Subgroups — #31897448

Order 18176.

460 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588193	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22586181	C₂	1	2	yes	maximal
4	#22586181	C₂	1	2	yes	maximal
5	C₄×#125×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14556039	—	2	1	no	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
10	#17985754	—	2	1	no	—
11	#17985754	—	2	1	no	—
12	C₂×#125×C₇₁	C₄	1	1	yes	characteristic
13	#389×C₇₁	C₄	1	1	yes	characteristic
14	#389×C₇₁	C₄	1	1	yes	characteristic
15	#387×C₇₁	—	2	1	no	—
16	#387×C₇₁	—	2	1	no	—
17	#382×C₇₁	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
19	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
20	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₄³×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₇₁⋊C₈	—	2	2	no	—
24	C₂²×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	Q₈	1	4	yes	—
28	#14556039	—	4	2	no	—
29	C₄×C₇₁⋊C₈	—	2	4	no	—
30	#125×C₇₁	—	2	2	no	—
31	#125×C₇₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	—	2	1	no	—
41	C₂×C₄²×C₇₁	—	2	1	no	—
42	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
49	#95×C₇₁	—	2	1	no	—
50	#95×C₇₁	—	2	1	no	—
51	#95×C₇₁	—	2	1	no	—
52	#95×C₇₁	—	2	1	no	—
53	#95×C₇₁	—	2	1	no	—
54	#95×C₇₁	—	2	1	no	—
55	C₂×C₇₁⋊C₈	—	2	4	no	—
56	C₂×C₇₁⋊C₈	—	2	4	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₂×C₇₁⋊C₈	—	4	2	no	—
59	C₂×C₇₁⋊C₈	—	4	2	no	—
60	C₂×C₇₁⋊C₈	—	4	2	no	—
61	C₂²×C₄×C₇₁	—	2	1	no	—
62	C₂²×C₄×C₇₁	—	2	1	no	—
63	C₂²×C₄×C₇₁	—	2	1	no	—
64	C₂²×C₄×C₇₁	—	2	1	no	—
65	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₇₁	—	2	4	no	—
73	C₄⋊C₄×C₇₁	—	2	4	no	—
74	C₄⋊C₄×C₇₁	—	2	1	no	—
75	C₄⋊C₄×C₇₁	—	2	1	no	—
76	C₄⋊C₄×C₇₁	—	2	2	no	—
77	C₄⋊C₄×C₇₁	—	2	2	no	—
78	C₄⋊C₄×C₇₁	—	4	2	no	—
79	C₄⋊C₄×C₇₁	—	4	2	no	—
80	C₄⋊C₄×C₇₁	M₁₆	1	2	yes	—
81	C₄⋊C₄×C₇₁	M₁₆	1	2	yes	—
82	C₄²×C₇₁	—	2	2	no	—
83	C₄²×C₇₁	—	2	1	no	—
84	C₄²×C₇₁	—	2	1	no	—
85	C₄²×C₇₁	—	2	2	no	—
86	C₄²×C₇₁	—	2	2	no	—
87	C₄²×C₇₁	—	2	1	no	—
88	C₄²×C₇₁	—	2	1	no	—
89	C₄²×C₇₁	—	4	1	no	—
90	C₄²×C₇₁	—	4	2	no	—
91	C₄²×C₇₁	—	4	1	no	—
92	C₄²×C₇₁	—	4	1	no	—
93	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
94	C₄²×C₇₁	#41	1	2	yes	—
95	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	2	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	4	1	no	—
112	C₂×C₄×C₇₁	—	4	1	no	—
113	C₂×C₄×C₇₁	—	4	1	no	—
114	C₂×C₄×C₇₁	—	4	1	no	—
115	C₂×C₄×C₇₁	—	4	1	no	—
116	C₂×C₄×C₇₁	—	4	1	no	—
117	C₂×C₄×C₇₁	—	4	1	no	—
118	C₂×C₄×C₇₁	—	4	1	no	—
119	C₂×C₄×C₇₁	—	4	1	no	—
120	C₂×C₄×C₇₁	—	4	1	no	—
121	C₂×C₄×C₇₁	#104	1	2	yes	—
122	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇₁	#131	1	2	yes	—
129	C₂×C₄×C₇₁	#135	1	2	yes	—
130	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
132	C₇₁⋊C₈	—	4	4	no	—
133	C₇₁⋊C₈	—	4	4	no	—
134	C₂²×C₇₁	#391	1	1	yes	characteristic
135	C₂²×C₇₁	#407	1	1	yes	characteristic
136	C₂²×C₇₁	#408	1	1	yes	characteristic
137	C₂²×C₇₁	#416	1	1	yes	characteristic
138	C₂²×C₇₁	#417	1	1	yes	characteristic
139	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
140	C₂²×C₇₁	#421	1	1	yes	characteristic
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	2	2	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	—	4	2	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	—	4	2	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	—	8	1	no	—
150	C₄×C₇₁	—	8	1	no	—
151	C₄×C₇₁	#446	1	2	yes	—
152	C₄×C₇₁	#446	1	2	yes	—
153	C₄×C₇₁	#478	1	2	yes	—
154	C₄×C₇₁	#479	1	2	yes	—
155	#9867	—	71	1	no	maximal, Sylow
156	C₂×C₇₁	#1488	1	1	yes	characteristic
157	C₂×C₇₁	#1489	1	1	yes	characteristic
158	C₂×C₇₁	#1494	1	1	yes	characteristic
159	C₂×C₇₁	#1495	1	1	yes	characteristic
160	C₂×C₇₁	#1985	1	1	yes	characteristic
161	C₂×C₇₁	#1986	1	1	yes	characteristic
162	C₂×C₇₁	#1989	1	1	yes	characteristic
163	C₄×#125	D₇₁	1	1	yes	characteristic
164	#1767	—	71	1	no	—
165	C₄×C₄⋊C₈	—	71	1	no	—
166	#1279	—	71	2	no	—
167	#1279	—	71	2	no	—
168	C₇₁	#9867	1	1	yes	characteristic, complemented, Sylow
169	C₂×#125	Dic₇₁	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	71	2	no	—
171	C₂×C₄⋊C₈	—	142	1	no	—
172	C₂×#97	—	142	1	no	—
173	C₂×C₄×C₈	—	71	2	no	—
174	#389	Dic₇₁	1	1	yes	characteristic
175	#389	Dic₇₁	1	1	yes	characteristic
176	#387	—	2	1	no	—
177	#387	—	2	1	no	—
178	#382	Dic₇₁	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	2	1	no	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
182	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
183	C₄³	C₂×D₇₁	1	1	yes	characteristic
184	#336	—	142	1	no	—
185	#336	—	142	1	no	—
186	C₂²×C₈	—	142	2	no	—
187	C₂²×C₈	—	142	2	no	—
188	#125	—	2	2	no	—
189	#125	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
207	C₄⋊C₈	—	71	4	no	—
208	C₄⋊C₈	—	142	4	no	—
209	C₄⋊C₈	—	142	2	no	—
210	#97	—	284	2	no	—
211	C₄×C₈	—	142	4	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
229	C₂×C₈	—	284	2	no	—
230	C₂×C₈	—	284	2	no	—
231	C₂×C₈	—	284	2	no	—
232	C₂×C₈	—	284	2	no	—
233	C₂×C₈	—	142	4	no	—
234	C₂×C₈	—	142	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	#11765849	1	2	yes	—
244	C₄⋊C₄	#11765849	1	2	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	1	no	—
256	C₄²	Q₈×D₇₁	1	2	yes	—
257	C₄²	#11765866	1	2	yes	—
258	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	#14556055	1	2	yes	—
285	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
286	C₂×C₄	#14556104	1	2	yes	—
287	C₂×C₄	#14556117	1	1	yes	characteristic
288	C₂×C₄	#14556126	1	1	yes	characteristic
289	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
290	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
291	C₂×C₄	#14556135	1	2	yes	—
292	C₂×C₄	#14556148	1	2	yes	—
293	C₂×C₄	#14556153	1	1	yes	characteristic
294	C₂×C₄	#14556158	1	1	yes	characteristic
295	C₈	—	284	4	no	—
296	C₈	—	284	4	no	—
297	C₂²	#17985906	1	1	yes	characteristic
298	C₂²	#17985963	1	1	yes	characteristic
299	C₂²	#17985966	1	1	yes	characteristic
300	C₂²	#17985995	1	1	yes	characteristic
301	C₂²	#17986000	1	1	yes	characteristic
302	C₂²	C₂×#14556055	1	1	yes	characteristic
303	C₂²	#17986016	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	#17986130	1	2	yes	—
315	C₄	#17986130	1	2	yes	—
316	C₄	#17986296	1	2	yes	—
317	C₄	#17986303	1	2	yes	—
318	C₂	#22586774	1	1	yes	characteristic
319	C₂	#22586779	1	1	yes	characteristic
320	C₂	#22586800	1	1	yes	characteristic
321	C₂	#22586805	1	1	yes	characteristic
322	C₂	#22589281	1	1	yes	characteristic
323	C₂	#22589285	1	1	yes	characteristic
324	C₂	#22589298	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#125×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#125	D₇₁
hypocenter	C₇₁	#9867
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
168. C₇₁	#9867	semidirect

Maximal subgroups

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

subgroup	index
#22588193	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22586181	2
C₄×#125×C₇₁	2
#9867	71