Subgroups — #31888499

Order 18176.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597316	C₂	1	1	yes	maximal, characteristic
2	#22588092	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586928	C₂	1	1	yes	maximal, characteristic
4	#22586630	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586240	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586233	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1500	C₂	1	1	yes	maximal, characteristic
8	#17986130	C₄	1	1	yes	characteristic, complemented
9	Q₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic, complemented
10	#17986077	C₄	1	1	yes	characteristic, complemented
11	#17986071	—	2	1	no	—
12	D₄×C₇₁⋊C₈	—	2	1	no	—
13	D₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic, complemented
14	#17986060	—	2	1	no	—
15	#17986056	C₂²	1	1	yes	characteristic
16	#17985955	—	2	1	no	—
17	C₄×#11765849	C₂²	1	1	yes	characteristic
18	C₂×#14556039	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	#17985754	—	2	1	no	—
21	#17985754	—	2	1	no	—
22	#17985741	C₄	1	2	yes	complemented
23	#17985739	C₄	1	2	yes	complemented
24	#17985721	—	2	1	no	—
25	#17985720	—	2	1	no	—
26	#529×C₇₁	C₂²	1	1	yes	characteristic
27	#431×C₇₁	C₂²	1	1	yes	characteristic
28	#376×C₇₁	C₂²	1	1	yes	characteristic
29	#328×C₇₁	C₄	1	2	yes	—
30	#327×C₇₁	C₄	1	2	yes	—
31	C₂×#11765849	—	2	1	no	—
32	C₂×#11765849	—	2	1	no	—
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	—	2	1	no	—
35	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
37	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
38	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
39	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
40	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
42	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
44	#14556039	—	2	1	no	—
45	#14556039	—	2	1	no	—
46	#14556039	—	2	1	no	—
47	#14556039	C₂×C₄	1	2	yes	—
48	#14556039	C₂×C₄	1	2	yes	—
49	C₄×C₇₁⋊C₈	—	2	1	no	—
50	C₄×C₇₁⋊C₈	—	2	1	no	—
51	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
55	#126×C₇₁	—	2	1	no	—
56	#125×C₇₁	C₂×C₄	1	1	yes	characteristic
57	#124×C₇₁	C₂×C₄	1	1	yes	characteristic
58	#123×C₇₁	—	2	1	no	—
59	C₂²⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
60	#121×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₇₁	—	2	1	no	—
63	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
64	#117×C₇₁	—	2	1	no	—
65	C₂×C₄²×C₇₁	—	2	1	no	—
66	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₁	—	2	1	no	—
68	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₇₁	—	2	1	no	—
70	#95×C₇₁	—	2	1	no	—
71	#95×C₇₁	C₂×C₄	1	2	yes	—
72	#11765849	—	4	2	no	—
73	#11765849	—	4	2	no	—
74	C₂×C₇₁⋊C₈	—	2	1	no	—
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	—	2	2	no	—
77	C₂×C₇₁⋊C₈	—	2	2	no	—
78	C₂×C₇₁⋊C₈	—	4	1	no	—
79	C₂×C₇₁⋊C₈	—	4	1	no	—
80	C₂×C₇₁⋊C₈	—	4	2	no	—
81	C₂×C₇₁⋊C₈	—	4	1	no	—
82	C₂×C₇₁⋊C₈	—	4	1	no	—
83	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Q₈×C₇₁	C₄²	1	1	yes	characteristic
88	C₂×D₄×C₇₁	—	2	1	no	—
89	C₂×D₄×C₇₁	C₄²	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	—	2	1	no	—
91	C₂²×C₄×C₇₁	—	2	2	no	—
92	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
95	C₂×C₈×C₇₁	—	2	2	no	—
96	C₄⋊C₄×C₇₁	—	2	1	no	—
97	C₄⋊C₄×C₇₁	—	2	1	no	—
98	C₄⋊C₄×C₇₁	—	2	1	no	—
99	C₄⋊C₄×C₇₁	—	2	1	no	—
100	C₄⋊C₄×C₇₁	C₄²	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₄²	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₂²⋊C₄×C₇₁	—	2	1	no	—
105	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂²⋊C₄×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	—	2	1	no	—
108	C₄²×C₇₁	—	4	1	no	—
109	C₄²×C₇₁	—	4	1	no	—
110	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₁	#41	1	1	yes	characteristic
114	C₂³×C₇₁	—	2	1	no	—
115	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₇₁	—	4	1	no	—
117	D₄×C₇₁	—	4	1	no	—
118	D₄×C₇₁	—	4	2	no	—
119	C₂×C₄×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	2	1	no	—
122	C₂×C₄×C₇₁	—	2	1	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	—	2	1	no	—
125	C₂×C₄×C₇₁	—	4	1	no	—
126	C₂×C₄×C₇₁	—	4	1	no	—
127	C₂×C₄×C₇₁	—	4	1	no	—
128	C₂×C₄×C₇₁	—	4	2	no	—
129	C₂×C₄×C₇₁	—	4	2	no	—
130	C₂×C₄×C₇₁	—	4	2	no	—
131	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
137	C₈×C₇₁	—	4	2	no	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₇₁⋊C₈	—	4	2	no	—
143	C₇₁⋊C₈	—	4	2	no	—
144	C₇₁⋊C₈	—	4	1	no	—
145	C₇₁⋊C₈	—	4	1	no	—
146	C₂²×C₇₁	—	2	1	no	—
147	C₂²×C₇₁	—	2	1	no	—
148	C₂²×C₇₁	—	4	1	no	—
149	C₂²×C₇₁	—	4	1	no	—
150	C₂²×C₇₁	—	4	1	no	—
151	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₇₁	#444	1	2	yes	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	—	4	1	no	—
159	C₄×C₇₁	—	4	1	no	—
160	C₄×C₇₁	—	4	1	no	—
161	C₄×C₇₁	—	8	2	no	—
162	C₄×C₇₁	#421	1	1	yes	characteristic
163	C₄×C₇₁	#421	1	1	yes	characteristic
164	#8248	—	71	1	no	maximal, Sylow
165	C₂×C₇₁	—	2	2	no	—
166	C₂×C₇₁	—	8	1	no	—
167	C₂×C₇₁	#1757	1	1	yes	characteristic
168	C₂×C₇₁	#1759	1	1	yes	characteristic
169	C₂×C₇₁	#1765	1	1	yes	characteristic
170	#2966	—	71	1	no	—
171	#1746	—	71	1	no	—
172	#1524	—	71	1	no	—
173	#1500	D₇₁	1	1	yes	characteristic
174	#1451	—	71	1	no	—
175	#1302	—	71	1	no	—
176	#1300	—	71	1	no	—
177	C₇₁	#8248	1	1	yes	characteristic, complemented, Sylow
178	#529	C₂×D₇₁	1	1	yes	characteristic
179	#446	—	71	1	no	—
180	C₈×Q₈	—	71	1	no	—
181	#436	—	71	1	no	—
182	#435	—	142	1	no	—
183	C₈×D₄	—	71	1	no	—
184	C₈×D₄	—	142	1	no	—
185	#433	—	142	1	no	—
186	#432	—	71	1	no	—
187	#431	C₂×D₇₁	1	1	yes	characteristic
188	#405	—	142	1	no	—
189	C₄×M₁₆	—	71	1	no	—
190	C₂×#97	—	71	1	no	—
191	C₂×C₄×C₈	—	71	1	no	—
192	#376	C₂×D₇₁	1	1	yes	characteristic
193	#336	—	142	1	no	—
194	#336	—	142	1	no	—
195	#330	—	71	2	no	—
196	#329	—	71	2	no	—
197	#328	Dic₇₁	1	2	yes	—
198	#327	Dic₇₁	1	2	yes	—
199	#322	—	142	1	no	—
200	#322	—	142	1	no	—
201	C₂×M₁₆	—	142	1	no	—
202	C₂×M₁₆	—	142	1	no	—
203	C₂²×C₈	—	71	1	no	—
204	C₂²×C₈	—	71	1	no	—
205	C₂²×C₈	—	142	1	no	—
206	C₂²×C₈	—	142	1	no	—
207	#126	—	2	1	no	—
208	#125	C₄×D₇₁	1	1	yes	characteristic
209	#124	C₄×D₇₁	1	1	yes	characteristic
210	#123	—	2	1	no	—
211	C₂²⋊Q₈	C₂×Dic₇₁	1	1	yes	characteristic
212	#121	C₂×Dic₇₁	1	1	yes	characteristic
213	C₄×Q₈	C₄×D₇₁	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	C₄×D₇₁	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	71	1	no	—
221	C₄⋊C₈	—	71	1	no	—
222	C₄⋊C₈	—	142	1	no	—
223	C₄⋊C₈	—	142	1	no	—
224	C₂²⋊C₈	—	142	1	no	—
225	C₂²⋊C₈	—	142	1	no	—
226	C₂²⋊C₈	—	142	1	no	—
227	C₂²⋊C₈	C₂×Dic₇₁	1	2	yes	—
228	#97	—	71	2	no	—
229	#97	—	71	2	no	—
230	#97	—	142	1	no	—
231	#97	—	142	1	no	—
232	#97	—	142	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	71	1	no	—
235	C₄×C₈	—	71	1	no	—
236	C₄×C₈	—	71	1	no	—
237	C₄×C₈	—	71	1	no	—
238	C₄×C₈	—	142	1	no	—
239	C₄×C₈	—	142	1	no	—
240	#95	—	2	1	no	—
241	#95	C₂×Dic₇₁	1	2	yes	—
242	C₂×Q₈	C₄×Dic₇₁	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₄×Dic₇₁	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
249	C₂²×C₄	#11765864	1	1	yes	characteristic
250	M₁₆	—	284	2	no	—
251	M₁₆	—	284	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	284	1	no	—
254	C₂×C₈	—	284	1	no	—
255	C₂×C₈	—	284	2	no	—
256	C₂×C₈	—	284	1	no	—
257	C₂×C₈	—	284	1	no	—
258	C₂×C₈	—	71	1	no	—
259	C₂×C₈	—	71	1	no	—
260	C₂×C₈	—	71	1	no	—
261	C₂×C₈	—	71	1	no	—
262	C₂×C₈	—	142	1	no	—
263	C₂×C₈	—	142	1	no	—
264	C₂×C₈	—	142	2	no	—
265	C₂×C₈	—	142	2	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₄×Dic₇₁	1	1	yes	characteristic
271	C₄⋊C₄	C₄×Dic₇₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
282	C₄²	D₄×D₇₁	1	1	yes	characteristic
283	C₄²	#11765864	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
302	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
303	C₂×C₄	#14556084	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	284	2	no	—
309	C₈	—	284	2	no	—
310	C₈	—	284	1	no	—
311	C₈	—	284	1	no	—
312	C₈	—	142	1	no	—
313	C₈	—	142	1	no	—
314	C₈	—	142	1	no	—
315	C₈	—	142	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂²⋊C₄×Dic₇₁	1	1	yes	characteristic
322	C₂²	#17986121	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	8	2	no	—
332	C₄	#17986014	1	1	yes	characteristic
333	C₄	#17986014	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	#22588142	1	1	yes	characteristic
337	C₂	#22588152	1	1	yes	characteristic
338	C₂	#22588187	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1500	C₂
socle	C₂²×C₇₁	C₄×C₂²⋊C₄
prime radical	#121×C₇₁	C₂×C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1500	D₇₁
hypocenter	C₇₁	#8248
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

Complements

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

N	H ≅ G/N	type
2. #22588092	C₂	semidirect
4. #22586630	C₂	semidirect
5. #22586240	C₂	semidirect
6. #22586233	C₂	semidirect
8. #17986130	C₄	semidirect
9. Q₈×C₇₁⋊C₈	C₄	semidirect
10. #17986077	C₄	semidirect
13. D₄×C₇₁⋊C₈	C₄	semidirect
22. #17985741	C₄	semidirect
23. #17985739	C₄	semidirect
177. C₇₁	#8248	semidirect

Maximal subgroups

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

subgroup	index
#22597316	2
#22588092	2
#22586928	2
#22586630	2
#22586240	2
#22586233	2
C₇₁×#1500	2
#8248	71