Subgroups — #31890725

Order 18176.

414 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	#22597051	C₂	1	1	yes	maximal, characteristic
2	#22588477	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	#17986023	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
16	#17985967	—	2	1	no	—
17	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	#17985763	—	2	1	no	—
19	#17985763	—	2	1	no	—
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	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
47	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
48	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
49	(C₂×D₇₁)⋊C₈	—	2	2	no	—
50	(C₂×D₇₁)⋊C₈	—	2	2	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	C₂×M₁₆×C₇₁	—	2	1	no	—
56	C₂×M₁₆×C₇₁	—	2	1	no	—
57	C₂²×C₈×C₇₁	—	2	1	no	—
58	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₁	—	2	1	no	—
60	C₄⋊C₈×C₇₁	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
65	#11765865	—	4	1	no	—
66	#11765865	—	4	1	no	—
67	#11765865	—	4	1	no	—
68	#11765865	—	4	1	no	—
69	C₂×Dic₁₄₂	—	2	1	no	—
70	C₂×D₂₈₄	—	2	1	no	—
71	C₂×D₂₈₄	—	2	1	no	—
72	C₂×D₂₈₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₂²⋊D₇₁	—	2	2	no	—
74	C₂×C₄×D₇₁	—	2	2	no	—
75	#11765849	—	4	1	no	—
76	#11765849	—	4	1	no	—
77	#11765848	—	4	4	no	—
78	C₂×C₇₁⋊C₈	—	2	1	no	—
79	C₂×C₇₁⋊C₈	—	2	1	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	C₈×D₇₁	—	4	4	no	—
90	C₄⋊Dic₇₁	—	2	1	no	—
91	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
92	(C₂×D₇₁)⋊C₄	—	2	2	no	—
93	(C₂×D₇₁)⋊C₄	—	2	2	no	—
94	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
97	M₁₆×C₇₁	—	4	1	no	—
98	M₁₆×C₇₁	—	4	1	no	—
99	M₁₆×C₇₁	—	4	1	no	—
100	M₁₆×C₇₁	—	4	1	no	—
101	C₂×C₈×C₇₁	—	2	2	no	—
102	C₂×C₈×C₇₁	—	4	1	no	—
103	C₂×C₈×C₇₁	—	4	1	no	—
104	C₂×C₈×C₇₁	—	8	1	no	—
105	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇₁	#41	1	1	yes	characteristic
109	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×D₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	4	1	no	—
114	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
122	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂²⋊D₇₁	—	4	2	no	—
125	C₂²⋊D₇₁	—	4	2	no	—
126	C₂×Dic₇₁	—	2	2	no	—
127	D₂₈₄	—	4	2	no	—
128	D₂₈₄	—	4	1	no	—
129	D₂₈₄	—	4	1	no	—
130	D₂₈₄	—	4	1	no	—
131	D₂₈₄	—	4	1	no	—
132	C₄×D₇₁	—	4	2	no	—
133	C₄×D₇₁	—	4	2	no	—
134	Dic₁₄₂	—	4	1	no	—
135	Dic₁₄₂	—	4	1	no	—
136	C₈×C₇₁	—	4	2	no	—
137	C₈×C₇₁	—	8	1	no	—
138	C₈×C₇₁	—	8	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₇₁⋊C₈	—	2	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	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	—	2	1	no	—
147	C₂²×C₇₁	#380	1	1	yes	characteristic
148	C₂²×C₇₁	#478	1	1	yes	characteristic
149	C₂²×C₇₁	#478	1	1	yes	characteristic
150	C₂×D₇₁	—	4	2	no	—
151	C₂×D₇₁	—	4	2	no	—
152	C₂×D₇₁	—	4	2	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	#416	1	1	yes	characteristic
157	C₄×C₇₁	#418	1	1	yes	characteristic
158	C₄×C₇₁	#463	1	1	yes	characteristic
159	C₄×C₇₁	#463	1	1	yes	characteristic
160	C₄×C₇₁	#472	1	1	yes	characteristic
161	C₄×C₇₁	#473	1	1	yes	characteristic
162	Dic₇₁	—	4	2	no	—
163	#8688	—	71	1	no	maximal, Sylow
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	#1804	1	1	yes	characteristic
167	C₂×C₇₁	#1813	1	1	yes	characteristic
168	C₂×C₇₁	#1816	1	1	yes	characteristic
169	D₇₁	—	8	2	no	—
170	#2931	—	71	1	no	—
171	#1837	—	71	1	no	—
172	C₂×#335	—	71	1	no	—
173	#1547	—	71	1	no	—
174	#1547	—	71	1	no	—
175	#1292	—	71	1	no	—
176	#1292	D₇₁	1	1	yes	characteristic, complemented
177	C₇₁	#8688	1	1	yes	characteristic, complemented, Sylow
178	C₂×#131	—	142	1	no	—
179	#532	—	71	1	no	—
180	#436	—	71	2	no	—
181	#436	—	142	1	no	—
182	C₈×D₄	—	71	2	no	—
183	C₈×D₄	—	142	1	no	—
184	#423	—	71	1	no	—
185	#423	C₂×D₇₁	1	1	yes	characteristic
186	#423	C₂×D₇₁	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	71	1	no	—
188	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
189	#408	—	142	1	no	—
190	C₂×C₄×C₈	—	71	1	no	—
191	#341	—	142	1	no	—
192	#341	—	142	1	no	—
193	#335	—	142	1	no	—
194	#335	—	142	1	no	—
195	#332	—	142	1	no	—
196	#332	—	142	1	no	—
197	#331	—	142	1	no	—
198	#331	—	142	1	no	—
199	C₂×#41	—	142	1	no	—
200	#131	—	284	2	no	—
201	#131	—	284	2	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	142	2	no	—
205	C₂×M₁₆	—	142	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	71	1	no	—
208	C₂²×C₈	—	71	1	no	—
209	C₂²×C₈	—	142	2	no	—
210	C₂²×C₈	—	142	1	no	—
211	C₄⋊Q₈	—	71	1	no	—
212	#127	—	71	1	no	—
213	#124	—	142	1	no	—
214	#121	—	71	2	no	—
215	#121	—	142	1	no	—
216	C₄×D₄	—	71	2	no	—
217	C₄×D₄	—	142	1	no	—
218	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	142	1	no	—
225	C₄⋊C₈	—	142	1	no	—
226	C₄⋊C₈	—	142	1	no	—
227	C₄⋊C₈	—	142	1	no	—
228	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
229	C₂²⋊C₈	—	142	2	no	—
230	C₂²⋊C₈	—	142	2	no	—
231	C₄×C₈	—	71	1	no	—
232	C₄×C₈	—	71	1	no	—
233	C₄×C₈	—	71	1	no	—
234	C₄×C₈	—	71	1	no	—
235	#41	—	284	1	no	—
236	#41	—	284	1	no	—
237	#41	—	284	1	no	—
238	#41	—	284	1	no	—
239	C₂×Q₈	—	142	1	no	—
240	C₂×D₄	—	71	2	no	—
241	C₂×D₄	—	142	2	no	—
242	C₂×D₄	—	142	1	no	—
243	C₂×D₄	—	142	1	no	—
244	C₂²×C₄	—	142	2	no	—
245	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
246	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
247	C₂²×C₄	#11765865	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	284	4	no	—
253	M₁₆	—	284	1	no	—
254	M₁₆	—	284	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	8	1	no	—
259	C₂×C₈	—	284	4	no	—
260	C₂×C₈	—	284	1	no	—
261	C₂×C₈	—	284	1	no	—
262	C₂×C₈	—	568	1	no	—
263	C₂×C₈	—	71	1	no	—
264	C₂×C₈	—	71	1	no	—
265	C₂×C₈	—	71	1	no	—
266	C₂×C₈	—	71	1	no	—
267	C₂×C₈	—	142	1	no	—
268	C₂×C₈	—	142	1	no	—
269	C₂×C₈	—	142	1	no	—
270	C₂×C₈	—	142	1	no	—
271	C₄⋊C₄	—	71	2	no	—
272	C₄⋊C₄	—	142	1	no	—
273	C₂²⋊C₄	—	142	2	no	—
274	C₂²⋊C₄	—	142	2	no	—
275	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
276	C₄²	D₄×D₇₁	1	1	yes	characteristic
277	C₄²	D₄×D₇₁	1	1	yes	characteristic
278	C₄²	#11765865	1	1	yes	characteristic
279	C₂³	—	142	2	no	—
280	C₂³	#14556106	1	1	yes	characteristic
281	Q₈	—	284	1	no	—
282	Q₈	—	284	1	no	—
283	D₄	—	284	2	no	—
284	D₄	—	284	2	no	—
285	D₄	—	284	2	no	—
286	D₄	—	284	1	no	—
287	D₄	—	284	1	no	—
288	D₄	—	284	1	no	—
289	D₄	—	284	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	284	2	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	142	2	no	—
296	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
298	C₂×C₄	#14556085	1	1	yes	characteristic
299	C₂×C₄	#14556101	1	1	yes	characteristic
300	C₂×C₄	#14556101	1	1	yes	characteristic
301	C₂×C₄	#14556106	1	1	yes	characteristic
302	C₂×C₄	#14556149	1	1	yes	characteristic
303	C₂×C₄	#14556151	1	1	yes	characteristic
304	C₂×C₄	#14556152	1	1	yes	characteristic
305	C₂×C₄	#14556155	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	8	1	no	—
308	C₈	—	8	1	no	—
309	C₈	—	568	1	no	—
310	C₈	—	568	1	no	—
311	C₈	—	142	1	no	—
312	C₈	—	142	1	no	—
313	C₈	—	142	1	no	—
314	C₈	—	142	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	284	2	no	—
318	C₂²	—	284	2	no	—
319	C₂²	—	284	2	no	—
320	C₂²	#17985858	1	1	yes	characteristic
321	C₂²	#17986298	1	1	yes	characteristic
322	C₂²	#17986302	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	284	2	no	—
327	C₄	#17985994	1	1	yes	characteristic
328	C₄	#17986002	1	1	yes	characteristic
329	C₄	#17986213	1	1	yes	characteristic
330	C₄	#17986217	1	1	yes	characteristic
331	C₄	#17986268	1	1	yes	characteristic
332	C₄	#17986273	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	568	2	no	—
336	C₂	#22588349	1	1	yes	characteristic
337	C₂	#22588389	1	1	yes	characteristic
338	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	#8688
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. #22588477	C₂	semidirect
3. C₂×#17985751	C₂	semidirect
6. #22586214	C₂	semidirect
7. C₇₁×#1292	C₂	semidirect
176. #1292	D₇₁	semidirect
177. C₇₁	#8688	semidirect

Maximal subgroups

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

subgroup	index
#22597051	2
#22588477	2
C₂×#17985751	2
#22587018	2
#22587015	2
#22586214	2
C₇₁×#1292	2
#8688	71