Subgroups — #31891211

Order 18176.

362 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588456	C₂	1	2	yes	maximal
2	#22588085	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985749	C₂	1	2	yes	maximal
5	C₂×#335×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	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
9	#17985753	—	2	4	no	—
10	#17985753	C₄	1	2	yes	—
11	#17985753	C₄	1	2	yes	—
12	#17985752	—	2	2	no	—
13	#17985749	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#335×C₇₁	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₁	D₄	1	2	yes	—
21	C₂×C₄×Dic₇₁	Q₈	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₁	—	2	2	no	—
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	D₄	1	2	yes	—
33	C₄×C₈×C₇₁	Q₈	1	2	yes	—
34	C₂²×Dic₇₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	4	4	no	—
39	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
40	C₄×Dic₇₁	—	2	2	no	—
41	C₄×Dic₇₁	—	2	2	no	—
42	C₄×Dic₇₁	—	2	2	no	—
43	C₄×Dic₇₁	—	2	2	no	—
44	C₄×Dic₇₁	—	4	2	no	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	4	4	no	—
52	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#131	1	2	yes	—
67	C₂×C₄×C₇₁	#131	1	2	yes	—
68	C₂×C₄×C₇₁	#135	1	2	yes	—
69	C₂×C₄×C₇₁	#135	1	2	yes	—
70	C₂×Dic₇₁	—	2	4	no	—
71	C₂×Dic₇₁	—	2	4	no	—
72	C₂×Dic₇₁	—	2	4	no	—
73	C₈×C₇₁	—	2	2	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₈×C₇₁	—	8	2	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	2	yes	—
80	C₂²×C₇₁	#427	1	2	yes	—
81	C₂²×C₇₁	#430	1	2	yes	—
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	#424	1	2	yes	—
85	C₄×C₇₁	#424	1	1	yes	characteristic
86	C₄×C₇₁	#424	1	1	yes	characteristic
87	C₄×C₇₁	#495	1	1	yes	characteristic
88	C₄×C₇₁	#495	1	1	yes	characteristic
89	C₄×C₇₁	#499	1	2	yes	—
90	Dic₇₁	—	4	4	no	—
91	#8811	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	#1572	1	4	yes	—
93	C₂×C₇₁	#1843	1	1	yes	characteristic
94	C₂×C₇₁	#1846	1	1	yes	characteristic
95	C₂×C₇₁	#1849	1	1	yes	characteristic
96	#1831	—	71	2	no	—
97	#1744	—	71	1	no	—
98	C₂×#335	—	71	1	no	—
99	C₂×#335	D₇₁	1	1	yes	characteristic
100	C₂×#334	—	71	2	no	—
101	C₇₁	#8811	1	1	yes	characteristic, complemented, Sylow
102	C₂×C₄⋊C₈	—	71	2	no	—
103	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
104	C₂×C₄×C₈	—	71	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₄³	—	71	1	no	—
107	#336	—	71	2	no	—
108	#336	—	71	2	no	—
109	#336	—	142	4	no	—
110	#335	—	2	2	no	—
111	#335	—	142	2	no	—
112	#334	—	142	4	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	71	2	no	—
115	C₂²×C₈	—	142	2	no	—
116	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
117	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₄²	—	71	2	no	—
119	C₂×C₄²	—	71	2	no	—
120	C₂×C₄²	—	71	2	no	—
121	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	142	2	no	—
125	C₄⋊C₈	—	142	2	no	—
126	C₄×C₈	—	71	2	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	Dic₁₄₂	1	2	yes	—
129	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
130	C₂²×C₄	—	71	4	no	—
131	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
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	—	142	2	no	—
142	C₂×C₈	—	142	2	no	—
143	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
144	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
145	C₄²	—	284	2	no	—
146	C₄²	—	142	2	no	—
147	C₄²	—	142	2	no	—
148	C₄²	—	142	2	no	—
149	C₄²	—	142	2	no	—
150	C₄²	D₄×D₇₁	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₄²	Q₈×D₇₁	1	1	yes	characteristic
153	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
154	C₂³	C₄⋊C₄×D₇₁	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	142	4	no	—
157	C₂×C₄	—	142	4	no	—
158	C₂×C₄	—	142	4	no	—
159	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
161	C₂×C₄	#14556124	1	1	yes	characteristic
162	C₂×C₄	#14556126	1	1	yes	characteristic
163	C₂×C₄	#14556127	1	1	yes	characteristic
164	C₂×C₄	#14556127	1	1	yes	characteristic
165	C₂×C₄	#14556134	1	2	yes	—
166	C₂×C₄	#14556136	1	2	yes	—
167	C₂×C₄	#14556147	1	2	yes	—
168	C₂×C₄	#14556151	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	8	2	no	—
172	C₈	—	568	2	no	—
173	C₈	—	142	4	no	—
174	C₂²	#17985868	1	1	yes	characteristic
175	C₂²	#17986024	1	2	yes	—
176	C₂²	#17986036	1	2	yes	—
177	C₂²	#17986046	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	284	4	no	—
181	C₄	#17986024	1	2	yes	—
182	C₄	#17986025	1	1	yes	characteristic
183	C₄	#17986027	1	1	yes	characteristic
184	C₄	#17986382	1	1	yes	characteristic
185	C₄	#17986384	1	1	yes	characteristic
186	C₄	#17986400	1	2	yes	—
187	C₂	#22587119	1	4	yes	—
188	C₂	#22588493	1	1	yes	characteristic
189	C₂	#22588503	1	1	yes	characteristic
190	C₂	#22588513	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#8811
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

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

Maximal subgroups

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

subgroup	index
#22588456	2
#22588085	2
C₂×#17985752	2
C₂×#17985749	2
C₂×#335×C₇₁	2
#8811	71