Subgroups — #31891210

Order 18176.

362 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588457	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985751	C₂	1	2	yes	maximal
4	C₂×#17985749	C₂	1	2	yes	maximal
5	C₇₁×#1831	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×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985754	—	2	2	no	—
10	#17985754	C₄	1	2	yes	—
11	#17985754	C₄	1	2	yes	—
12	#17985751	—	2	4	no	—
13	#17985749	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	#336×C₇₁	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
24	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
25	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₇₁	—	2	2	no	—
27	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	D₄	1	2	yes	—
30	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₁	—	2	2	no	—
33	C₄⋊C₈×C₇₁	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₂×C₇₁⋊C₈	—	4	4	no	—
40	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	—	4	4	no	—
49	C₄²×C₇₁	—	2	2	no	—
50	C₄²×C₇₁	—	2	2	no	—
51	C₄²×C₇₁	—	2	4	no	—
52	C₄²×C₇₁	—	4	2	no	—
53	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#131	1	2	yes	—
72	C₂×C₄×C₇₁	#131	1	2	yes	—
73	C₂×C₄×C₇₁	#135	1	4	yes	—
74	C₈×C₇₁	—	8	2	no	—
75	C₇₁⋊C₈	—	2	4	no	—
76	C₇₁⋊C₈	—	2	4	no	—
77	C₇₁⋊C₈	—	8	2	no	—
78	C₂²×C₇₁	#384	1	1	yes	characteristic
79	C₂²×C₇₁	#424	1	1	yes	characteristic
80	C₂²×C₇₁	#424	1	1	yes	characteristic
81	C₂²×C₇₁	#427	1	2	yes	—
82	C₂²×C₇₁	#430	1	2	yes	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	—	4	2	no	—
86	C₄×C₇₁	—	4	2	no	—
87	C₄×C₇₁	#424	1	2	yes	—
88	C₄×C₇₁	#424	1	2	yes	—
89	C₄×C₇₁	#495	1	2	yes	—
90	C₄×C₇₁	#499	1	2	yes	—
91	#8811	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	#1572	1	2	yes	—
93	C₂×C₇₁	#1572	1	2	yes	—
94	C₂×C₇₁	#1843	1	1	yes	characteristic
95	C₂×C₇₁	#1846	1	1	yes	characteristic
96	C₂×C₇₁	#1849	1	1	yes	characteristic
97	#1831	—	71	1	no	—
98	#1831	D₇₁	1	1	yes	characteristic
99	#1744	—	71	1	no	—
100	C₂×#335	—	71	2	no	—
101	C₂×#334	—	71	2	no	—
102	C₇₁	#8811	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	71	2	no	—
104	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
105	C₂×C₄×C₈	—	71	1	no	—
106	C₂×C₄×C₈	—	71	1	no	—
107	C₄³	C₂×D₇₁	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	#336	—	71	2	no	—
110	#336	—	71	2	no	—
111	#336	—	142	2	no	—
112	#335	—	142	4	no	—
113	#334	—	142	4	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	71	2	no	—
116	C₂²×C₈	—	71	2	no	—
117	C₂²×C₈	—	142	2	no	—
118	C₂×C₄²	Dic₁₄₂	1	2	yes	—
119	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
122	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	142	2	no	—
126	C₄⋊C₈	—	142	2	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	—	71	2	no	—
129	C₄×C₈	—	71	2	no	—
130	C₄×C₈	—	71	2	no	—
131	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
132	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	284	2	no	—
139	C₂×C₈	—	284	4	no	—
140	C₂×C₈	—	71	4	no	—
141	C₂×C₈	—	71	4	no	—
142	C₂×C₈	—	142	2	no	—
143	C₂×C₈	—	142	2	no	—
144	C₂×C₈	—	142	2	no	—
145	C₂×C₈	—	142	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₄×D₇₁	1	1	yes	characteristic
151	C₄²	D₄×D₇₁	1	1	yes	characteristic
152	C₄²	Q₈×D₇₁	1	1	yes	characteristic
153	C₄²	Q₈×D₇₁	1	1	yes	characteristic
154	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
163	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
164	C₂×C₄	#14556124	1	1	yes	characteristic
165	C₂×C₄	#14556126	1	1	yes	characteristic
166	C₂×C₄	#14556127	1	1	yes	characteristic
167	C₂×C₄	#14556127	1	1	yes	characteristic
168	C₂×C₄	#14556134	1	2	yes	—
169	C₂×C₄	#14556136	1	2	yes	—
170	C₂×C₄	#14556148	1	4	yes	—
171	C₈	—	8	2	no	—
172	C₈	—	568	2	no	—
173	C₈	—	142	4	no	—
174	C₈	—	142	4	no	—
175	C₂²	#17985868	1	1	yes	characteristic
176	C₂²	#17986025	1	1	yes	characteristic
177	C₂²	#17986027	1	1	yes	characteristic
178	C₂²	#17986035	1	2	yes	—
179	C₂²	#17986047	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#17986024	1	2	yes	—
185	C₄	#17986024	1	2	yes	—
186	C₄	#17986381	1	2	yes	—
187	C₄	#17986398	1	2	yes	—
188	C₂	#22587118	1	2	yes	—
189	C₂	#22587121	1	2	yes	—
190	C₂	#22588493	1	1	yes	characteristic
191	C₂	#22588502	1	1	yes	characteristic
192	C₂	#22588514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1831	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1831	D₇₁
hypocenter	C₇₁	#8811
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
102. C₇₁	#8811	semidirect

Maximal subgroups

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

subgroup	index
#22588457	2
#22588086	2
C₂×#17985751	2
C₂×#17985749	2
C₇₁×#1831	2
#8811	71