Subgroups — #31915933

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985809	C₂	1	4	yes	maximal, complemented
3	C₄×#14556060	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556060	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₇₁	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
10	#17985809	—	2	8	no	—
11	#17985762	—	2	1	no	—
12	C₂×#135×C₇₁	—	2	1	no	—
13	C₂×#135×C₇₁	C₂²	1	2	yes	—
14	#532×C₇₁	—	2	1	no	—
15	#495×C₇₁	—	2	1	no	—
16	#495×C₇₁	C₄	1	2	yes	complemented
17	#494×C₇₁	C₄	1	1	yes	characteristic, complemented
18	#493×C₇₁	C₄	1	1	yes	characteristic, complemented
19	#492×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×C₄×Dic₇₁	—	4	1	no	—
23	#14556068	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	#14556060	—	2	2	no	—
26	#14556060	D₄	1	4	yes	complemented
27	C₇₁⋊#95	—	4	1	no	—
28	C₂×#41×C₇₁	—	2	2	no	—
29	#135×C₇₁	—	2	4	no	—
30	#135×C₇₁	—	2	4	no	—
31	#135×C₇₁	—	4	4	no	—
32	C₂×Q₁₆×C₇₁	—	2	1	no	—
33	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₇₁	—	2	2	no	—
35	C₂×SD₁₆×C₇₁	—	2	2	no	—
36	C₂×D₈×C₇₁	—	2	1	no	—
37	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₁	—	2	1	no	—
41	#127×C₇₁	—	2	1	no	—
42	#124×C₇₁	—	2	2	no	—
43	#121×C₇₁	—	4	1	no	—
44	#121×C₇₁	—	4	1	no	—
45	C₄×D₄×C₇₁	—	4	1	no	—
46	C₄×D₄×C₇₁	—	4	1	no	—
47	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×Dic₇₁	—	4	2	no	—
53	C₄⋊Dic₇₁	—	2	4	no	—
54	C₄⋊Dic₇₁	—	4	2	no	—
55	C₄×Dic₇₁	—	8	2	no	—
56	C₇₁⋊C₁₆	—	4	2	no	—
57	#41×C₇₁	—	4	4	no	—
58	#41×C₇₁	—	4	4	no	—
59	C₂×Q₈×C₇₁	—	2	2	no	—
60	C₂×D₄×C₇₁	—	2	2	no	—
61	C₂×D₄×C₇₁	—	4	2	no	—
62	C₂×D₄×C₇₁	—	4	1	no	—
63	C₂×D₄×C₇₁	—	4	1	no	—
64	C₂²×C₄×C₇₁	—	4	2	no	—
65	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₇₁	—	2	4	no	—
69	Q₁₆×C₇₁	—	4	2	no	—
70	SD₁₆×C₇₁	—	4	4	no	—
71	SD₁₆×C₇₁	—	4	2	no	—
72	SD₁₆×C₇₁	—	4	2	no	—
73	D₈×C₇₁	—	2	4	no	—
74	D₈×C₇₁	—	4	2	no	—
75	C₂×C₈×C₇₁	—	2	2	no	—
76	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	#41	1	2	yes	—
82	C₄⋊C₄×C₇₁	—	4	1	no	—
83	C₄⋊C₄×C₇₁	—	4	1	no	—
84	C₂²⋊C₄×C₇₁	—	4	2	no	—
85	C₂²⋊C₄×C₇₁	—	4	2	no	—
86	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
89	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
90	C₂³×C₇₁	—	4	2	no	—
91	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₇₁	—	4	2	no	—
93	Q₈×C₇₁	—	4	2	no	—
94	D₄×C₇₁	—	4	2	no	—
95	D₄×C₇₁	—	4	2	no	—
96	D₄×C₇₁	—	8	2	no	—
97	D₄×C₇₁	—	8	4	no	—
98	D₄×C₇₁	—	8	2	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	4	1	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	—	8	4	no	—
103	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#121	1	2	yes	—
109	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×Dic₇₁	—	4	4	no	—
112	C₂×Dic₇₁	—	8	2	no	—
113	C₂×Dic₇₁	—	8	2	no	—
114	C₈×C₇₁	—	2	1	no	—
115	C₈×C₇₁	—	2	1	no	—
116	C₈×C₇₁	C₄×D₄	1	2	yes	—
117	C₈×C₇₁	#124	1	1	yes	characteristic
118	C₈×C₇₁	#127	1	1	yes	characteristic
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	—	2	1	no	—
121	C₂²×C₇₁	—	8	2	no	—
122	C₂²×C₇₁	—	8	2	no	—
123	C₂²×C₇₁	—	8	2	no	—
124	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
125	C₂²×C₇₁	#459	1	2	yes	—
126	C₄×C₇₁	—	2	1	no	—
127	C₄×C₇₁	—	2	1	no	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	—	8	2	no	—
130	C₄×C₇₁	#360	1	2	yes	complemented
131	C₄×C₇₁	#390	1	1	yes	characteristic
132	C₄×C₇₁	#418	1	1	yes	characteristic
133	C₄×C₇₁	#460	1	2	yes	—
134	Dic₇₁	—	8	4	no	—
135	#13255	—	71	1	no	maximal, Sylow
136	C₂×C₇₁	—	2	2	no	—
137	C₂×C₇₁	—	16	2	no	—
138	C₂×C₇₁	#1982	1	1	yes	characteristic
139	C₂×C₇₁	C₂×#360	1	1	yes	characteristic
140	C₂×C₇₁	#2145	1	1	yes	characteristic
141	#3147	D₇₁	1	1	yes	characteristic
142	#2150	—	71	1	no	—
143	C₂×#360	—	71	4	no	—
144	C₄×#106	—	71	1	no	—
145	C₇₁	#13255	1	1	yes	characteristic, complemented, Sylow
146	C₂×#135	—	2	1	no	—
147	C₂×#135	C₂×D₇₁	1	2	yes	—
148	#532	—	2	1	no	—
149	C₂×M₃₂	—	71	2	no	—
150	#495	—	2	1	no	—
151	#495	Dic₇₁	1	2	yes	complemented
152	#494	Dic₇₁	1	1	yes	characteristic, complemented
153	#493	Dic₇₁	1	1	yes	characteristic, complemented
154	#492	—	2	1	no	—
155	C₂×#106	—	71	2	no	—
156	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	142	1	no	—
158	C₄⋊C₁₆	—	142	1	no	—
159	C₄⋊C₁₆	—	142	1	no	—
160	#360	—	142	8	no	—
161	#340	—	142	1	no	—
162	C₂×#41	—	2	2	no	—
163	#135	—	2	4	no	—
164	#135	—	2	4	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
172	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	1	no	—
175	#127	—	2	1	no	—
176	#124	—	2	2	no	—
177	#121	—	4	1	no	—
178	#121	—	4	1	no	—
179	C₄×D₄	—	4	1	no	—
180	C₄×D₄	—	4	1	no	—
181	C₂×C₄⋊C₄	—	142	2	no	—
182	C₂×C₄²	—	284	1	no	—
183	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
184	M₃₂	—	142	4	no	—
185	C₂×C₁₆	—	142	2	no	—
186	#106	—	71	4	no	—
187	#106	—	142	2	no	—
188	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
189	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
192	#95	—	284	1	no	—
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	284	2	no	—
202	C₂²×C₄	#11765850	1	1	yes	characteristic
203	C₂²×C₄	#11765850	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
205	Q₁₆	—	2	4	no	—
206	Q₁₆	—	4	2	no	—
207	SD₁₆	—	4	4	no	—
208	SD₁₆	—	4	2	no	—
209	SD₁₆	—	4	2	no	—
210	D₈	—	2	4	no	—
211	D₈	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
216	C₂×C₈	D₄×D₇₁	1	2	yes	—
217	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
218	C₂×C₈	#11765864	1	2	yes	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	284	2	no	—
222	C₄⋊C₄	—	142	4	no	—
223	C₂²⋊C₄	—	4	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₄²	—	568	2	no	—
226	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
227	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
228	C₄²	#11765853	1	1	yes	characteristic
229	C₄²	#11765853	1	1	yes	characteristic
230	C₁₆	—	284	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×#11765850	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	4	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	284	4	no	—
245	C₂×C₄	—	568	2	no	—
246	C₂×C₄	—	568	2	no	—
247	C₂×C₄	#14556049	1	1	yes	characteristic
248	C₂×C₄	#14556049	1	1	yes	characteristic
249	C₂×C₄	#14556049	1	1	yes	characteristic
250	C₂×C₄	#14556049	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
252	C₂×C₄	#14556099	1	2	yes	—
253	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
254	C₂×C₄	#14556154	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	D₄×Dic₇₁	1	2	yes	—
258	C₈	#14556115	1	1	yes	characteristic
259	C₈	#14556124	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×#14556049	1	1	yes	characteristic
266	C₂²	#17986191	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	—	568	4	no	—
272	C₄	#17985809	1	2	yes	complemented
273	C₄	#17985902	1	1	yes	characteristic
274	C₄	#17986004	1	1	yes	characteristic
275	C₄	#17986196	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	16	2	no	—
278	C₂	#22589268	1	1	yes	characteristic
279	C₂	C₂×#17985809	1	1	yes	characteristic
280	C₂	#22590076	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#3147	C₂
socle	C₂²×C₇₁	C₂×#102
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#3147	D₇₁
hypocenter	C₇₁	#13255
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22590096	C₂	semidirect
2. C₂×#17985809	C₂	semidirect
3. C₄×#14556060	C₂	semidirect
16. #495×C₇₁	C₄	semidirect
17. #494×C₇₁	C₄	semidirect
18. #493×C₇₁	C₄	semidirect
26. #14556060	D₄	semidirect
130. C₄×C₇₁	#360	semidirect
145. C₇₁	#13255	semidirect
151. #495	Dic₇₁	semidirect
152. #494	Dic₇₁	semidirect
153. #493	Dic₇₁	semidirect
272. C₄	#17985809	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#17985809	2
C₄×#14556060	2
C₇₁×#3147	2
#13255	71