Subgroups — #31912559

Order 18176.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985784	C₂	1	1	yes	maximal, characteristic
3	#22589939	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17985775	C₂	1	1	yes	maximal, characteristic, complemented
5	#22588177	C₂	1	1	yes	maximal, characteristic
6	#22586430	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986417	—	2	2	no	—
10	#17986417	C₄	1	4	yes	complemented
11	C₂×#14556067	—	2	1	no	—
12	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₁	—	2	1	no	—
15	C₂×#14556056	C₄	1	1	yes	characteristic
16	#17985996	C₄	1	1	yes	characteristic
17	#17985954	C₂²	1	1	yes	characteristic
18	#17985784	C₄	1	4	yes	complemented
19	#17985775	—	2	2	no	—
20	#17985774	—	2	1	no	—
21	C₁₆×Dic₇₁	—	2	1	no	—
22	#17985764	C₄	1	1	yes	characteristic
23	#17985759	C₄	1	1	yes	characteristic
24	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
28	#14556134	—	2	1	no	—
29	#14556134	—	2	1	no	—
30	#14556134	C₂×C₄	1	2	yes	—
31	#14556134	C₂×C₄	1	2	yes	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
34	C₂×#11765848	—	2	1	no	—
35	C₂×C₈×D₇₁	—	2	1	no	—
36	#14556087	C₂×C₄	1	1	yes	characteristic
37	#14556067	—	2	4	no	—
38	C₂×C₇₁⋊C₁₆	—	2	1	no	—
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
43	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
45	C₁₆×D₇₁	—	2	4	no	—
46	#14556056	C₈	1	4	yes	—
47	#14556053	C₈	1	2	yes	—
48	#14556050	C₈	1	2	yes	—
49	#14556038	—	2	1	no	—
50	C₈×Dic₇₁	—	2	1	no	—
51	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₇₁	—	2	1	no	—
55	M₃₂×C₇₁	—	2	1	no	—
56	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
57	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₇₁	—	2	1	no	—
59	C₂×C₁₆×C₇₁	—	2	1	no	—
60	C₄×C₈×C₇₁	—	2	1	no	—
61	C₄×C₈×C₇₁	—	2	1	no	—
62	#11765865	—	2	1	no	—
63	#11765865	—	2	1	no	—
64	#11765865	C₄²	1	2	yes	—
65	#11765865	C₂×C₈	1	2	yes	—
66	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
67	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	#11765849	—	2	1	no	—
71	#11765849	—	2	1	no	—
72	#11765849	C₄²	1	2	yes	—
73	#11765849	C₂×C₈	1	2	yes	—
74	#11765848	—	2	2	no	—
75	#11765848	—	2	2	no	—
76	C₂×C₇₁⋊C₈	—	2	1	no	—
77	C₂×C₇₁⋊C₈	—	2	1	no	—
78	C₈×D₇₁	—	2	2	no	—
79	C₈×D₇₁	—	2	2	no	—
80	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₇₁	—	2	1	no	—
82	C₄×Dic₇₁	—	2	1	no	—
83	C₇₁⋊C₁₆	—	2	2	no	—
84	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
85	C₇₁⋊C₁₆	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₇₁	—	2	1	no	—
87	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	—	2	1	no	—
89	C₂×C₈×C₇₁	—	2	1	no	—
90	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
91	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
96	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
97	C₄²×C₇₁	—	2	1	no	—
98	C₄²×C₇₁	C₂×C₈	1	2	yes	—
99	C₁₆×C₇₁	—	2	2	no	—
100	C₁₆×C₇₁	—	2	2	no	—
101	C₂³×C₇₁	#131	1	1	yes	characteristic
102	C₂²×D₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	4	2	no	—
105	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
111	C₂²⋊D₇₁	—	2	2	no	—
112	C₂²⋊D₇₁	—	2	2	no	—
113	C₂×Dic₇₁	—	2	1	no	—
114	C₂×Dic₇₁	—	2	2	no	—
115	D₂₈₄	—	2	1	no	—
116	D₂₈₄	C₄×C₈	1	2	yes	—
117	C₄×D₇₁	—	2	2	no	—
118	C₄×D₇₁	—	2	2	no	—
119	Dic₁₄₂	—	2	1	no	—
120	Dic₁₄₂	C₄×C₈	1	2	yes	—
121	C₈×C₇₁	—	2	2	no	—
122	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₇₁	#117	1	1	yes	characteristic
124	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
125	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
126	C₇₁⋊C₈	—	2	2	no	—
127	C₇₁⋊C₈	—	2	2	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	#405	1	1	yes	characteristic
130	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
132	C₂×D₇₁	—	2	2	no	—
133	C₂×D₇₁	—	4	2	no	—
134	C₄×C₇₁	—	4	2	no	—
135	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
138	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
139	Dic₇₁	—	2	2	no	—
140	Dic₇₁	—	4	2	no	—
141	#12615	—	71	1	no	maximal, Sylow
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	—	2	1	no	—
144	C₂×C₇₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₇₁	#2171	1	2	yes	—
146	D₇₁	—	4	2	no	—
147	C₂×#504	—	71	1	no	—
148	C₂×#350	—	71	1	no	—
149	#2108	—	71	1	no	—
150	C₂×#346	—	71	1	no	—
151	#1764	—	71	1	no	—
152	#1379	—	71	1	no	—
153	#1376	D₇₁	1	1	yes	characteristic, complemented
154	C₇₁	#12615	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	71	1	no	—
156	#504	—	71	4	no	—
157	#504	—	142	2	no	—
158	C₂×M₃₂	—	142	1	no	—
159	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
160	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
161	C₂²×C₁₆	—	71	1	no	—
162	C₂²×C₁₆	—	71	1	no	—
163	C₂²×C₁₆	—	142	1	no	—
164	C₂×#104	—	71	1	no	—
165	#416	—	71	1	no	—
166	#405	—	71	1	no	—
167	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
168	#350	—	71	4	no	—
169	#346	—	142	2	no	—
170	#346	—	142	1	no	—
171	C₄×C₁₆	—	142	1	no	—
172	#341	—	71	1	no	—
173	#339	—	71	1	no	—
174	C₂×#41	—	71	1	no	—
175	#131	—	71	2	no	—
176	#131	—	71	2	no	—
177	#131	—	142	1	no	—
178	#131	—	142	1	no	—
179	C₂×M₁₆	—	71	1	no	—
180	C₂×M₁₆	—	71	1	no	—
181	C₂×M₁₆	—	142	1	no	—
182	C₂²×C₈	—	142	1	no	—
183	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
185	#117	—	71	1	no	—
186	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	142	4	no	—
190	M₃₂	C₄×D₇₁	1	2	yes	—
191	M₃₂	C₄×D₇₁	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	71	2	no	—
195	C₂×C₁₆	—	71	2	no	—
196	C₂×C₁₆	—	71	1	no	—
197	C₂×C₁₆	—	71	1	no	—
198	C₂×C₁₆	—	71	2	no	—
199	C₂×C₁₆	—	142	4	no	—
200	C₂×C₁₆	—	142	1	no	—
201	C₂×C₁₆	—	142	1	no	—
202	#104	—	71	4	no	—
203	#103	—	71	2	no	—
204	#102	—	71	2	no	—
205	#97	—	142	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	142	1	no	—
209	#41	—	71	2	no	—
210	#41	—	71	2	no	—
211	#41	—	142	1	no	—
212	#41	—	142	1	no	—
213	C₂×Q₈	—	71	1	no	—
214	C₂×D₄	—	71	1	no	—
215	C₂×D₄	—	142	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	142	1	no	—
218	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
219	M₁₆	—	71	2	no	—
220	M₁₆	—	71	2	no	—
221	M₁₆	—	142	1	no	—
222	M₁₆	—	142	1	no	—
223	M₁₆	—	142	2	no	—
224	M₁₆	—	142	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	142	1	no	—
228	C₂×C₈	—	142	1	no	—
229	C₂×C₈	—	142	2	no	—
230	C₂×C₈	—	142	2	no	—
231	C₂×C₈	C₈×D₇₁	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
234	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
235	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
236	C₂×C₈	#11765864	1	1	yes	characteristic
237	C₂×C₈	#11765866	1	1	yes	characteristic
238	C₄⋊C₄	—	71	2	no	—
239	C₂²⋊C₄	—	142	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	142	1	no	—
242	C₄²	C₈×D₇₁	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	71	2	no	—
246	C₁₆	—	71	2	no	—
247	C₁₆	—	142	2	no	—
248	C₂³	—	142	1	no	—
249	C₂³	#14556136	1	1	yes	characteristic
250	Q₈	—	71	2	no	—
251	Q₈	—	142	1	no	—
252	D₄	—	71	2	no	—
253	D₄	—	142	2	no	—
254	D₄	—	142	2	no	—
255	D₄	—	142	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	142	2	no	—
259	C₂×C₄	—	142	1	no	—
260	C₂×C₄	—	142	2	no	—
261	C₂×C₄	—	142	2	no	—
262	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
263	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	#14556090	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
267	C₂×C₄	#14556136	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	142	2	no	—
270	C₈	—	142	2	no	—
271	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
272	C₈	#14556086	1	1	yes	characteristic
273	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
274	C₈	#14556090	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	284	2	no	—
277	C₂²	—	142	2	no	—
278	C₂²	#17985956	1	1	yes	characteristic
279	C₂²	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
280	C₂²	#17986064	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	284	2	no	—
283	C₄	—	142	2	no	—
284	C₄	#17985846	1	1	yes	characteristic
285	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
286	C₄	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
287	C₄	#17986064	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	284	2	no	—
291	C₂	#22588110	1	1	yes	characteristic
292	C₂	#22590181	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1376	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1376	D₇₁
hypocenter	C₇₁	#12615
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. #22589939	C₂	semidirect
4. C₂×#17985775	C₂	semidirect
6. #22586430	C₂	semidirect
7. C₇₁×#1376	C₂	semidirect
10. #17986417	C₄	semidirect
18. #17985784	C₄	semidirect
153. #1376	D₇₁	semidirect
154. C₇₁	#12615	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986417	2
C₂×#17985784	2
#22589939	2
C₂×#17985775	2
#22588177	2
#22586430	2
C₇₁×#1376	2
#12615	71