Subgroups — #32150989

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597317	C₂	1	1	yes	maximal, characteristic
2	#22590249	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590171	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590164	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₈₄	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590146	C₂	1	2	yes	maximal, complemented
8	#22590142	C₂	1	1	yes	maximal, characteristic, complemented
9	#22589935	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986536	C₄	1	1	yes	characteristic
13	C₂×#14556067	C₂²	1	2	yes	—
14	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
16	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
17	#17986130	C₄	1	1	yes	characteristic
18	#17986076	C₄	1	1	yes	characteristic
19	#17986069	C₄	1	2	yes	—
20	C₈×D₂₈₄	C₂²	1	2	yes	—
21	C₈×D₂₈₄	C₂²	1	1	yes	characteristic
22	#17986059	C₄	1	2	yes	—
23	#17986055	C₄	1	1	yes	characteristic
24	#17985953	C₂²	1	2	yes	—
25	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
28	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₇₁)⋊C₁₆	C₂²	1	2	yes	—
31	#17985775	C₂²	1	2	yes	—
32	#17985774	C₂²	1	2	yes	—
33	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
41	C₂×#11765848	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₇₁	C₂³	1	2	yes	—
45	#14556123	C₈	1	2	yes	—
46	#14556119	C₈	1	1	yes	characteristic
47	#14556117	C₈	1	1	yes	characteristic
48	#14556113	C₈	1	2	yes	—
49	#14556104	C₈	1	1	yes	characteristic
50	#14556103	C₈	1	1	yes	characteristic
51	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
52	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
53	#14556085	C₂×C₄	1	2	yes	—
54	C₄²×D₇₁	C₂×C₄	1	1	yes	characteristic
55	#14556067	—	2	4	no	—
56	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₇₁	—	2	2	no	—
60	C₁₆×D₇₁	—	2	2	no	—
61	C₁₆×D₇₁	—	2	4	no	—
62	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
63	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
66	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
67	#14556039	C₂×C₄	1	2	yes	—
68	#14556038	C₂×C₄	1	2	yes	—
69	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
75	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₁	C₂³	1	2	yes	—
80	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
81	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
86	#11765848	—	2	4	no	—
87	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₇₁	—	2	1	no	—
91	C₈×D₇₁	—	2	1	no	—
92	C₈×D₇₁	—	2	2	no	—
93	C₈×D₇₁	—	2	2	no	—
94	C₈×D₇₁	—	2	2	no	—
95	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
97	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
100	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
102	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
103	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
104	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
107	C₇₁⋊C₁₆	—	2	1	no	—
108	C₇₁⋊C₁₆	—	2	2	no	—
109	C₇₁⋊C₁₆	#41	1	2	yes	—
110	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
111	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
116	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
118	C₁₆×C₇₁	—	2	1	no	—
119	C₁₆×C₇₁	—	2	2	no	—
120	C₁₆×C₇₁	#41	1	2	yes	—
121	C₂²×D₇₁	C₂²×C₈	1	2	yes	—
122	C₂²×D₇₁	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₇₁	—	2	2	no	—
124	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
133	D₂₈₄	—	2	4	no	—
134	D₂₈₄	—	2	2	no	—
135	C₄×D₇₁	—	2	2	no	—
136	C₄×D₇₁	—	2	1	no	—
137	C₄×D₇₁	—	2	2	no	—
138	C₄×D₇₁	—	2	1	no	—
139	C₄×D₇₁	—	2	2	no	—
140	C₄×D₇₁	—	2	2	no	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	—	2	2	no	—
143	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
145	C₇₁⋊C₈	—	2	1	no	—
146	C₇₁⋊C₈	—	2	2	no	—
147	C₇₁⋊C₈	#131	1	2	yes	—
148	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₇₁	—	2	1	no	—
150	C₂×D₇₁	—	2	1	no	—
151	C₂×D₇₁	—	2	2	no	—
152	C₂×D₇₁	—	2	2	no	—
153	C₂×D₇₁	—	2	2	no	—
154	C₂×D₇₁	#504	1	2	yes	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	2	no	—
157	C₄×C₇₁	—	2	2	no	—
158	C₄×C₇₁	#504	1	2	yes	—
159	C₄×C₇₁	C₄×#41	1	1	yes	characteristic
160	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	2	1	no	—
163	Dic₇₁	—	2	1	no	—
164	Dic₇₁	—	2	2	no	—
165	#33480	—	71	1	no	maximal, Sylow
166	C₂×C₇₁	C₈×#41	1	1	yes	characteristic
167	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
169	D₇₁	—	2	2	no	—
170	D₇₁	—	4	2	no	—
171	#2966	—	71	1	no	—
172	#2183	—	71	1	no	—
173	Q₈×C₁₆	D₇₁	1	1	yes	characteristic, complemented
174	#2169	—	71	1	no	—
175	#2168	—	71	2	no	—
176	D₄×C₁₆	—	71	1	no	—
177	D₄×C₁₆	—	71	2	no	—
178	#2164	—	71	2	no	—
179	#2163	—	71	1	no	—
180	#2108	—	71	2	no	—
181	C₂×C₄×C₁₆	—	71	1	no	—
182	C₇₁	#33480	1	1	yes	characteristic, complemented, Sylow
183	#529	—	71	1	no	—
184	C₂×M₃₂	—	71	2	no	—
185	C₂²×C₁₆	—	71	1	no	—
186	C₂²×C₁₆	—	71	1	no	—
187	C₂²×C₁₆	—	71	2	no	—
188	#446	—	71	1	no	—
189	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
190	#436	—	71	1	no	—
191	#435	—	71	2	no	—
192	C₈×D₄	—	71	2	no	—
193	C₈×D₄	—	71	1	no	—
194	#433	—	71	2	no	—
195	#432	—	71	1	no	—
196	#405	—	71	2	no	—
197	C₂×C₄×C₈	—	71	1	no	—
198	C₄⋊C₁₆	—	71	1	no	—
199	C₄⋊C₁₆	—	71	2	no	—
200	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
202	C₂²⋊C₁₆	—	71	2	no	—
203	C₂²⋊C₁₆	—	71	2	no	—
204	C₂²⋊C₁₆	—	71	2	no	—
205	#346	—	71	2	no	—
206	#346	—	71	2	no	—
207	C₄×C₁₆	—	71	1	no	—
208	C₄×C₁₆	—	71	1	no	—
209	C₄×C₁₆	—	71	1	no	—
210	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
211	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
212	C₂×M₁₆	—	71	2	no	—
213	C₂²×C₈	—	71	1	no	—
214	C₂²×C₈	—	71	1	no	—
215	C₂²×C₈	—	71	2	no	—
216	#126	—	71	2	no	—
217	#125	—	71	1	no	—
218	#124	—	71	1	no	—
219	#123	—	71	2	no	—
220	C₂²⋊Q₈	—	71	1	no	—
221	#121	—	71	1	no	—
222	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
223	C₄×D₄	—	71	2	no	—
224	C₄×D₄	—	71	1	no	—
225	#117	—	71	2	no	—
226	C₂×C₄²	—	71	1	no	—
227	M₃₂	—	142	4	no	—
228	C₂×C₁₆	—	71	1	no	—
229	C₂×C₁₆	—	71	1	no	—
230	C₂×C₁₆	—	71	2	no	—
231	C₂×C₁₆	—	142	2	no	—
232	C₂×C₁₆	—	142	2	no	—
233	C₂×C₁₆	—	142	4	no	—
234	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
236	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
237	C₄⋊C₈	—	71	1	no	—
238	C₄⋊C₈	—	71	2	no	—
239	C₄⋊C₈	C₄×D₇₁	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
241	C₂²⋊C₈	—	71	2	no	—
242	C₂²⋊C₈	—	71	2	no	—
243	C₂²⋊C₈	—	71	2	no	—
244	#97	—	71	2	no	—
245	#97	—	71	2	no	—
246	C₄×C₈	—	71	1	no	—
247	C₄×C₈	—	71	1	no	—
248	C₄×C₈	—	71	1	no	—
249	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₇₁	1	2	yes	—
251	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
252	C₂×D₄	—	71	2	no	—
253	C₂×D₄	—	71	1	no	—
254	C₂²×C₄	—	71	1	no	—
255	C₂²×C₄	—	71	2	no	—
256	C₂²×C₄	—	71	1	no	—
257	C₂²×C₄	—	71	1	no	—
258	M₁₆	—	142	4	no	—
259	C₂×C₈	—	71	1	no	—
260	C₂×C₈	—	71	1	no	—
261	C₂×C₈	—	71	2	no	—
262	C₂×C₈	—	142	2	no	—
263	C₂×C₈	—	142	1	no	—
264	C₂×C₈	—	142	2	no	—
265	C₂×C₈	—	142	1	no	—
266	C₂×C₈	—	142	2	no	—
267	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
269	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
270	C₄⋊C₄	—	71	2	no	—
271	C₄⋊C₄	—	71	2	no	—
272	C₄⋊C₄	—	71	1	no	—
273	C₄⋊C₄	—	71	2	no	—
274	C₄⋊C₄	C₈×D₇₁	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
276	C₂²⋊C₄	—	71	2	no	—
277	C₂²⋊C₄	—	71	2	no	—
278	C₂²⋊C₄	—	71	2	no	—
279	C₂²⋊C₄	—	71	2	no	—
280	C₂²⋊C₄	—	71	2	no	—
281	C₄²	—	71	1	no	—
282	C₄²	—	71	1	no	—
283	C₄²	—	71	1	no	—
284	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₇₁	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	71	2	no	—
289	C₁₆	—	142	1	no	—
290	C₁₆	—	142	2	no	—
291	C₁₆	#11765866	1	2	yes	—
292	C₂³	—	71	1	no	—
293	C₂³	—	71	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	142	4	no	—
296	D₄	—	142	2	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	71	2	no	—
301	C₂×C₄	—	142	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	—	142	2	no	—
304	C₂×C₄	—	142	1	no	—
305	C₂×C₄	—	142	1	no	—
306	C₂×C₄	—	142	2	no	—
307	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	71	2	no	—
315	C₈	—	142	1	no	—
316	C₈	—	142	2	no	—
317	C₈	#41×D₇₁	1	1	yes	characteristic
318	C₈	C₂×#11765866	1	1	yes	characteristic
319	C₂²	—	71	2	no	—
320	C₂²	—	142	2	no	—
321	C₂²	—	142	2	no	—
322	C₂²	—	142	2	no	—
323	C₂²	—	142	1	no	—
324	C₂²	—	142	1	no	—
325	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	142	1	no	—
330	C₄	—	142	2	no	—
331	C₄	—	142	1	no	—
332	C₄	—	142	1	no	—
333	C₄	#17986415	1	2	yes	—
334	C₄	C₄×#11765866	1	1	yes	characteristic
335	C₄	#131×D₇₁	1	1	yes	characteristic
336	C₂	—	284	2	no	—
337	C₂	—	142	2	no	—
338	C₂	C₈×#11765866	1	1	yes	characteristic
339	C₂	#504×D₇₁	1	1	yes	characteristic
340	C₂	C₂×#17986415	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	Q₈×C₁₆×C₇₁	C₂
prime radical	#14556103	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	Q₈×C₁₆	D₇₁
hypocenter	C₇₁	#33480
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22590249	C₂	semidirect
3. #22590171	C₂	semidirect
4. #22590164	C₂	semidirect
5. C₁₆×D₂₈₄	C₂	semidirect
6. C₁₆×D₂₈₄	C₂	semidirect
7. #22590146	C₂	semidirect
8. #22590142	C₂	semidirect
9. #22589935	C₂	semidirect
10. C₄×C₁₆×D₇₁	C₂	semidirect
11. Q₈×C₁₆×C₇₁	C₂	semidirect
173. Q₈×C₁₆	D₇₁	semidirect
182. C₇₁	#33480	semidirect

Maximal subgroups

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

subgroup	index
#22597317	2
#22590249	2
#22590171	2
#22590164	2
C₁₆×D₂₈₄	2
#22590146	2
#22590142	2
#22589935	2
C₄×C₁₆×D₇₁	2
Q₈×C₁₆×C₇₁	2
#33480	71