Subgroups — #31915480

Order 18176.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985825	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22589071	C₂	1	2	yes	maximal
4	#22586431	C₂	1	2	yes	maximal
5	C₂×#366×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
9	#17985895	—	2	2	no	—
10	#17985825	C₄	1	4	yes	complemented
11	C₁₆×Dic₇₁	—	2	2	no	—
12	#17985761	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₁	C₂²	1	2	yes	—
15	#366×C₇₁	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×Dic₇₁⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
23	#14556062	—	2	2	no	—
24	#14556062	C₂×C₄	1	4	yes	—
25	C₈×Dic₇₁	—	2	1	no	—
26	C₈×Dic₇₁	—	2	1	no	—
27	C₇₁⋊#95	—	4	2	no	—
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
33	#107×C₇₁	—	2	2	no	—
34	#107×C₇₁	—	2	2	no	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
37	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	Dic₇₁⋊C₄	—	8	4	no	—
44	C₄×Dic₇₁	—	2	1	no	—
45	C₄×Dic₇₁	—	2	1	no	—
46	C₇₁⋊C₁₆	—	2	2	no	—
47	C₂²×C₄×C₇₁	—	4	2	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₄⋊C₄×C₇₁	—	4	2	no	—
57	C₁₆×C₇₁	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	8	2	no	—
60	C₂×C₄×C₇₁	—	8	2	no	—
61	C₂×C₄×C₇₁	—	8	2	no	—
62	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
68	C₂×Dic₇₁	—	2	1	no	—
69	C₂×Dic₇₁	—	2	1	no	—
70	C₂×Dic₇₁	—	2	1	no	—
71	C₂×Dic₇₁	—	2	1	no	—
72	C₂×Dic₇₁	—	4	4	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₂×Dic₇₁	—	8	2	no	—
75	C₂×Dic₇₁	D₁₆	1	2	yes	—
76	C₂×Dic₇₁	Q₃₂	1	2	yes	—
77	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₇₁	#125	1	1	yes	characteristic
81	C₇₁⋊C₈	—	2	2	no	—
82	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₇₁	#466	1	1	yes	characteristic
84	C₂²×C₇₁	#500	1	1	yes	characteristic
85	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₇₁	#508	1	1	yes	characteristic
88	C₂²×C₇₁	#508	1	1	yes	characteristic
89	C₄×C₇₁	—	16	2	no	—
90	C₄×C₇₁	#389	1	1	yes	characteristic
91	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
92	C₄×C₇₁	#467	1	1	yes	characteristic
93	C₄×C₇₁	#499	1	1	yes	characteristic
94	Dic₇₁	—	2	2	no	—
95	Dic₇₁	—	2	2	no	—
96	Dic₇₁	—	8	4	no	—
97	#13165	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	#1942	1	1	yes	characteristic
99	C₂×C₇₁	C₄×D₁₆	1	1	yes	characteristic
100	C₂×C₇₁	C₄×Q₃₂	1	1	yes	characteristic
101	C₂×C₇₁	#2215	1	1	yes	characteristic
102	C₂×C₇₁	#2216	1	1	yes	characteristic
103	C₂×C₇₁	#2252	1	1	yes	characteristic
104	C₂×C₇₁	#2253	1	1	yes	characteristic
105	C₂×#366	—	71	1	no	—
106	C₂×#366	D₇₁	1	1	yes	characteristic
107	C₂×C₄×C₁₆	—	71	1	no	—
108	#1942	—	71	2	no	—
109	#1380	—	71	2	no	—
110	C₇₁	#13165	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₁₆	—	71	1	no	—
112	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
113	C₂×#107	—	71	2	no	—
114	C₂×#107	C₂×D₇₁	1	2	yes	—
115	C₂×C₄×C₈	—	71	1	no	—
116	#389	—	142	2	no	—
117	#366	—	2	2	no	—
118	#366	—	71	4	no	—
119	C₄×C₁₆	—	142	2	no	—
120	#340	—	142	2	no	—
121	C₂²×C₈	—	71	1	no	—
122	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	284	2	no	—
125	C₂×C₄⋊C₄	—	142	2	no	—
126	C₂×C₄²	—	71	1	no	—
127	C₂×C₁₆	—	71	2	no	—
128	C₂×C₁₆	—	142	1	no	—
129	C₂×C₁₆	—	142	1	no	—
130	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
131	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	71	4	no	—
136	#107	—	142	2	no	—
137	C₄×C₈	—	142	1	no	—
138	C₄×C₈	—	142	1	no	—
139	#95	—	284	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	284	2	no	—
142	C₂²×C₄	—	71	1	no	—
143	C₂²×C₄	—	71	1	no	—
144	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
145	C₂×C₈	—	71	2	no	—
146	C₂×C₈	—	142	1	no	—
147	C₂×C₈	—	142	1	no	—
148	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
152	C₂×C₈	#11765864	1	1	yes	characteristic
153	C₂×C₈	#11765866	1	1	yes	characteristic
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	284	2	no	—
157	C₄⋊C₄	—	568	4	no	—
158	C₄⋊C₄	—	142	4	no	—
159	C₄²	—	142	1	no	—
160	C₄²	—	142	1	no	—
161	C₁₆	—	142	2	no	—
162	C₁₆	Dic₇₁⋊C₄	1	4	yes	complemented
163	C₂³	D₈×D₇₁	1	1	yes	characteristic
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	284	4	no	—
168	C₂×C₄	—	568	2	no	—
169	C₂×C₄	—	568	2	no	—
170	C₂×C₄	—	71	2	no	—
171	C₂×C₄	—	71	2	no	—
172	C₂×C₄	—	142	1	no	—
173	C₂×C₄	—	142	1	no	—
174	C₂×C₄	—	142	1	no	—
175	C₂×C₄	—	142	1	no	—
176	C₂×C₄	#14556090	1	1	yes	characteristic
177	C₂×C₄	#14556099	1	1	yes	characteristic
178	C₂×C₄	#14556127	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
180	C₂×C₄	#14556147	1	1	yes	characteristic
181	C₂×C₄	#14556151	1	1	yes	characteristic
182	C₈	—	142	2	no	—
183	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
184	C₈	#14556093	1	1	yes	characteristic
185	C₈	#14556104	1	1	yes	characteristic
186	C₈	#14556119	1	1	yes	characteristic
187	C₂²	#17986081	1	1	yes	characteristic
188	C₂²	#17986232	1	1	yes	characteristic
189	C₂²	#17986403	1	1	yes	characteristic
190	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
191	C₂²	Q₃₂×D₇₁	1	1	yes	characteristic
192	C₂²	#17986428	1	1	yes	characteristic
193	C₂²	#17986432	1	1	yes	characteristic
194	C₄	—	16	2	no	—
195	C₄	—	568	4	no	—
196	C₄	—	142	2	no	—
197	C₄	—	142	2	no	—
198	C₄	#17985895	1	1	yes	characteristic
199	C₄	#17986093	1	1	yes	characteristic
200	C₄	#17986237	1	1	yes	characteristic
201	C₄	#17986400	1	1	yes	characteristic
202	C₂	#22589071	1	1	yes	characteristic
203	C₂	#22590186	1	1	yes	characteristic
204	C₂	#22590198	1	1	yes	characteristic
205	C₂	#22590423	1	1	yes	characteristic
206	C₂	#22590428	1	1	yes	characteristic
207	C₂	#22590614	1	1	yes	characteristic
208	C₂	#22590617	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#366×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#366	D₇₁
hypocenter	C₇₁	#13165
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
10. #17985825	C₄	semidirect
57. C₁₆×C₇₁	C₄⋊C₄	semidirect
110. C₇₁	#13165	semidirect
162. C₁₆	Dic₇₁⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985825	2
C₂×C₁₆×Dic₇₁	2
#22589071	2
#22586431	2
C₂×#366×C₇₁	2
#13165	71