Subgroups — #31914332

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×#17986415	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17986114	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985784	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586421	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586417	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2113	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556148	C₄	1	1	yes	characteristic
9	C₂×#14556134	C₂²	1	1	yes	characteristic
10	#17986415	—	2	2	no	—
11	#17986415	—	2	2	no	—
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	C₂×#14556067	—	2	1	no	—
14	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₁	—	2	1	no	—
16	#17986114	—	2	2	no	—
17	#17986114	—	2	2	no	—
18	C₂×#14556063	C₄	1	1	yes	characteristic
19	C₂×#14556055	C₄	1	1	yes	characteristic
20	C₂×#14556054	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
22	#17985784	—	2	2	no	—
23	#17985783	—	2	2	no	—
24	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₇₁	—	2	1	no	—
28	C₂²⋊C₁₆×C₇₁	—	2	1	no	—
29	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
30	#14556148	—	2	2	no	—
31	#14556148	—	2	2	no	—
32	C₂×#11765855	C₈	1	1	yes	characteristic
33	C₂×#11765853	C₈	1	1	yes	characteristic
34	C₂×#11765852	C₈	1	1	yes	characteristic
35	C₂×#11765850	C₈	1	1	yes	characteristic
36	#14556134	—	2	1	no	—
37	#14556134	—	2	1	no	—
38	#14556134	D₄	1	2	yes	—
39	#14556134	D₄	1	2	yes	—
40	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
41	C₂×#11765848	—	2	1	no	—
42	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₁	—	2	1	no	—
44	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
45	#14556068	—	2	1	no	—
46	#14556068	D₄	1	2	yes	—
47	#14556067	—	2	4	no	—
48	C₂×C₇₁⋊C₁₆	—	2	1	no	—
49	C₂×C₇₁⋊C₁₆	—	2	1	no	—
50	C₂×C₇₁⋊C₁₆	—	2	1	no	—
51	C₂×C₇₁⋊C₁₆	—	2	1	no	—
52	C₁₆×D₇₁	—	2	4	no	—
53	#14556063	—	2	2	no	—
54	#14556055	—	2	2	no	—
55	#14556054	—	2	2	no	—
56	C₈×Dic₇₁	—	2	1	no	—
57	C₈×Dic₇₁	—	2	1	no	—
58	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
59	#131×C₇₁	—	2	1	no	—
60	#131×C₇₁	—	2	1	no	—
61	#131×C₇₁	—	2	1	no	—
62	#131×C₇₁	—	2	1	no	—
63	C₂×M₁₆×C₇₁	—	2	1	no	—
64	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₇₁	—	2	1	no	—
66	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
67	M₃₂×C₇₁	—	2	1	no	—
68	M₃₂×C₇₁	—	2	1	no	—
69	C₂×C₁₆×C₇₁	—	2	1	no	—
70	C₂×C₁₆×C₇₁	—	2	1	no	—
71	C₂×C₁₆×C₇₁	—	2	1	no	—
72	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
73	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
74	#11765865	—	2	1	no	—
75	#11765865	—	2	1	no	—
76	#11765865	C₂²⋊C₄	1	2	yes	—
77	#11765865	M₁₆	1	2	yes	—
78	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
79	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
80	C₂×C₂²⋊D₇₁	—	2	1	no	—
81	C₂²×Dic₇₁	—	2	1	no	—
82	C₂×C₄×D₇₁	—	2	1	no	—
83	#11765855	—	2	2	no	—
84	#11765853	—	2	2	no	—
85	#11765852	—	2	2	no	—
86	#11765850	—	2	2	no	—
87	#11765849	—	2	1	no	—
88	#11765849	C₂²⋊C₄	1	2	yes	—
89	#11765848	—	2	2	no	—
90	#11765848	—	2	2	no	—
91	C₂×C₇₁⋊C₈	—	2	1	no	—
92	C₂×C₇₁⋊C₈	—	2	1	no	—
93	C₂×C₇₁⋊C₈	—	2	1	no	—
94	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
95	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
96	C₈×D₇₁	—	2	2	no	—
97	C₈×D₇₁	—	2	2	no	—
98	C₄×Dic₇₁	—	2	1	no	—
99	C₄×Dic₇₁	—	2	1	no	—
100	C₇₁⋊C₁₆	—	2	2	no	—
101	C₇₁⋊C₁₆	—	4	1	no	—
102	#41×C₇₁	—	2	1	no	—
103	#41×C₇₁	—	2	1	no	—
104	#41×C₇₁	—	2	1	no	—
105	#41×C₇₁	—	2	1	no	—
106	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
107	C₂×D₄×C₇₁	—	2	1	no	—
108	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
109	C₂²×C₄×C₇₁	—	2	1	no	—
110	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₇₁	—	2	1	no	—
112	M₁₆×C₇₁	—	2	1	no	—
113	M₁₆×C₇₁	—	4	1	no	—
114	M₁₆×C₇₁	—	4	1	no	—
115	C₂×C₈×C₇₁	—	2	1	no	—
116	C₂×C₈×C₇₁	—	4	1	no	—
117	C₂×C₈×C₇₁	—	4	1	no	—
118	C₂×C₈×C₇₁	C₂×D₄	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	#41	1	1	yes	characteristic
123	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
124	C₁₆×C₇₁	—	2	1	no	—
125	C₁₆×C₇₁	—	2	1	no	—
126	C₁₆×C₇₁	—	4	1	no	—
127	C₂³×C₇₁	—	2	1	no	—
128	C₂³×C₇₁	#131	1	1	yes	characteristic
129	C₂²×D₇₁	—	2	1	no	—
130	Q₈×C₇₁	—	2	1	no	—
131	Q₈×C₇₁	—	2	1	no	—
132	D₄×C₇₁	—	2	1	no	—
133	D₄×C₇₁	—	2	1	no	—
134	D₄×C₇₁	—	4	1	no	—
135	D₄×C₇₁	—	4	1	no	—
136	C₂×C₄×C₇₁	—	2	1	no	—
137	C₂×C₄×C₇₁	—	4	1	no	—
138	C₂×C₄×C₇₁	—	4	1	no	—
139	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
145	C₂²⋊D₇₁	—	2	2	no	—
146	C₂²⋊D₇₁	—	2	2	no	—
147	C₂×Dic₇₁	—	2	1	no	—
148	C₂×Dic₇₁	—	2	2	no	—
149	C₂×Dic₇₁	—	4	1	no	—
150	C₂×Dic₇₁	—	4	1	no	—
151	D₂₈₄	—	2	1	no	—
152	D₂₈₄	C₂²⋊C₈	1	2	yes	—
153	C₄×D₇₁	—	2	2	no	—
154	C₄×D₇₁	—	2	2	no	—
155	Dic₁₄₂	—	2	1	no	—
156	Dic₁₄₂	C₂²⋊C₈	1	2	yes	—
157	C₈×C₇₁	—	4	1	no	—
158	C₈×C₇₁	#120	1	1	yes	characteristic
159	C₈×C₇₁	#121	1	1	yes	characteristic
160	C₈×C₇₁	#121	1	1	yes	characteristic
161	C₈×C₇₁	#123	1	1	yes	characteristic
162	C₇₁⋊C₈	—	2	2	no	—
163	C₇₁⋊C₈	—	2	1	no	—
164	C₇₁⋊C₈	—	2	1	no	—
165	C₂²×C₇₁	—	2	1	no	—
166	C₂²×C₇₁	—	2	1	no	—
167	C₂²×C₇₁	—	4	1	no	—
168	C₂²×C₇₁	—	4	1	no	—
169	C₂²×C₇₁	—	4	1	no	—
170	C₂²×C₇₁	#408	1	1	yes	characteristic
171	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
172	C₂²×C₇₁	#435	1	1	yes	characteristic
173	C₂×D₇₁	—	2	2	no	—
174	C₂×D₇₁	—	4	1	no	—
175	C₂×D₇₁	—	4	1	no	—
176	C₄×C₇₁	—	4	1	no	—
177	C₄×C₇₁	#386	1	1	yes	characteristic
178	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
180	C₄×C₇₁	#435	1	1	yes	characteristic
181	Dic₇₁	—	2	2	no	—
182	Dic₇₁	—	4	2	no	—
183	#12980	—	71	1	no	maximal, Sylow
184	C₂×C₇₁	—	2	1	no	—
185	C₂×C₇₁	—	2	1	no	—
186	C₂×C₇₁	—	8	1	no	—
187	C₂×C₇₁	#1869	1	1	yes	characteristic
188	C₂×C₇₁	#2170	1	1	yes	characteristic
189	C₂×C₇₁	#2171	1	1	yes	characteristic
190	D₇₁	—	4	2	no	—
191	C₂×#504	—	71	1	no	—
192	C₂×#443	—	71	1	no	—
193	C₂×#350	—	71	1	no	—
194	C₂×#350	—	71	1	no	—
195	#2113	D₇₁	1	1	yes	characteristic, complemented
196	#1376	—	71	1	no	—
197	#1374	—	71	1	no	—
198	C₇₁	#12980	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	71	1	no	—
200	C₂×#131	—	71	1	no	—
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	—	142	1	no	—
206	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
207	C₂²×C₁₆	—	71	1	no	—
208	C₂²×C₁₆	—	142	1	no	—
209	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
210	#443	—	142	2	no	—
211	#443	—	142	2	no	—
212	C₂×#108	—	71	1	no	—
213	C₂×#104	—	71	1	no	—
214	C₂×#104	—	71	1	no	—
215	C₂×C₄×C₈	—	71	1	no	—
216	#350	—	142	2	no	—
217	#350	—	142	2	no	—
218	C₂²⋊C₁₆	—	2	1	no	—
219	C₂²⋊C₁₆	—	2	1	no	—
220	C₂×#41	—	71	1	no	—
221	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
222	#135	—	142	2	no	—
223	#135	—	142	2	no	—
224	C₂×Q₁₆	—	71	1	no	—
225	C₂×SD₁₆	—	71	1	no	—
226	C₂×SD₁₆	—	71	1	no	—
227	C₂×D₈	—	71	1	no	—
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	—	2	1	no	—
231	#131	—	2	1	no	—
232	#131	—	71	2	no	—
233	#131	—	71	2	no	—
234	#131	—	142	1	no	—
235	#131	—	142	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	71	1	no	—
238	C₂×M₁₆	—	142	1	no	—
239	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	71	1	no	—
242	C₂²×C₈	—	142	1	no	—
243	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₂×C₄²	—	71	1	no	—
245	M₃₂	—	2	1	no	—
246	M₃₂	—	2	1	no	—
247	M₃₂	—	71	2	no	—
248	M₃₂	—	142	4	no	—
249	M₃₂	—	142	1	no	—
250	C₂×C₁₆	—	2	1	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	142	4	no	—
254	C₂×C₁₆	—	142	1	no	—
255	C₂×C₁₆	—	142	1	no	—
256	C₂×C₁₆	—	142	1	no	—
257	C₂×C₁₆	—	142	1	no	—
258	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
260	#108	—	142	2	no	—
261	#104	—	142	2	no	—
262	#104	—	142	2	no	—
263	C₄×C₈	—	142	1	no	—
264	C₄×C₈	—	142	1	no	—
265	#41	—	2	1	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	2	1	no	—
269	#41	—	71	2	no	—
270	#41	—	71	2	no	—
271	#41	—	142	1	no	—
272	#41	—	142	1	no	—
273	C₂×Q₈	—	71	1	no	—
274	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	71	1	no	—
277	C₂×D₄	—	142	1	no	—
278	C₂×D₄	C₈×D₇₁	1	1	yes	characteristic
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	142	1	no	—
281	C₂²×C₄	—	142	1	no	—
282	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
283	Q₁₆	—	142	2	no	—
284	SD₁₆	—	142	2	no	—
285	SD₁₆	—	142	2	no	—
286	D₈	—	142	2	no	—
287	M₁₆	—	2	1	no	—
288	M₁₆	—	2	1	no	—
289	M₁₆	—	4	1	no	—
290	M₁₆	—	4	1	no	—
291	M₁₆	—	71	2	no	—
292	M₁₆	—	142	1	no	—
293	M₁₆	—	142	2	no	—
294	M₁₆	—	142	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	71	1	no	—
299	C₂×C₈	—	71	1	no	—
300	C₂×C₈	—	142	1	no	—
301	C₂×C₈	—	142	1	no	—
302	C₂×C₈	—	142	1	no	—
303	C₂×C₈	—	142	2	no	—
304	C₂×C₈	—	142	2	no	—
305	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
307	C₂×C₈	D₄×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	#11765865	1	1	yes	characteristic
311	C₄²	—	142	1	no	—
312	C₄²	—	142	1	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	4	1	no	—
316	C₁₆	—	284	1	no	—
317	C₁₆	—	142	2	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	142	1	no	—
320	C₂³	#14556134	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	71	2	no	—
324	Q₈	—	142	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	2	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	71	2	no	—
330	D₄	—	142	2	no	—
331	D₄	—	142	2	no	—
332	D₄	—	142	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	284	1	no	—
337	C₂×C₄	—	284	1	no	—
338	C₂×C₄	—	142	2	no	—
339	C₂×C₄	—	142	1	no	—
340	C₂×C₄	—	142	2	no	—
341	C₂×C₄	—	142	2	no	—
342	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
343	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
344	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
347	C₂×C₄	#14556136	1	1	yes	characteristic
348	C₈	—	4	1	no	—
349	C₈	—	142	2	no	—
350	C₈	—	142	1	no	—
351	C₈	—	142	1	no	—
352	C₈	#14556096	1	1	yes	characteristic
353	C₈	#14556099	1	1	yes	characteristic
354	C₈	#14556101	1	1	yes	characteristic
355	C₈	#14556110	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	284	1	no	—
362	C₂²	—	284	1	no	—
363	C₂²	—	142	2	no	—
364	C₂²	#17985964	1	1	yes	characteristic
365	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
366	C₂²	#17986073	1	1	yes	characteristic
367	C₄	—	4	1	no	—
368	C₄	—	284	2	no	—
369	C₄	—	142	2	no	—
370	C₄	#17985875	1	1	yes	characteristic
371	C₄	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
372	C₄	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
373	C₄	#17986068	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	8	1	no	—
377	C₂	—	284	2	no	—
378	C₂	#22588629	1	1	yes	characteristic
379	C₂	#22590175	1	1	yes	characteristic
380	C₂	#22590180	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2113	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#14556148	C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2113	D₇₁
hypocenter	C₇₁	#12980
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#17986415	C₂	semidirect
3. C₂×#17985784	C₂	semidirect
4. C₂×#17985783	C₂	semidirect
5. #22586421	C₂	semidirect
6. #22586417	C₂	semidirect
7. C₇₁×#2113	C₂	semidirect
195. #2113	D₇₁	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×#17986415	2
C₂×#17986114	2
C₂×#17985784	2
C₂×#17985783	2
#22586421	2
#22586417	2
C₇₁×#2113	2
#12980	71