Subgroups — #31890705

Order 18176.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597051	C₂	1	1	yes	maximal, characteristic
2	#22588511	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17985751	C₂	1	1	yes	maximal, characteristic, complemented
4	#22587018	C₂	1	1	yes	maximal, characteristic
5	#22587015	C₂	1	1	yes	maximal, characteristic
6	#22586214	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556136	—	2	1	no	—
9	#17986549	C₂²	1	1	yes	characteristic
10	#17986076	—	2	1	no	—
11	#17986076	C₄	1	2	yes	—
12	#17986064	—	2	1	no	—
13	#17986064	C₄	1	2	yes	—
14	C₂×#14556063	—	2	1	no	—
15	C₂×#14556063	—	2	1	no	—
16	#17986023	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	#17985967	—	2	1	no	—
19	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	#17985751	—	2	1	no	—
21	#17985751	—	2	1	no	—
22	#17985746	—	2	1	no	—
23	#17985744	—	2	1	no	—
24	#17985742	—	2	1	no	—
25	#17985742	—	2	1	no	—
26	#423×C₇₁	C₂²	1	1	yes	characteristic
27	#423×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×#11765865	—	2	1	no	—
30	#14556136	—	4	2	no	—
31	#14556136	—	4	2	no	—
32	C₂×#11765849	—	2	1	no	—
33	C₂×#11765848	—	2	2	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	C₂×C₈×D₇₁	—	2	2	no	—
38	#14556128	D₄	1	1	yes	characteristic
39	#14556125	D₄	1	1	yes	characteristic
40	#14556116	—	2	1	no	—
41	#14556102	—	2	1	no	—
42	#14556102	Q₈	1	2	yes	—
43	C₄×D₂₈₄	—	2	1	no	—
44	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
45	#14556063	—	4	2	no	—
46	#14556063	—	4	2	no	—
47	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
48	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
49	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
50	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
51	(C₂×D₇₁)⋊C₈	—	2	2	no	—
52	(C₂×D₇₁)⋊C₈	—	2	2	no	—
53	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
56	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₇₁	—	2	1	no	—
58	C₂×M₁₆×C₇₁	—	2	1	no	—
59	C₂²×C₈×C₇₁	—	2	1	no	—
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₁	—	2	1	no	—
62	C₄⋊C₈×C₇₁	—	2	1	no	—
63	C₄⋊C₈×C₇₁	—	2	1	no	—
64	C₄⋊C₈×C₇₁	—	2	1	no	—
65	C₄⋊C₈×C₇₁	—	2	1	no	—
66	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
67	#11765865	—	4	1	no	—
68	#11765865	—	4	1	no	—
69	#11765865	—	4	1	no	—
70	#11765865	—	4	1	no	—
71	C₂×Dic₁₄₂	—	2	1	no	—
72	C₂×D₂₈₄	—	2	1	no	—
73	C₂×D₂₈₄	—	2	1	no	—
74	C₂×D₂₈₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₂²⋊D₇₁	—	2	2	no	—
76	C₂×C₄×D₇₁	—	2	2	no	—
77	#11765849	—	4	1	no	—
78	#11765849	—	4	1	no	—
79	#11765848	—	4	4	no	—
80	C₂×C₇₁⋊C₈	—	2	1	no	—
81	C₂×C₇₁⋊C₈	—	2	1	no	—
82	C₂×C₇₁⋊C₈	—	4	1	no	—
83	C₂×C₇₁⋊C₈	—	4	1	no	—
84	C₂×C₇₁⋊C₈	—	8	1	no	—
85	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
90	C₂×C₇₁⋊C₈	SD₁₆	1	2	yes	—
91	C₈×D₇₁	—	4	4	no	—
92	C₄⋊Dic₇₁	—	2	1	no	—
93	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
94	(C₂×D₇₁)⋊C₄	—	2	2	no	—
95	(C₂×D₇₁)⋊C₄	—	2	2	no	—
96	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
99	M₁₆×C₇₁	—	4	1	no	—
100	M₁₆×C₇₁	—	4	1	no	—
101	M₁₆×C₇₁	—	4	1	no	—
102	M₁₆×C₇₁	—	4	1	no	—
103	C₂×C₈×C₇₁	—	2	2	no	—
104	C₂×C₈×C₇₁	—	4	1	no	—
105	C₂×C₈×C₇₁	—	4	1	no	—
106	C₂×C₈×C₇₁	—	8	1	no	—
107	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₁	#41	1	1	yes	characteristic
111	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²×D₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	4	1	no	—
116	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
119	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
126	C₂²⋊D₇₁	—	4	2	no	—
127	C₂²⋊D₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	2	2	no	—
129	D₂₈₄	—	4	2	no	—
130	D₂₈₄	—	4	1	no	—
131	D₂₈₄	—	4	1	no	—
132	D₂₈₄	—	4	1	no	—
133	D₂₈₄	—	4	1	no	—
134	C₄×D₇₁	—	4	2	no	—
135	C₄×D₇₁	—	4	2	no	—
136	Dic₁₄₂	—	4	1	no	—
137	Dic₁₄₂	—	4	1	no	—
138	C₈×C₇₁	—	4	2	no	—
139	C₈×C₇₁	—	8	1	no	—
140	C₈×C₇₁	—	8	1	no	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₇₁⋊C₈	—	2	1	no	—
143	C₇₁⋊C₈	—	8	1	no	—
144	C₇₁⋊C₈	—	8	1	no	—
145	C₇₁⋊C₈	#102	1	2	yes	—
146	C₇₁⋊C₈	#102	1	2	yes	—
147	C₂²×C₇₁	—	2	1	no	—
148	C₂²×C₇₁	—	2	1	no	—
149	C₂²×C₇₁	#380	1	1	yes	characteristic
150	C₂²×C₇₁	#474	1	1	yes	characteristic
151	C₂²×C₇₁	#476	1	1	yes	characteristic
152	C₂×D₇₁	—	4	2	no	—
153	C₂×D₇₁	—	4	2	no	—
154	C₂×D₇₁	—	4	2	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	2	1	no	—
158	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
159	C₄×C₇₁	#417	1	1	yes	characteristic
160	C₄×C₇₁	#459	1	1	yes	characteristic
161	C₄×C₇₁	#461	1	1	yes	characteristic
162	C₄×C₇₁	#472	1	1	yes	characteristic
163	C₄×C₇₁	#473	1	1	yes	characteristic
164	Dic₇₁	—	4	2	no	—
165	#8684	—	71	1	no	maximal, Sylow
166	C₂×C₇₁	—	2	1	no	—
167	C₂×C₇₁	—	2	1	no	—
168	C₂×C₇₁	#1802	1	1	yes	characteristic
169	C₂×C₇₁	#1813	1	1	yes	characteristic
170	C₂×C₇₁	#1816	1	1	yes	characteristic
171	D₇₁	—	8	2	no	—
172	#2931	—	71	1	no	—
173	#1848	—	71	1	no	—
174	C₂×#335	—	71	1	no	—
175	#1547	—	71	1	no	—
176	#1547	—	71	1	no	—
177	#1292	—	71	1	no	—
178	#1292	D₇₁	1	1	yes	characteristic, complemented
179	C₇₁	#8684	1	1	yes	characteristic, complemented, Sylow
180	C₂×#131	—	142	1	no	—
181	#532	—	71	1	no	—
182	#436	—	71	2	no	—
183	#436	—	142	1	no	—
184	C₈×D₄	—	71	2	no	—
185	C₈×D₄	—	142	1	no	—
186	C₂×#108	—	142	1	no	—
187	C₂×#108	—	142	1	no	—
188	#423	—	71	1	no	—
189	#423	C₂×D₇₁	1	1	yes	characteristic
190	#423	C₂×D₇₁	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	71	1	no	—
192	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
193	#408	—	142	1	no	—
194	C₂×C₄×C₈	—	71	1	no	—
195	#335	—	142	1	no	—
196	#335	—	142	1	no	—
197	#332	—	142	1	no	—
198	#332	—	142	1	no	—
199	#331	—	142	1	no	—
200	#331	—	142	1	no	—
201	C₂×#41	—	142	1	no	—
202	#131	—	284	2	no	—
203	#131	—	284	2	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	142	2	no	—
207	C₂×M₁₆	—	142	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	71	1	no	—
210	C₂²×C₈	—	71	1	no	—
211	C₂²×C₈	—	142	2	no	—
212	C₂²×C₈	—	142	1	no	—
213	C₄⋊Q₈	—	71	1	no	—
214	#127	—	71	1	no	—
215	#124	—	142	1	no	—
216	#121	—	71	2	no	—
217	#121	—	142	1	no	—
218	C₄×D₄	—	71	2	no	—
219	C₄×D₄	—	142	1	no	—
220	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
221	#108	—	284	2	no	—
222	#108	—	284	2	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	142	1	no	—
229	C₄⋊C₈	—	142	1	no	—
230	C₄⋊C₈	—	142	1	no	—
231	C₄⋊C₈	—	142	1	no	—
232	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
233	C₂²⋊C₈	—	142	2	no	—
234	C₂²⋊C₈	—	142	2	no	—
235	C₄×C₈	—	71	1	no	—
236	C₄×C₈	—	71	1	no	—
237	C₄×C₈	—	71	1	no	—
238	C₄×C₈	—	71	1	no	—
239	#41	—	284	1	no	—
240	#41	—	284	1	no	—
241	#41	—	284	1	no	—
242	#41	—	284	1	no	—
243	C₂×Q₈	—	142	1	no	—
244	C₂×D₄	—	71	2	no	—
245	C₂×D₄	—	142	2	no	—
246	C₂×D₄	—	142	1	no	—
247	C₂×D₄	—	142	1	no	—
248	C₂²×C₄	—	142	2	no	—
249	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
250	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
251	C₂²×C₄	#11765865	1	1	yes	characteristic
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	284	4	no	—
257	M₁₆	—	284	1	no	—
258	M₁₆	—	284	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₂×C₈	—	284	4	no	—
264	C₂×C₈	—	284	1	no	—
265	C₂×C₈	—	284	1	no	—
266	C₂×C₈	—	568	1	no	—
267	C₂×C₈	—	71	1	no	—
268	C₂×C₈	—	71	1	no	—
269	C₂×C₈	—	71	2	no	—
270	C₂×C₈	—	71	2	no	—
271	C₂×C₈	—	71	1	no	—
272	C₂×C₈	—	71	1	no	—
273	C₂×C₈	—	142	1	no	—
274	C₂×C₈	—	142	1	no	—
275	C₄⋊C₄	—	71	2	no	—
276	C₄⋊C₄	—	142	1	no	—
277	C₂²⋊C₄	—	142	2	no	—
278	C₂²⋊C₄	—	142	2	no	—
279	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
280	C₄²	D₄×D₇₁	1	1	yes	characteristic
281	C₄²	D₄×D₇₁	1	1	yes	characteristic
282	C₄²	#11765865	1	1	yes	characteristic
283	C₂³	—	142	2	no	—
284	C₂³	#14556106	1	1	yes	characteristic
285	Q₈	—	284	1	no	—
286	Q₈	—	284	1	no	—
287	D₄	—	284	2	no	—
288	D₄	—	284	2	no	—
289	D₄	—	284	2	no	—
290	D₄	—	284	1	no	—
291	D₄	—	284	1	no	—
292	D₄	—	284	1	no	—
293	D₄	—	284	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	284	2	no	—
299	C₂×C₄	—	142	2	no	—
300	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
302	C₂×C₄	#14556085	1	1	yes	characteristic
303	C₂×C₄	#14556101	1	1	yes	characteristic
304	C₂×C₄	#14556101	1	1	yes	characteristic
305	C₂×C₄	#14556106	1	1	yes	characteristic
306	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
308	C₂×C₄	#14556155	1	1	yes	characteristic
309	C₂×C₄	#14556160	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	568	1	no	—
314	C₈	—	568	1	no	—
315	C₈	—	71	2	no	—
316	C₈	—	71	2	no	—
317	C₈	—	142	1	no	—
318	C₈	—	142	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	284	2	no	—
322	C₂²	—	284	2	no	—
323	C₂²	—	284	2	no	—
324	C₂²	#17985858	1	1	yes	characteristic
325	C₂²	#17986278	1	1	yes	characteristic
326	C₂²	#17986288	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	284	2	no	—
331	C₄	#102×D₇₁	1	1	yes	characteristic
332	C₄	#17985999	1	1	yes	characteristic
333	C₄	#17986193	1	1	yes	characteristic
334	C₄	#17986203	1	1	yes	characteristic
335	C₄	#17986268	1	1	yes	characteristic
336	C₄	#17986273	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	568	2	no	—
340	C₂	#22588339	1	1	yes	characteristic
341	C₂	#22588389	1	1	yes	characteristic
342	C₂	#22588403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985858
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1292	C₂
socle	C₂²×C₇₁	#380
prime radical	#17986549	C₂²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1292	D₇₁
hypocenter	C₇₁	#8684
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #22588511	C₂	semidirect
3. C₂×#17985751	C₂	semidirect
6. #22586214	C₂	semidirect
7. C₇₁×#1292	C₂	semidirect
178. #1292	D₇₁	semidirect
179. C₇₁	#8684	semidirect

Maximal subgroups

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

subgroup	index
#22597051	2
#22588511	2
C₂×#17985751	2
#22587018	2
#22587015	2
#22586214	2
C₇₁×#1292	2
#8684	71