Subgroups — #32152977

Order 18176.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22590627	C₂	1	4	yes	maximal, complemented
3	#22590620	C₂	1	2	yes	maximal, complemented
4	#22590613	C₂	1	2	yes	maximal, complemented
5	#22590129	C₂	1	4	yes	maximal
6	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556128	—	2	1	no	—
9	C₂×#14556068	C₂²	1	2	yes	—
10	#17986404	—	2	1	no	—
11	#17986404	—	2	1	no	—
12	#17986404	C₂²	1	2	yes	—
13	#17986404	C₂²	1	2	yes	—
14	C₂×#14556062	—	2	1	no	—
15	C₂×#14556062	C₂²	1	2	yes	—
16	#17985826	C₂²	1	8	yes	—
17	#17985824	C₂²	1	8	yes	—
18	#17985775	C₂²	1	2	yes	—
19	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
20	#533×C₇₁	—	2	1	no	—
21	#500×C₇₁	C₂²	1	1	yes	characteristic
22	#500×C₇₁	C₂²	1	1	yes	characteristic
23	#499×C₇₁	—	2	1	no	—
24	#499×C₇₁	C₂²	1	2	yes	—
25	#498×C₇₁	—	2	1	no	—
26	#427×C₇₁	—	2	1	no	—
27	#427×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₄₂	—	4	1	no	—
30	#14556128	—	2	2	no	—
31	#14556128	—	2	2	no	—
32	#14556128	—	4	1	no	—
33	#14556128	—	4	1	no	—
34	C₂×C₄⋊Dic₇₁	—	2	2	no	—
35	C₂×C₄⋊Dic₇₁	—	4	1	no	—
36	#14556068	Q₈	1	8	yes	complemented
37	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
38	#14556062	—	2	2	no	—
39	#14556062	—	2	2	no	—
40	#14556062	—	2	2	no	—
41	#14556062	C₂³	1	4	yes	—
42	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₇₁	—	2	1	no	—
45	C₄⋊Q₈×C₇₁	—	2	1	no	—
46	#125×C₇₁	—	2	2	no	—
47	C₂²⋊Q₈×C₇₁	—	4	1	no	—
48	C₂²⋊Q₈×C₇₁	—	4	1	no	—
49	C₄×Q₈×C₇₁	—	4	1	no	—
50	C₄×Q₈×C₇₁	—	4	1	no	—
51	#117×C₇₁	—	2	2	no	—
52	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
53	#107×C₇₁	—	2	1	no	—
54	#107×C₇₁	—	2	1	no	—
55	#107×C₇₁	C₂³	1	2	yes	—
56	#107×C₇₁	C₂³	1	2	yes	—
57	#106×C₇₁	—	2	2	no	—
58	#106×C₇₁	—	2	2	no	—
59	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₄₂	—	4	2	no	—
64	C₂×Dic₁₄₂	—	4	2	no	—
65	C₂×Dic₁₄₂	—	8	4	no	—
66	C₂²×Dic₇₁	—	4	2	no	—
67	C₄⋊Dic₇₁	—	2	4	no	—
68	C₄⋊Dic₇₁	—	4	2	no	—
69	C₄⋊Dic₇₁	—	4	2	no	—
70	C₄⋊Dic₇₁	—	4	2	no	—
71	C₇₁⋊C₁₆	C₂×Q₈	1	8	yes	complemented
72	C₂×Q₈×C₇₁	—	4	1	no	—
73	C₂×Q₈×C₇₁	—	4	1	no	—
74	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
78	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₇₁	—	2	2	no	—
84	C₄⋊C₄×C₇₁	—	2	2	no	—
85	C₄⋊C₄×C₇₁	—	4	2	no	—
86	C₄⋊C₄×C₇₁	—	4	2	no	—
87	C₄⋊C₄×C₇₁	—	4	1	no	—
88	C₄⋊C₄×C₇₁	—	4	1	no	—
89	C₂²⋊C₄×C₇₁	—	4	2	no	—
90	C₄²×C₇₁	—	4	2	no	—
91	C₄²×C₇₁	D₈	1	1	yes	characteristic
92	C₄²×C₇₁	D₈	1	1	yes	characteristic
93	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₇₁	—	8	2	no	—
97	Q₈×C₇₁	—	8	2	no	—
98	C₂×C₄×C₇₁	—	2	2	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	4	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₇₁	—	4	4	no	—
111	C₂×Dic₇₁	—	8	4	no	—
112	Dic₁₄₂	—	8	4	no	—
113	Dic₁₄₂	—	8	4	no	—
114	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
115	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
116	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
117	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
118	C₂²×C₇₁	—	2	1	no	—
119	C₂²×C₇₁	—	2	1	no	—
120	C₂²×C₇₁	#500	1	2	yes	—
121	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	8	2	no	—
125	C₄×C₇₁	—	8	2	no	—
126	C₄×C₇₁	#500	1	2	yes	—
127	C₄×C₇₁	#509	1	2	yes	—
128	C₄×C₇₁	#510	1	2	yes	—
129	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
131	Dic₇₁	—	8	4	no	—
132	#33731	—	71	1	no	maximal, Sylow
133	C₂×C₇₁	—	2	2	no	—
134	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#509	1	1	yes	characteristic
136	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
137	#3160	D₇₁	1	1	yes	characteristic
138	C₂×#500	—	71	1	no	—
139	#2255	—	71	4	no	—
140	#2254	—	71	2	no	—
141	#2252	—	71	2	no	—
142	#2159	—	71	4	no	—
143	C₄×M₃₂	—	71	1	no	—
144	C₇₁	#33731	1	1	yes	characteristic, complemented, Sylow
145	#533	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	142	1	no	—
147	C₂×M₃₂	—	71	2	no	—
148	#500	—	71	2	no	—
149	#500	—	71	2	no	—
150	#500	—	142	1	no	—
151	#500	—	142	1	no	—
152	#500	C₂×D₇₁	1	1	yes	characteristic
153	#500	C₂×D₇₁	1	1	yes	characteristic
154	#499	—	2	1	no	—
155	#499	C₂×D₇₁	1	2	yes	—
156	#498	—	2	1	no	—
157	#427	—	2	1	no	—
158	#427	C₂×D₇₁	1	2	yes	—
159	C₂×#107	—	71	2	no	—
160	C₂×#107	—	142	1	no	—
161	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
162	#367	—	71	8	no	—
163	#366	—	71	8	no	—
164	#346	—	71	2	no	—
165	C₄×C₁₆	—	71	2	no	—
166	C₂²×Q₈	—	284	1	no	—
167	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	—	284	1	no	—
172	C₄⋊Q₈	—	284	1	no	—
173	C₄⋊Q₈	—	142	2	no	—
174	C₄⋊Q₈	—	142	2	no	—
175	#125	—	2	2	no	—
176	C₂²⋊Q₈	—	4	1	no	—
177	C₂²⋊Q₈	—	4	1	no	—
178	C₄×Q₈	—	4	1	no	—
179	C₄×Q₈	—	4	1	no	—
180	#117	—	2	2	no	—
181	C₂×C₄⋊C₄	—	284	1	no	—
182	C₂×C₄⋊C₄	—	142	2	no	—
183	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
184	M₃₂	—	71	8	no	—
185	C₂×C₁₆	—	71	4	no	—
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	71	4	no	—
189	#107	—	142	2	no	—
190	#107	—	142	2	no	—
191	#107	—	142	2	no	—
192	#107	C₂²×D₇₁	1	2	yes	—
193	#107	C₂²×D₇₁	1	2	yes	—
194	#106	—	2	2	no	—
195	#106	—	2	2	no	—
196	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	284	2	no	—
203	C₂×Q₈	—	284	2	no	—
204	C₂×Q₈	—	568	4	no	—
205	C₂²×C₄	—	284	2	no	—
206	C₂²×C₄	#11765850	1	1	yes	characteristic
207	C₂²×C₄	#11765850	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
209	C₂×C₈	#11765855	1	4	yes	—
210	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
213	C₂×C₈	Q₈×D₇₁	1	4	yes	—
214	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	284	2	no	—
222	C₄⋊C₄	—	284	2	no	—
223	C₄⋊C₄	—	284	2	no	—
224	C₄⋊C₄	—	142	4	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	#11765850	1	1	yes	characteristic
228	C₄²	#11765850	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
231	C₁₆	—	71	8	no	—
232	C₂³	C₂×#11765850	1	1	yes	characteristic
233	Q₈	—	8	2	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	568	4	no	—
236	Q₈	—	568	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	284	4	no	—
242	C₂×C₄	—	568	4	no	—
243	C₂×C₄	#14556126	1	2	yes	—
244	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
245	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
246	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
247	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
248	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
249	C₂×C₄	C₂×#11765855	1	2	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
251	C₈	#14556126	1	2	yes	—
252	C₈	C₂×#11765855	1	2	yes	—
253	C₈	C₂×#11765855	1	2	yes	—
254	C₈	C₂×Q₈×D₇₁	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#17986402	1	2	yes	—
258	C₂²	C₂²×#11765850	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	568	4	no	—
264	C₄	#17986402	1	2	yes	—
265	C₄	#17986434	1	2	yes	—
266	C₄	#17986439	1	2	yes	—
267	C₄	C₂×#14556126	1	1	yes	characteristic
268	C₄	C₂²×#11765855	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#17986402	1	1	yes	characteristic
271	C₂	C₂×#17986434	1	1	yes	characteristic
272	C₂	C₂×#17986439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#33731
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22590627	C₂	semidirect
3. #22590620	C₂	semidirect
4. #22590613	C₂	semidirect
36. #14556068	Q₈	semidirect
71. C₇₁⋊C₁₆	C₂×Q₈	semidirect
144. C₇₁	#33731	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
#22590627	2
#22590620	2
#22590613	2
#22590129	2
C₄×#14556068	2
C₇₁×#3160	2
#33731	71