Subgroups — #31901700

Order 18176.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589289	C₂	1	2	yes	maximal
2	#22589058	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985748	C₂	1	2	yes	maximal
4	#22586175	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×Dic₇₁⋊C₈	—	2	1	no	—
9	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985761	—	2	2	no	—
11	#17985761	—	2	2	no	—
12	#17985753	—	2	1	no	—
13	#17985748	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	#500×C₇₁	—	2	4	no	—
16	C₂×#107×C₇₁	—	2	1	no	—
17	C₂×#107×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	—	2	1	no	—
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	—	2	1	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	Dic₇₁⋊C₈	—	4	2	no	—
28	C₇₁⋊#95	—	2	1	no	—
29	C₂²×Q₈×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₇₁	—	2	2	no	—
33	C₄⋊Q₈×C₇₁	—	2	2	no	—
34	C₄⋊Q₈×C₇₁	—	2	2	no	—
35	C₄⋊Q₈×C₇₁	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	#107×C₇₁	—	2	4	no	—
40	#107×C₇₁	—	2	4	no	—
41	C₄×C₈×C₇₁	—	2	2	no	—
42	C₂²×Dic₇₁	—	2	1	no	—
43	C₂²×Dic₇₁	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	4	1	no	—
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₂×C₇₁⋊C₈	—	4	1	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	1	no	—
50	C₄⋊Dic₇₁	—	2	1	no	—
51	C₄⋊Dic₇₁	—	2	2	no	—
52	C₄⋊Dic₇₁	Q₁₆	1	2	yes	—
53	C₄×Dic₇₁	—	4	1	no	—
54	C₄×Dic₇₁	—	4	1	no	—
55	C₂×Q₈×C₇₁	—	2	4	no	—
56	C₂×Q₈×C₇₁	—	4	2	no	—
57	C₂×Q₈×C₇₁	—	8	2	no	—
58	C₂×Q₈×C₇₁	—	8	2	no	—
59	C₂²×C₄×C₇₁	—	4	2	no	—
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
68	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₇₁	—	2	4	no	—
70	C₄⋊C₄×C₇₁	—	2	4	no	—
71	C₄⋊C₄×C₇₁	—	4	2	no	—
72	C₄⋊C₄×C₇₁	—	4	2	no	—
73	C₄²×C₇₁	C₂×D₄	1	2	yes	—
74	C₄²×C₇₁	#41	1	2	yes	—
75	C₂³×C₇₁	#121	1	1	yes	characteristic
76	Q₈×C₇₁	—	8	4	no	—
77	Q₈×C₇₁	—	8	4	no	—
78	C₂×C₄×C₇₁	—	2	1	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	4	4	no	—
81	C₂×C₄×C₇₁	—	8	2	no	—
82	C₂×C₄×C₇₁	—	8	2	no	—
83	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#121	1	2	yes	—
85	C₂×C₄×C₇₁	#123	1	2	yes	—
86	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#137	1	2	yes	—
92	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
94	C₂×Dic₇₁	—	2	2	no	—
95	C₂×Dic₇₁	—	4	1	no	—
96	C₂×Dic₇₁	—	4	1	no	—
97	C₂×Dic₇₁	—	4	1	no	—
98	C₂×Dic₇₁	—	4	1	no	—
99	C₂×Dic₇₁	—	4	1	no	—
100	C₈×C₇₁	—	2	2	no	—
101	C₈×C₇₁	—	2	2	no	—
102	C₇₁⋊C₈	—	8	1	no	—
103	C₇₁⋊C₈	—	8	2	no	—
104	C₂²×C₇₁	#394	1	1	yes	characteristic
105	C₂²×C₇₁	#461	1	1	yes	characteristic
106	C₂²×C₇₁	#462	1	1	yes	characteristic
107	C₂²×C₇₁	#467	1	1	yes	characteristic
108	C₂²×C₇₁	#470	1	1	yes	characteristic
109	C₂²×C₇₁	#472	1	1	yes	characteristic
110	C₂²×C₇₁	#472	1	1	yes	characteristic
111	C₄×C₇₁	—	2	2	no	—
112	C₄×C₇₁	—	8	4	no	—
113	C₄×C₇₁	#448	1	1	yes	characteristic
114	C₄×C₇₁	#451	1	1	yes	characteristic
115	C₄×C₇₁	#462	1	2	yes	—
116	C₄×C₇₁	#464	1	2	yes	—
117	C₄×C₇₁	#487	1	1	yes	characteristic
118	C₄×C₇₁	#491	1	1	yes	characteristic
119	Dic₇₁	—	4	2	no	—
120	Dic₇₁	—	8	1	no	—
121	#10543	—	71	1	no	maximal, Sylow
122	C₂×C₇₁	#1675	1	1	yes	characteristic
123	C₂×C₇₁	#1676	1	1	yes	characteristic
124	C₂×C₇₁	#1695	1	1	yes	characteristic
125	C₂×C₇₁	#1696	1	1	yes	characteristic
126	C₂×C₇₁	#2043	1	1	yes	characteristic
127	C₂×C₇₁	#2045	1	1	yes	characteristic
128	C₂×C₇₁	#2056	1	1	yes	characteristic
129	C₂×#500	D₇₁	1	1	yes	characteristic
130	#1987	—	71	2	no	—
131	#1939	—	71	1	no	—
132	C₂×#333	—	71	2	no	—
133	#1277	—	71	1	no	—
134	C₇₁	#10543	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
136	#500	—	2	4	no	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	—	2	1	no	—
139	C₂×C₄⋊C₈	—	71	1	no	—
140	C₂×C₄⋊C₈	—	71	2	no	—
141	C₂×C₄⋊C₈	—	142	1	no	—
142	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	71	1	no	—
144	#340	—	142	2	no	—
145	#340	—	142	2	no	—
146	#336	—	142	1	no	—
147	#333	—	142	4	no	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×C₈	—	142	2	no	—
150	C₂²×C₈	—	142	1	no	—
151	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	C₂²⋊D₇₁	1	4	yes	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	71	1	no	—
160	C₂×C₄⋊C₄	—	71	1	no	—
161	C₂×C₄²	—	142	1	no	—
162	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
163	#107	—	2	4	no	—
164	#107	—	2	4	no	—
165	C₄⋊C₈	—	284	2	no	—
166	C₄⋊C₈	—	142	2	no	—
167	C₄⋊C₈	—	142	2	no	—
168	C₄⋊C₈	—	142	2	no	—
169	C₄×C₈	—	2	2	no	—
170	#95	—	142	1	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	8	2	no	—
174	C₂×Q₈	—	8	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	142	1	no	—
177	C₂²×C₄	—	142	1	no	—
178	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
180	C₂²×C₄	#11765864	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	284	2	no	—
186	C₂×C₈	—	284	1	no	—
187	C₂×C₈	—	284	2	no	—
188	C₂×C₈	—	284	1	no	—
189	C₂×C₈	—	284	2	no	—
190	C₂×C₈	—	284	1	no	—
191	C₂×C₈	#11765852	1	2	yes	—
192	C₂×C₈	#11765855	1	2	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	71	2	no	—
198	C₄⋊C₄	—	142	2	no	—
199	C₄⋊C₄	—	142	1	no	—
200	C₄²	—	284	1	no	—
201	C₄²	—	284	1	no	—
202	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
203	C₄²	#11765864	1	2	yes	—
204	C₂³	#14556099	1	1	yes	characteristic
205	Q₈	—	8	4	no	—
206	Q₈	—	8	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	—	284	1	no	—
213	C₂×C₄	—	284	1	no	—
214	C₂×C₄	—	284	1	no	—
215	C₂×C₄	—	284	1	no	—
216	C₂×C₄	—	284	1	no	—
217	C₂×C₄	—	142	2	no	—
218	C₂×C₄	#14556097	1	1	yes	characteristic
219	C₂×C₄	#14556100	1	2	yes	—
220	C₂×C₄	#14556109	1	2	yes	—
221	C₂×C₄	#14556115	1	1	yes	characteristic
222	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
223	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
224	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
225	C₂×C₄	#14556151	1	1	yes	characteristic
226	C₂×C₄	#14556159	1	1	yes	characteristic
227	C₂×C₄	#14556159	1	1	yes	characteristic
228	C₂×C₄	#14556160	1	2	yes	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	568	1	no	—
232	C₈	—	568	2	no	—
233	C₂²	#17985913	1	1	yes	characteristic
234	C₂²	#17986201	1	1	yes	characteristic
235	C₂²	#17986206	1	1	yes	characteristic
236	C₂²	#17986237	1	1	yes	characteristic
237	C₂²	#17986254	1	1	yes	characteristic
238	C₂²	#17986266	1	1	yes	characteristic
239	C₂²	#17986266	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	8	4	no	—
242	C₄	—	284	2	no	—
243	C₄	—	568	1	no	—
244	C₄	#17986138	1	1	yes	characteristic
245	C₄	#17986155	1	1	yes	characteristic
246	C₄	#17986207	1	2	yes	—
247	C₄	#17986223	1	2	yes	—
248	C₄	#17986349	1	1	yes	characteristic
249	C₄	#17986367	1	1	yes	characteristic
250	C₂	#22587736	1	1	yes	characteristic
251	C₂	#22587741	1	1	yes	characteristic
252	C₂	#22587860	1	1	yes	characteristic
253	C₂	#22587865	1	1	yes	characteristic
254	C₂	#22589583	1	1	yes	characteristic
255	C₂	#22589593	1	1	yes	characteristic
256	C₂	#22589660	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556099
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#121
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#10543
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
134. C₇₁	#10543	semidirect

Maximal subgroups

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

subgroup	index
#22589289	2
#22589058	2
C₂×#17985748	2
#22586175	2
C₂×#500×C₇₁	2
#10543	71