Subgroups — #31914331

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986416	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589954	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586422	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#443×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986416	—	2	2	no	—
9	#17986416	—	2	2	no	—
10	C₂×#14556068	—	2	1	no	—
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
14	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#17985783	—	2	2	no	—
17	#17985783	—	2	2	no	—
18	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
19	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
20	C₂×#135×C₇₁	C₄	1	1	yes	characteristic
21	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
23	#443×C₇₁	—	2	2	no	—
24	#443×C₇₁	—	2	2	no	—
25	C₂×#108×C₇₁	C₄	1	1	yes	characteristic
26	C₂×#104×C₇₁	C₄	1	1	yes	characteristic
27	C₂×#104×C₇₁	C₄	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	#14556068	—	2	4	no	—
30	#14556068	—	2	1	no	—
31	#14556068	—	2	1	no	—
32	#14556068	—	2	1	no	—
33	#14556068	D₄	1	2	yes	—
34	C₂×C₇₁⋊C₁₆	—	2	4	no	—
35	C₂×C₇₁⋊C₁₆	—	2	1	no	—
36	C₂×C₇₁⋊C₁₆	—	2	1	no	—
37	C₂×C₇₁⋊C₁₆	—	2	1	no	—
38	C₂×C₇₁⋊C₁₆	—	2	1	no	—
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	—	2	1	no	—
42	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
43	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
46	#135×C₇₁	—	2	2	no	—
47	#135×C₇₁	—	2	2	no	—
48	C₂×Q₁₆×C₇₁	C₈	1	1	yes	characteristic
49	C₂×SD₁₆×C₇₁	C₈	1	1	yes	characteristic
50	C₂×SD₁₆×C₇₁	C₈	1	1	yes	characteristic
51	C₂×D₈×C₇₁	C₈	1	1	yes	characteristic
52	#131×C₇₁	—	2	1	no	—
53	#131×C₇₁	—	2	1	no	—
54	#131×C₇₁	—	2	1	no	—
55	#131×C₇₁	—	2	1	no	—
56	#131×C₇₁	—	2	1	no	—
57	#131×C₇₁	—	2	1	no	—
58	#131×C₇₁	D₄	1	2	yes	—
59	#131×C₇₁	D₄	1	2	yes	—
60	C₂×M₁₆×C₇₁	—	2	1	no	—
61	C₂×M₁₆×C₇₁	—	2	1	no	—
62	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₁	—	2	1	no	—
65	C₂²×C₈×C₇₁	—	2	1	no	—
66	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
69	#108×C₇₁	—	2	2	no	—
70	#104×C₇₁	—	2	2	no	—
71	#104×C₇₁	—	2	2	no	—
72	C₄×C₈×C₇₁	—	2	1	no	—
73	C₄×C₈×C₇₁	—	2	1	no	—
74	C₇₁⋊C₁₆	—	2	1	no	—
75	C₇₁⋊C₁₆	—	2	2	no	—
76	C₇₁⋊C₁₆	—	2	1	no	—
77	C₇₁⋊C₁₆	—	4	1	no	—
78	C₇₁⋊C₁₆	—	4	1	no	—
79	#41×C₇₁	—	2	1	no	—
80	#41×C₇₁	—	2	1	no	—
81	#41×C₇₁	—	2	1	no	—
82	#41×C₇₁	—	2	1	no	—
83	#41×C₇₁	—	2	1	no	—
84	#41×C₇₁	—	2	1	no	—
85	#41×C₇₁	C₂²⋊C₄	1	2	yes	—
86	#41×C₇₁	M₁₆	1	2	yes	—
87	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
88	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
89	C₂×D₄×C₇₁	—	2	1	no	—
90	C₂×D₄×C₇₁	—	2	1	no	—
91	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
92	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	—	2	1	no	—
94	C₂²×C₄×C₇₁	—	2	1	no	—
95	C₂²×C₄×C₇₁	—	2	1	no	—
96	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	Q₁₆×C₇₁	—	2	2	no	—
98	SD₁₆×C₇₁	—	2	2	no	—
99	SD₁₆×C₇₁	—	2	2	no	—
100	D₈×C₇₁	—	2	2	no	—
101	M₁₆×C₇₁	—	2	1	no	—
102	M₁₆×C₇₁	—	2	2	no	—
103	M₁₆×C₇₁	—	2	2	no	—
104	M₁₆×C₇₁	—	2	1	no	—
105	M₁₆×C₇₁	—	2	1	no	—
106	M₁₆×C₇₁	—	4	1	no	—
107	M₁₆×C₇₁	—	4	1	no	—
108	M₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
109	C₂×C₈×C₇₁	—	2	1	no	—
110	C₂×C₈×C₇₁	—	2	1	no	—
111	C₂×C₈×C₇₁	—	2	1	no	—
112	C₂×C₈×C₇₁	—	2	2	no	—
113	C₂×C₈×C₇₁	—	2	2	no	—
114	C₂×C₈×C₇₁	—	2	1	no	—
115	C₂×C₈×C₇₁	—	4	1	no	—
116	C₂×C₈×C₇₁	—	4	1	no	—
117	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
118	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
119	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
124	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
125	C₄²×C₇₁	—	2	1	no	—
126	C₄²×C₇₁	—	2	1	no	—
127	C₂³×C₇₁	—	2	1	no	—
128	C₂³×C₇₁	—	2	1	no	—
129	C₂³×C₇₁	#131	1	1	yes	characteristic
130	Q₈×C₇₁	—	2	1	no	—
131	Q₈×C₇₁	—	2	1	no	—
132	Q₈×C₇₁	—	2	1	no	—
133	Q₈×C₇₁	C₂²⋊C₈	1	2	yes	—
134	D₄×C₇₁	—	2	2	no	—
135	D₄×C₇₁	—	2	2	no	—
136	D₄×C₇₁	—	2	1	no	—
137	D₄×C₇₁	—	2	1	no	—
138	D₄×C₇₁	—	2	1	no	—
139	D₄×C₇₁	—	4	1	no	—
140	D₄×C₇₁	—	4	1	no	—
141	D₄×C₇₁	C₂²⋊C₈	1	2	yes	—
142	C₂×C₄×C₇₁	—	2	1	no	—
143	C₂×C₄×C₇₁	—	2	2	no	—
144	C₂×C₄×C₇₁	—	2	2	no	—
145	C₂×C₄×C₇₁	—	2	2	no	—
146	C₂×C₄×C₇₁	—	2	1	no	—
147	C₂×C₄×C₇₁	—	4	1	no	—
148	C₂×C₄×C₇₁	—	4	1	no	—
149	C₂×C₄×C₇₁	—	4	1	no	—
150	C₂×C₄×C₇₁	—	4	1	no	—
151	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
156	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
157	C₈×C₇₁	—	2	2	no	—
158	C₈×C₇₁	—	2	1	no	—
159	C₈×C₇₁	—	2	1	no	—
160	C₈×C₇₁	—	4	1	no	—
161	C₈×C₇₁	#120	1	1	yes	characteristic
162	C₈×C₇₁	#121	1	1	yes	characteristic
163	C₈×C₇₁	#121	1	1	yes	characteristic
164	C₈×C₇₁	#123	1	1	yes	characteristic
165	C₂²×C₇₁	—	2	1	no	—
166	C₂²×C₇₁	—	2	1	no	—
167	C₂²×C₇₁	—	2	2	no	—
168	C₂²×C₇₁	—	4	1	no	—
169	C₂²×C₇₁	—	4	1	no	—
170	C₂²×C₇₁	—	4	1	no	—
171	C₂²×C₇₁	—	4	1	no	—
172	C₂²×C₇₁	—	4	1	no	—
173	C₂²×C₇₁	#408	1	1	yes	characteristic
174	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
175	C₂²×C₇₁	#435	1	1	yes	characteristic
176	C₄×C₇₁	—	2	2	no	—
177	C₄×C₇₁	—	4	2	no	—
178	C₄×C₇₁	—	4	1	no	—
179	C₄×C₇₁	#386	1	1	yes	characteristic
180	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
182	C₄×C₇₁	#435	1	1	yes	characteristic
183	#12980	—	71	1	no	maximal, Sylow
184	C₂×C₇₁	—	2	1	no	—
185	C₂×C₇₁	—	2	1	no	—
186	C₂×C₇₁	—	4	2	no	—
187	C₂×C₇₁	—	8	1	no	—
188	C₂×C₇₁	#1869	1	1	yes	characteristic
189	C₂×C₇₁	#2170	1	1	yes	characteristic
190	C₂×C₇₁	#2171	1	1	yes	characteristic
191	C₂×#504	—	71	1	no	—
192	C₂×#443	D₇₁	1	1	yes	characteristic
193	C₂×#350	—	71	1	no	—
194	C₂×#350	—	71	1	no	—
195	#2113	—	71	1	no	—
196	#1376	—	71	1	no	—
197	#1374	—	71	1	no	—
198	C₇₁	#12980	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	Dic₇₁	1	1	yes	characteristic
200	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
201	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
202	#504	—	142	2	no	—
203	#504	—	142	2	no	—
204	C₂×M₃₂	—	71	1	no	—
205	C₂×M₃₂	—	71	1	no	—
206	C₂×M₃₂	—	142	1	no	—
207	C₂²×C₁₆	—	71	1	no	—
208	C₂²×C₁₆	—	71	1	no	—
209	C₂²×C₁₆	—	142	1	no	—
210	#443	—	2	2	no	—
211	#443	—	2	2	no	—
212	C₂×#108	Dic₇₁	1	1	yes	characteristic
213	C₂×#104	Dic₇₁	1	1	yes	characteristic
214	C₂×#104	Dic₇₁	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
216	#350	—	142	2	no	—
217	#350	—	142	2	no	—
218	C₂²⋊C₁₆	—	142	1	no	—
219	C₂²⋊C₁₆	—	142	1	no	—
220	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
221	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
222	#135	—	2	2	no	—
223	#135	—	2	2	no	—
224	C₂×Q₁₆	C₇₁⋊C₈	1	1	yes	characteristic
225	C₂×SD₁₆	C₇₁⋊C₈	1	1	yes	characteristic
226	C₂×SD₁₆	C₇₁⋊C₈	1	1	yes	characteristic
227	C₂×D₈	C₇₁⋊C₈	1	1	yes	characteristic
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	—	2	1	no	—
231	#131	—	2	1	no	—
232	#131	—	2	1	no	—
233	#131	—	2	1	no	—
234	#131	C₂²⋊D₇₁	1	2	yes	—
235	#131	C₂²⋊D₇₁	1	2	yes	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
239	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
243	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
245	M₃₂	—	71	2	no	—
246	M₃₂	—	142	1	no	—
247	M₃₂	—	142	1	no	—
248	M₃₂	—	142	4	no	—
249	M₃₂	—	142	1	no	—
250	C₂×C₁₆	—	71	1	no	—
251	C₂×C₁₆	—	71	1	no	—
252	C₂×C₁₆	—	142	1	no	—
253	C₂×C₁₆	—	142	1	no	—
254	C₂×C₁₆	—	142	1	no	—
255	C₂×C₁₆	—	142	1	no	—
256	C₂×C₁₆	—	142	4	no	—
257	C₂×C₁₆	—	142	1	no	—
258	C₂×C₁₆	—	142	1	no	—
259	C₂×C₁₆	—	142	1	no	—
260	#108	—	2	2	no	—
261	#104	—	2	2	no	—
262	#104	—	2	2	no	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	—	2	1	no	—
265	#41	—	2	1	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	2	1	no	—
269	#41	—	2	1	no	—
270	#41	—	2	1	no	—
271	#41	C₂²⋊Dic₇₁	1	2	yes	—
272	#41	#11765849	1	2	yes	—
273	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
274	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
278	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
283	Q₁₆	—	2	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	2	2	no	—
286	D₈	—	2	2	no	—
287	M₁₆	—	2	1	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	2	1	no	—
291	M₁₆	—	2	1	no	—
292	M₁₆	—	4	1	no	—
293	M₁₆	—	4	1	no	—
294	M₁₆	C₂²⋊Dic₇₁	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
304	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
309	C₂×C₈	#11765864	1	1	yes	characteristic
310	C₂×C₈	#11765864	1	1	yes	characteristic
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₁₆	—	284	1	no	—
314	C₁₆	—	284	1	no	—
315	C₁₆	—	142	1	no	—
316	C₁₆	—	142	1	no	—
317	C₁₆	—	142	2	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	#14556135	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	D₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
343	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
344	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
346	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
347	C₂×C₄	#14556135	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	1	no	—
352	C₈	#14556096	1	1	yes	characteristic
353	C₈	#14556100	1	1	yes	characteristic
354	C₈	#14556100	1	1	yes	characteristic
355	C₈	#14556109	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	#17985966	1	1	yes	characteristic
365	C₂²	D₄×C₇₁⋊C₈	1	1	yes	characteristic
366	C₂²	#17986071	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	1	no	—
370	C₄	#17985879	1	1	yes	characteristic
371	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
372	C₄	D₄×C₇₁⋊C₈	1	1	yes	characteristic
373	C₄	#17986071	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	4	2	no	—
377	C₂	—	8	1	no	—
378	C₂	#22588633	1	1	yes	characteristic
379	C₂	#22590177	1	1	yes	characteristic
380	C₂	#22590182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#443×C₇₁	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#135×C₇₁	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#443	D₇₁
hypocenter	C₇₁	#12980
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#17986416	C₂	semidirect
2. C₂×#17985783	C₂	semidirect
3. C₂×#17985783	C₂	semidirect
4. #22589954	C₂	semidirect
5. #22586422	C₂	semidirect
6. #22586418	C₂	semidirect
198. C₇₁	#12980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986416	2
C₂×#17985783	2
#22589954	2
#22586422	2
#22586418	2
C₂×#443×C₇₁	2
#12980	71