—	67	2	no	—
219	#461	—	67	2	no	—
220	#451	—	134	2	no	—
221	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
222	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
223	C₄×Q₁₆	—	134	1	no	—
224	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
225	C₄×D₈	—	67	1	no	—
226	C₄×D₈	—	134	1	no	—
227	#436	—	67	2	no	—
228	#419	—	134	2	no	—
229	#335	—	67	1	no	—
230	#335	C₂×D₆₇	1	1	yes	characteristic
231	#332	—	67	2	no	—
232	#332	—	67	2	no	—
233	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
234	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
235	D₄⋊C₈	—	67	2	no	—
236	C₈²	C₂×D₆₇	1	1	yes	characteristic
237	C₂²×Q₈	—	134	2	no	—
238	C₂×Q₁₆	—	134	2	no	—
239	C₂×Q₁₆	—	134	1	no	—
240	C₂×Q₁₆	—	134	1	no	—
241	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic
242	C₂×SD₁₆	—	134	2	no	—
243	C₂×D₈	—	67	1	no	—
244	C₂×M₁₆	—	134	2	no	—
245	C₄⋊Q₈	—	67	2	no	—
246	C₄⋊Q₈	—	67	2	no	—
247	C₄⋊Q₈	—	134	2	no	—
248	#126	—	268	2	no	—
249	#124	—	134	2	no	—
250	C₂²⋊Q₈	—	268	2	no	—
251	C₂²⋊Q₈	—	134	2	no	—
252	C₂²⋊Q₈	—	134	2	no	—
253	C₄×Q₈	—	134	2	no	—
254	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
255	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
256	C₄×D₄	—	67	2	no	—
257	C₄×D₄	—	134	2	no	—
258	#117	—	134	2	no	—
259	#107	—	67	1	no	—
260	#107	—	134	2	no	—
261	#107	—	134	1	no	—
262	#107	—	134	1	no	—
263	#107	D₂₆₈	1	1	yes	characteristic
264	#106	—	268	2	no	—
265	#106	—	134	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	67	2	no	—
269	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
271	#103	—	134	2	no	—
272	#103	—	134	2	no	—
273	#103	—	134	2	no	—
274	#103	—	134	2	no	—
275	#103	—	134	2	no	—
276	#103	D₂₆₈	1	1	yes	characteristic
277	#103	D₂₆₈	1	1	yes	characteristic
278	#102	—	67	2	no	—
279	#102	—	134	2	no	—
280	C₂²⋊C₈	—	134	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
286	C₂×Q₈	—	268	2	no	—
287	C₂×Q₈	—	268	4	no	—
288	C₂×Q₈	—	536	2	no	—
289	C₂×Q₈	—	134	2	no	—
290	C₂×Q₈	—	134	2	no	—
291	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
292	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
293	C₂×D₄	—	67	2	no	—
294	C₂²×C₄	—	268	2	no	—
295	C₂²×C₄	—	134	2	no	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	268	2	no	—
298	Q₁₆	—	536	1	no	—
299	Q₁₆	—	536	2	no	—
300	SD₁₆	—	536	2	no	—
301	D₈	—	134	2	no	—
302	M₁₆	—	536	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
310	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	268	2	no	—
314	C₄⋊C₄	—	268	2	no	—
315	C₄⋊C₄	—	67	2	no	—
316	C₄⋊C₄	—	134	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	—	134	2	no	—
319	C₄⋊C₄	—	134	2	no	—
320	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
321	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
322	C₂²⋊C₄	—	268	2	no	—
323	C₂²⋊C₄	—	268	2	no	—
324	C₂²⋊C₄	—	268	2	no	—
325	C₂²⋊C₄	—	134	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	268	2	no	—
329	C₄²	C₂³×D₆₇	1	1	yes	characteristic
330	C₂³	—	134	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	268	2	no	—
335	Q₈	—	268	2	no	—
336	Q₈	—	268	2	no	—
337	Q₈	—	268	2	no	—
338	Q₈	—	536	4	no	—
339	Q₈	#14524065	1	2	yes	—
340	D₄	—	134	2	no	—
341	D₄	—	134	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	268	2	no	—
347	C₂×C₄	—	268	2	no	—
348	C₂×C₄	—	268	2	no	—
349	C₂×C₄	—	268	2	no	—
350	C₂×C₄	—	536	4	no	—
351	C₂×C₄	—	134	2	no	—
352	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
354	C₂×C₄	#14524089	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	268	2	no	—
362	C₈	#14524075	1	2	yes	—
363	C₂²	—	268	2	no	—
364	C₂²	—	268	2	no	—
365	C₂²	—	268	2	no	—
366	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	268	2	no	—
373	C₄	—	536	2	no	—
374	C₄	—	536	2	no	—
375	C₄	C₂×#14524065	1	1	yes	characteristic
376	C₄	C₂×#14524075	1	1	yes	characteristic
377	C₄	#16864201	1	1	yes	characteristic
378	C₄	#16864203	1	1	yes	characteristic
379	C₂	—	536	2	no	—
380	C₂	#22571876	1	1	yes	characteristic
381	C₂	#22571896	1	1	yes	characteristic
382	C₂	#22572014	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×D₅₃₆	D₄
hypercenter	C₈×Q₁₆	D₆₇
hypocenter	C₆₇	#19907
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22572494	C₂	semidirect
4. #22559806	C₂	semidirect
5. #22559757	C₂	semidirect
6. #22559603	C₂	semidirect
7. #22559579	C₂	semidirect
8. #22559481	C₂	semidirect
9. #22559475	C₂	semidirect
11. C₈×Q₁₆×C₆₇	C₂	semidirect
202. C₈×Q₁₆	D₆₇	semidirect
203. C₆₇	#19907	semidirect

Maximal subgroups

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

subgroup	index
#22573198	2
Q₈×D₅₃₆	2
#22572494	2
#22559806	2
#22559757	2
#22559603	2
#22559579	2
#22559481	2
#22559475	2
#22559095	2
C₈×Q₁₆×C₆₇	2
#19907	67