Subgroups — #31896791

Order 18176.

406 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586840	C₂	1	2	yes	maximal
4	#22586744	C₂	1	2	yes	maximal
5	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
6	#17986556	C₂²	1	2	yes	—
7	#17986404	C₄	1	1	yes	characteristic, complemented
8	#17986404	C₄	1	1	yes	characteristic, complemented
9	#17986401	—	2	1	no	—
10	#17986401	—	2	1	no	—
11	#17986397	C₄	1	2	yes	complemented
12	#17986037	—	2	1	no	—
13	#17986037	—	2	1	no	—
14	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₁	—	2	1	no	—
16	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
17	#17985753	—	2	1	no	—
18	#17985737	—	2	4	no	—
19	#17985731	—	2	4	no	—
20	#431×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#335×C₇₁	—	2	1	no	—
23	#334×C₇₁	—	2	1	no	—
24	C₂²×C₇₁⋊C₈	—	2	1	no	—
25	#14556128	C₂×C₄	1	2	yes	—
26	#14556128	C₂×C₄	1	2	yes	—
27	#14556120	—	2	2	no	—
28	#14556107	—	2	2	no	—
29	#14556107	—	2	2	no	—
30	C₄×Dic₁₄₂	—	2	2	no	—
31	C₄×Dic₁₄₂	—	2	2	no	—
32	#14556087	—	2	2	no	—
33	C₂×C₄⋊Dic₇₁	—	2	1	no	—
34	C₂×C₄×Dic₇₁	—	2	1	no	—
35	#14556062	—	2	1	no	—
36	#14556062	—	2	1	no	—
37	#14556062	—	2	1	no	—
38	#14556062	—	2	1	no	—
39	#14556060	—	2	2	no	—
40	#14556060	—	2	2	no	—
41	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
42	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
43	C₈×Dic₇₁	—	4	2	no	—
44	C₇₁⋊#95	D₄	1	2	yes	—
45	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₁	—	2	2	no	—
49	C₂²⋊C₈×C₇₁	—	2	2	no	—
50	C₄×C₈×C₇₁	—	2	2	no	—
51	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₂×Dic₁₄₂	—	2	2	no	—
56	C₂×Dic₁₄₂	—	2	2	no	—
57	C₂²×Dic₇₁	—	2	2	no	—
58	C₂×C₇₁⋊C₈	—	2	2	no	—
59	C₂×C₇₁⋊C₈	—	4	1	no	—
60	C₂×C₇₁⋊C₈	—	4	1	no	—
61	C₄⋊Dic₇₁	—	2	4	no	—
62	C₄⋊Dic₇₁	—	2	2	no	—
63	C₄⋊Dic₇₁	—	2	2	no	—
64	C₄⋊Dic₇₁	—	2	2	no	—
65	C₄⋊Dic₇₁	—	2	2	no	—
66	Dic₇₁⋊C₄	—	4	2	no	—
67	Dic₇₁⋊C₄	—	4	2	no	—
68	C₂²⋊Dic₇₁	—	4	2	no	—
69	C₄×Dic₇₁	—	4	2	no	—
70	C₄×Dic₇₁	—	4	1	no	—
71	C₄×Dic₇₁	—	4	1	no	—
72	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₇₁	—	2	1	no	—
76	C₂×C₈×C₇₁	—	2	1	no	—
77	C₂×C₈×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
83	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
84	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₁	#41	1	1	yes	characteristic
88	C₂³×C₇₁	#121	1	1	yes	characteristic
89	C₂×C₄×C₇₁	—	2	2	no	—
90	C₂×C₄×C₇₁	—	4	1	no	—
91	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
101	C₂×Dic₇₁	—	2	4	no	—
102	C₂×Dic₇₁	—	4	2	no	—
103	C₂×Dic₇₁	—	4	2	no	—
104	C₂×Dic₇₁	—	4	2	no	—
105	C₂×Dic₇₁	—	4	2	no	—
106	Dic₁₄₂	—	4	4	no	—
107	Dic₁₄₂	—	8	2	no	—
108	C₈×C₇₁	—	2	1	no	—
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	8	2	no	—
111	C₈×C₇₁	#103	1	2	yes	—
112	C₈×C₇₁	#103	1	2	yes	—
113	C₇₁⋊C₈	—	4	2	no	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	#390	1	1	yes	characteristic
117	C₂²×C₇₁	#461	1	1	yes	characteristic
118	C₂²×C₇₁	#462	1	1	yes	characteristic
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
123	C₄×C₇₁	#419	1	1	yes	characteristic
124	C₄×C₇₁	#443	1	2	yes	—
125	C₄×C₇₁	#461	1	1	yes	characteristic
126	C₄×C₇₁	#462	1	1	yes	characteristic
127	Dic₇₁	—	4	4	no	—
128	Dic₇₁	—	8	2	no	—
129	Dic₇₁	—	8	2	no	—
130	#9749	—	71	1	no	maximal, Sylow
131	C₂×C₇₁	—	2	1	no	—
132	C₂×C₇₁	—	2	1	no	—
133	C₂×C₇₁	#1974	1	1	yes	characteristic
134	C₂×C₇₁	#1976	1	1	yes	characteristic
135	C₂×C₇₁	#1981	1	1	yes	characteristic
136	#3160	—	71	1	no	—
137	C₈×C₄⋊C₄	—	71	1	no	—
138	#1564	D₇₁	1	1	yes	characteristic
139	#1503	—	71	2	no	—
140	#1481	—	71	2	no	—
141	C₇₁	#9749	1	1	yes	characteristic, complemented, Sylow
142	#533	—	71	2	no	—
143	#500	—	71	1	no	—
144	#500	—	71	1	no	—
145	#499	—	142	1	no	—
146	#499	—	142	1	no	—
147	#498	—	71	2	no	—
148	#431	C₂×D₇₁	1	2	yes	—
149	#427	—	142	1	no	—
150	#427	—	142	1	no	—
151	C₂×C₄⋊C₈	—	71	1	no	—
152	C₂×C₄×C₈	—	142	1	no	—
153	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
154	C₄×C₄⋊C₄	—	71	1	no	—
155	#336	—	142	1	no	—
156	#335	—	2	1	no	—
157	#334	—	2	1	no	—
158	#328	—	142	4	no	—
159	Q₈⋊C₈	—	142	4	no	—
160	C₂²×C₈	—	142	1	no	—
161	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
162	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
163	C₄⋊Q₈	—	71	2	no	—
164	C₄⋊Q₈	—	71	2	no	—
165	#125	—	142	2	no	—
166	C₂²⋊Q₈	—	142	2	no	—
167	C₂²⋊Q₈	—	142	2	no	—
168	C₄×Q₈	—	142	2	no	—
169	C₄×Q₈	—	142	2	no	—
170	#117	—	142	2	no	—
171	C₂×C₄⋊C₄	—	142	1	no	—
172	C₂×C₄²	—	142	1	no	—
173	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
174	#107	—	142	1	no	—
175	#107	—	142	1	no	—
176	#107	—	142	1	no	—
177	#107	—	142	1	no	—
178	#106	—	142	2	no	—
179	#106	—	142	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	71	2	no	—
182	C₄⋊C₈	—	142	1	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	284	2	no	—
186	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
187	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
190	#95	—	71	2	no	—
191	C₂×Q₈	—	142	2	no	—
192	C₂×Q₈	—	142	2	no	—
193	C₂²×C₄	—	142	2	no	—
194	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	C₂²×C₄	#11765866	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	284	1	no	—
201	C₂×C₈	—	284	1	no	—
202	C₂×C₈	—	142	2	no	—
203	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	#11765854	1	2	yes	—
208	C₂×C₈	Dic₂₈₄	1	2	yes	—
209	C₄⋊C₄	—	284	2	no	—
210	C₄⋊C₄	—	284	2	no	—
211	C₄⋊C₄	—	142	4	no	—
212	C₄⋊C₄	—	142	2	no	—
213	C₄⋊C₄	—	142	2	no	—
214	C₄⋊C₄	—	142	2	no	—
215	C₄⋊C₄	—	142	2	no	—
216	C₂²⋊C₄	—	284	2	no	—
217	C₄²	—	284	1	no	—
218	C₄²	—	284	2	no	—
219	C₄²	—	284	1	no	—
220	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
221	C₄²	D₄×D₇₁	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
223	C₄²	#11765866	1	1	yes	characteristic
224	C₂³	#14556103	1	1	yes	characteristic
225	Q₈	—	284	4	no	—
226	Q₈	—	568	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	284	2	no	—
230	C₂×C₄	—	284	2	no	—
231	C₂×C₄	—	284	2	no	—
232	C₂×C₄	—	284	2	no	—
233	C₂×C₄	—	142	4	no	—
234	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#14556090	1	1	yes	characteristic
236	C₂×C₄	#14556090	1	1	yes	characteristic
237	C₂×C₄	#14556103	1	1	yes	characteristic
238	C₂×C₄	#14556117	1	1	yes	characteristic
239	C₂×C₄	#14556124	1	1	yes	characteristic
240	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
242	C₂×C₄	#14556161	1	1	yes	characteristic
243	C₂×C₄	#14556161	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	8	2	no	—
247	C₈	—	284	2	no	—
248	C₈	#14556053	1	2	yes	—
249	C₈	#14556053	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#17985900	1	1	yes	characteristic
253	C₂²	#17986205	1	1	yes	characteristic
254	C₂²	#17986210	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	284	4	no	—
259	C₄	—	568	2	no	—
260	C₄	—	568	2	no	—
261	C₄	C₂×#14556053	1	1	yes	characteristic
262	C₄	#17986009	1	1	yes	characteristic
263	C₄	#17986115	1	2	yes	—
264	C₄	#17986205	1	1	yes	characteristic
265	C₄	#17986210	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#22589226	1	1	yes	characteristic
269	C₂	#22589236	1	1	yes	characteristic
270	C₂	#22589262	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556103
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1564	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1564	D₇₁
hypocenter	C₇₁	#9749
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
7. #17986404	C₄	semidirect
8. #17986404	C₄	semidirect
11. #17986397	C₄	semidirect
141. C₇₁	#9749	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
C₈×C₄⋊Dic₇₁	2
#22586840	2
#22586744	2
C₇₁×#1564	2
#9749	71