Subgroups — #31915634

Order 18176.

430 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986114	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985784	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17985783	C₂	1	1	yes	maximal, characteristic, complemented
5	#22589953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985774	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	C₂×#14556068	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17986114	—	2	2	no	—
13	#17986114	—	2	2	no	—
14	C₂×#14556063	C₄	1	1	yes	characteristic
15	C₂×#14556055	C₄	1	1	yes	characteristic
16	C₂×#14556054	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985819	—	2	2	no	—
19	#17985784	—	2	2	no	—
20	#17985783	—	2	2	no	—
21	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
22	#17985774	—	2	2	no	—
23	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	C₂²⋊C₁₆×C₇₁	—	2	2	no	—
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556148	—	2	2	no	—
29	#14556148	—	2	2	no	—
30	C₂×#11765855	C₈	1	1	yes	characteristic
31	C₂×#11765853	C₈	1	1	yes	characteristic
32	C₂×#11765852	C₈	1	1	yes	characteristic
33	C₂×#11765850	C₈	1	1	yes	characteristic
34	#14556134	—	2	1	no	—
35	#14556134	—	2	1	no	—
36	#14556134	—	2	1	no	—
37	#14556134	—	2	1	no	—
38	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
39	C₂×#11765848	—	2	1	no	—
40	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₇₁	—	2	1	no	—
42	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
43	#14556068	—	2	1	no	—
44	#14556068	—	2	1	no	—
45	C₂×C₇₁⋊C₁₆	—	2	1	no	—
46	C₂×C₇₁⋊C₁₆	—	2	1	no	—
47	C₂×C₇₁⋊C₁₆	—	2	1	no	—
48	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
49	#14556063	—	2	2	no	—
50	#14556055	—	2	2	no	—
51	#14556054	—	2	2	no	—
52	C₈×Dic₇₁	D₄	1	2	yes	—
53	C₈×Dic₇₁	D₄	1	2	yes	—
54	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
55	#131×C₇₁	—	2	1	no	—
56	#131×C₇₁	—	2	1	no	—
57	#131×C₇₁	—	2	1	no	—
58	#131×C₇₁	—	2	1	no	—
59	C₂×M₁₆×C₇₁	—	2	1	no	—
60	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₇₁	—	2	1	no	—
62	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	M₃₂×C₇₁	—	2	1	no	—
64	M₃₂×C₇₁	—	2	1	no	—
65	C₂×C₁₆×C₇₁	—	2	1	no	—
66	C₂×C₁₆×C₇₁	—	2	1	no	—
67	C₂×C₁₆×C₇₁	—	2	1	no	—
68	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
69	#11765865	—	2	1	no	—
70	#11765865	—	2	1	no	—
71	#11765865	—	2	1	no	—
72	#11765865	—	2	1	no	—
73	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₇₁	—	2	1	no	—
76	C₂²×Dic₇₁	—	2	1	no	—
77	C₂×C₄×D₇₁	—	2	1	no	—
78	#11765855	—	2	2	no	—
79	#11765853	—	2	2	no	—
80	#11765852	—	2	2	no	—
81	#11765850	—	2	2	no	—
82	#11765849	—	2	1	no	—
83	#11765849	—	2	1	no	—
84	#11765848	—	4	1	no	—
85	#11765848	—	4	1	no	—
86	C₂×C₇₁⋊C₈	—	2	1	no	—
87	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
89	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
90	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
91	C₈×D₇₁	—	4	1	no	—
92	C₈×D₇₁	—	4	1	no	—
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	C₂²⋊C₄	1	2	yes	—
95	C₇₁⋊C₁₆	—	2	2	no	—
96	C₇₁⋊C₁₆	—	4	1	no	—
97	#41×C₇₁	—	2	1	no	—
98	#41×C₇₁	—	2	1	no	—
99	#41×C₇₁	—	2	1	no	—
100	#41×C₇₁	—	2	1	no	—
101	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
102	C₂×D₄×C₇₁	—	2	1	no	—
103	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₁	—	2	1	no	—
105	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	M₁₆×C₇₁	—	2	1	no	—
107	M₁₆×C₇₁	—	2	1	no	—
108	M₁₆×C₇₁	—	4	1	no	—
109	M₁₆×C₇₁	—	4	1	no	—
110	C₂×C₈×C₇₁	—	2	1	no	—
111	C₂×C₈×C₇₁	—	4	1	no	—
112	C₂×C₈×C₇₁	—	4	1	no	—
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
119	C₁₆×C₇₁	—	2	2	no	—
120	C₁₆×C₇₁	—	4	1	no	—
121	C₂³×C₇₁	—	2	1	no	—
122	C₂³×C₇₁	#131	1	1	yes	characteristic
123	C₂²×D₇₁	—	2	1	no	—
124	Q₈×C₇₁	—	2	1	no	—
125	Q₈×C₇₁	—	2	1	no	—
126	D₄×C₇₁	—	2	1	no	—
127	D₄×C₇₁	—	2	1	no	—
128	D₄×C₇₁	—	4	1	no	—
129	D₄×C₇₁	—	4	1	no	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	—	4	1	no	—
132	C₂×C₄×C₇₁	—	4	1	no	—
133	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	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	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
139	C₂²⋊D₇₁	—	4	1	no	—
140	C₂²⋊D₇₁	—	4	1	no	—
141	C₂×Dic₇₁	—	2	2	no	—
142	C₂×Dic₇₁	—	2	1	no	—
143	C₂×Dic₇₁	—	4	2	no	—
144	D₂₈₄	—	2	1	no	—
145	D₂₈₄	—	2	1	no	—
146	C₄×D₇₁	—	4	1	no	—
147	C₄×D₇₁	—	4	1	no	—
148	Dic₁₄₂	—	2	1	no	—
149	Dic₁₄₂	—	2	1	no	—
150	C₈×C₇₁	—	4	1	no	—
151	C₈×C₇₁	#121	1	1	yes	characteristic
152	C₈×C₇₁	#121	1	1	yes	characteristic
153	C₈×C₇₁	#124	1	1	yes	characteristic
154	C₈×C₇₁	#127	1	1	yes	characteristic
155	C₇₁⋊C₈	—	4	1	no	—
156	C₇₁⋊C₈	C₂²⋊C₈	1	2	yes	—
157	C₇₁⋊C₈	C₂²⋊C₈	1	2	yes	—
158	C₂²×C₇₁	—	2	2	no	—
159	C₂²×C₇₁	—	4	2	no	—
160	C₂²×C₇₁	—	4	1	no	—
161	C₂²×C₇₁	#408	1	1	yes	characteristic
162	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
163	C₂²×C₇₁	#436	1	1	yes	characteristic
164	C₂×D₇₁	—	4	2	no	—
165	C₂×D₇₁	—	4	1	no	—
166	C₄×C₇₁	—	4	1	no	—
167	C₄×C₇₁	#390	1	1	yes	characteristic
168	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
170	C₄×C₇₁	#436	1	1	yes	characteristic
171	Dic₇₁	—	4	2	no	—
172	Dic₇₁	—	4	1	no	—
173	#13196	—	71	1	no	maximal, Sylow
174	C₂×C₇₁	—	2	1	no	—
175	C₂×C₇₁	—	2	1	no	—
176	C₂×C₇₁	—	8	1	no	—
177	C₂×C₇₁	#1954	1	1	yes	characteristic
178	C₂×C₇₁	#2171	1	2	yes	—
179	D₇₁	—	8	1	no	—
180	C₂×#443	—	71	1	no	—
181	C₂×#364	—	71	1	no	—
182	C₂×#350	—	71	1	no	—
183	C₂×#350	—	71	1	no	—
184	#2113	—	71	1	no	—
185	#2113	D₇₁	1	1	yes	characteristic, complemented
186	C₂×#346	—	71	1	no	—
187	C₇₁	#13196	1	1	yes	characteristic, complemented, Sylow
188	C₂×#135	—	71	1	no	—
189	C₂×#131	—	71	1	no	—
190	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
191	C₂×M₃₂	—	71	1	no	—
192	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
193	C₂²×C₁₆	—	71	1	no	—
194	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
195	#443	—	142	2	no	—
196	#443	—	142	2	no	—
197	C₂×#108	—	71	1	no	—
198	C₂×#104	—	71	1	no	—
199	C₂×#104	—	71	1	no	—
200	C₂×C₄×C₈	—	71	1	no	—
201	#364	—	142	2	no	—
202	#350	—	142	2	no	—
203	#350	—	142	2	no	—
204	C₂²⋊C₁₆	—	2	2	no	—
205	C₂²⋊C₁₆	—	142	2	no	—
206	#346	—	142	2	no	—
207	C₂×#41	—	71	1	no	—
208	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
209	#135	—	142	2	no	—
210	#135	—	142	2	no	—
211	C₂×Q₁₆	—	71	1	no	—
212	C₂×SD₁₆	—	71	1	no	—
213	C₂×SD₁₆	—	71	1	no	—
214	C₂×D₈	—	71	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	2	1	no	—
218	#131	—	2	1	no	—
219	#131	—	142	1	no	—
220	#131	—	142	1	no	—
221	#131	—	142	1	no	—
222	#131	—	142	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	71	1	no	—
225	C₂×M₁₆	—	142	1	no	—
226	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	71	1	no	—
229	C₂²×C₈	—	142	1	no	—
230	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×C₄²	—	71	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	142	1	no	—
235	M₃₂	—	142	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	71	2	no	—
240	C₂×C₁₆	—	142	1	no	—
241	C₂×C₁₆	—	142	1	no	—
242	C₂×C₁₆	—	142	1	no	—
243	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
244	#108	—	142	2	no	—
245	#104	—	142	2	no	—
246	#104	—	142	2	no	—
247	C₄×C₈	—	71	2	no	—
248	C₄×C₈	—	71	2	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	142	1	no	—
254	#41	—	142	1	no	—
255	#41	—	142	1	no	—
256	#41	—	142	1	no	—
257	C₂×Q₈	—	71	1	no	—
258	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	71	1	no	—
261	C₂×D₄	—	142	1	no	—
262	C₂×D₄	C₈×D₇₁	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	142	1	no	—
265	C₂²×C₄	—	142	1	no	—
266	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
267	Q₁₆	—	142	2	no	—
268	SD₁₆	—	142	2	no	—
269	SD₁₆	—	142	2	no	—
270	D₈	—	142	2	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	284	1	no	—
276	M₁₆	—	284	1	no	—
277	M₁₆	—	142	1	no	—
278	M₁₆	—	142	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	284	1	no	—
283	C₂×C₈	—	284	1	no	—
284	C₂×C₈	—	71	2	no	—
285	C₂×C₈	—	71	1	no	—
286	C₂×C₈	—	71	1	no	—
287	C₂×C₈	—	71	2	no	—
288	C₂×C₈	—	142	1	no	—
289	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
290	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
291	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
293	C₂×C₈	#11765864	1	1	yes	characteristic
294	C₂×C₈	#11765865	1	1	yes	characteristic
295	C₄²	—	71	2	no	—
296	C₄²	—	142	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	4	1	no	—
299	C₁₆	—	284	1	no	—
300	C₁₆	—	142	2	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	142	1	no	—
303	C₂³	#14556134	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	142	1	no	—
307	Q₈	—	142	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	284	1	no	—
313	D₄	—	284	1	no	—
314	D₄	—	142	1	no	—
315	D₄	—	142	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	284	2	no	—
320	C₂×C₄	—	284	1	no	—
321	C₂×C₄	—	284	1	no	—
322	C₂×C₄	—	142	1	no	—
323	C₂×C₄	—	142	2	no	—
324	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
325	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
327	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
328	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
329	C₂×C₄	#14556136	1	1	yes	characteristic
330	C₈	—	4	1	no	—
331	C₈	—	284	1	no	—
332	C₈	—	71	2	no	—
333	C₈	—	71	2	no	—
334	C₈	#14556100	1	1	yes	characteristic
335	C₈	#14556101	1	1	yes	characteristic
336	C₈	#14556114	1	1	yes	characteristic
337	C₈	#14556124	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	284	1	no	—
342	C₂²	—	284	2	no	—
343	C₂²	#17985964	1	1	yes	characteristic
344	C₂²	C₈×C₂²⋊D₇₁	1	1	yes	characteristic
345	C₂²	#17986074	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	284	2	no	—
348	C₄	—	284	1	no	—
349	C₄	#17985899	1	1	yes	characteristic
350	C₄	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
351	C₄	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
352	C₄	#17986075	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	8	1	no	—
356	C₂	—	568	1	no	—
357	C₂	#22589132	1	1	yes	characteristic
358	C₂	#22590180	1	2	yes	—

	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	#13196
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
2. C₂×#17985819	C₂	semidirect
3. C₂×#17985784	C₂	semidirect
4. C₂×#17985783	C₂	semidirect
5. #22589953	C₂	semidirect
6. C₂×#17985774	C₂	semidirect
7. C₇₁×#2113	C₂	semidirect
185. #2113	D₇₁	semidirect
187. C₇₁	#13196	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986114	2
C₂×#17985819	2
C₂×#17985784	2
C₂×#17985783	2
#22589953	2
C₂×#17985774	2
C₇₁×#2113	2
#13196	71