Subgroups — #31897464

Order 18176.

494 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	#22588457	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22586181	C₂	1	4	yes	maximal
4	C₄×#125×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄×C₇₁⋊C₈	—	2	2	no	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	#17985754	—	2	2	no	—
9	C₂×#125×C₇₁	C₄	1	1	yes	characteristic
10	#389×C₇₁	C₄	1	2	yes	—
11	#387×C₇₁	—	2	2	no	—
12	#382×C₇₁	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
19	C₄×C₇₁⋊C₈	—	2	8	no	—
20	C₄×C₇₁⋊C₈	—	2	2	no	—
21	C₄×C₇₁⋊C₈	Q₈	1	4	yes	—
22	#125×C₇₁	—	2	2	no	—
23	#125×C₇₁	C₈	1	4	yes	complemented
24	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₁	—	2	2	no	—
29	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	2	yes	—
32	C₂×C₄²×C₇₁	D₄	1	2	yes	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	#95×C₇₁	—	2	2	no	—
35	#95×C₇₁	—	2	2	no	—
36	#95×C₇₁	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	8	no	—
38	C₂×C₇₁⋊C₈	—	2	4	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	—	4	4	no	—
41	C₂²×C₄×C₇₁	—	2	2	no	—
42	C₂²×C₄×C₇₁	—	2	2	no	—
43	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
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×D₄	1	1	yes	characteristic
48	C₄⋊C₄×C₇₁	—	2	8	no	—
49	C₄⋊C₄×C₇₁	—	2	2	no	—
50	C₄⋊C₄×C₇₁	—	2	4	no	—
51	C₄⋊C₄×C₇₁	—	4	4	no	—
52	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
53	C₄²×C₇₁	—	2	4	no	—
54	C₄²×C₇₁	—	2	2	no	—
55	C₄²×C₇₁	—	2	4	no	—
56	C₄²×C₇₁	—	4	4	no	—
57	C₄²×C₇₁	C₂×C₈	1	2	yes	—
58	C₄²×C₇₁	M₁₆	1	2	yes	—
59	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
60	C₄²×C₇₁	#41	1	2	yes	—
61	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	4	no	—
68	C₂×C₄×C₇₁	—	4	2	no	—
69	C₂×C₄×C₇₁	—	4	2	no	—
70	C₂×C₄×C₇₁	—	4	2	no	—
71	C₂×C₄×C₇₁	—	4	2	no	—
72	C₂×C₄×C₇₁	—	4	2	no	—
73	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₇₁	#104	1	4	yes	—
75	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#131	1	2	yes	—
82	C₂×C₄×C₇₁	#135	1	4	yes	—
83	C₇₁⋊C₈	—	2	8	no	—
84	C₇₁⋊C₈	—	8	2	no	—
85	C₂²×C₇₁	#391	1	1	yes	characteristic
86	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₇₁	#408	1	1	yes	characteristic
88	C₂²×C₇₁	#416	1	2	yes	—
89	C₂²×C₇₁	C₂×#104	1	2	yes	—
90	C₄×C₇₁	—	2	4	no	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	2	4	no	—
93	C₄×C₇₁	—	4	4	no	—
94	C₄×C₇₁	—	8	2	no	—
95	C₄×C₇₁	C₈×Q₈	1	2	yes	—
96	C₄×C₇₁	#446	1	2	yes	—
97	C₄×C₇₁	#488	1	2	yes	—
98	C₄×C₇₁	#499	1	2	yes	—
99	#9871	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	#1476	1	2	yes	—
101	C₂×C₇₁	#1494	1	2	yes	—
102	C₂×C₇₁	#1984	1	1	yes	characteristic
103	C₂×C₇₁	#1991	1	1	yes	characteristic
104	C₂×C₇₁	#1994	1	1	yes	characteristic
105	C₄×#125	D₇₁	1	1	yes	characteristic
106	#1831	—	71	1	no	—
107	C₄²×C₈	—	71	1	no	—
108	#1279	—	71	4	no	—
109	C₇₁	#9871	1	1	yes	characteristic, complemented, Sylow
110	C₂×#125	Dic₇₁	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	71	2	no	—
112	C₂×C₄×C₈	—	71	2	no	—
113	C₂×C₄×C₈	—	142	2	no	—
114	#389	Dic₇₁	1	2	yes	—
115	#387	—	2	2	no	—
116	#382	Dic₇₁	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	2	2	no	—
118	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
119	C₄³	C₂×D₇₁	1	1	yes	characteristic
120	#336	—	142	2	no	—
121	C₂²×C₈	—	142	2	no	—
122	C₂²×C₈	—	142	2	no	—
123	#125	—	2	2	no	—
124	#125	C₇₁⋊C₈	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
134	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
135	C₄⋊C₈	—	142	4	no	—
136	C₄×C₈	—	71	4	no	—
137	C₄×C₈	—	142	8	no	—
138	C₄×C₈	—	142	2	no	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
149	C₂×C₈	—	284	2	no	—
150	C₂×C₈	—	284	4	no	—
151	C₂×C₈	—	142	8	no	—
152	C₂×C₈	—	142	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
163	C₄²	#11765849	1	2	yes	—
164	C₄²	Q₈×D₇₁	1	2	yes	—
165	C₄²	#11765866	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	4	yes	—
179	C₂×C₄	#14556055	1	4	yes	—
180	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
181	C₂×C₄	#14556104	1	2	yes	—
182	C₂×C₄	#14556117	1	1	yes	characteristic
183	C₂×C₄	#14556126	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
186	C₂×C₄	#14556135	1	2	yes	—
187	C₂×C₄	#14556148	1	4	yes	—
188	C₈	—	568	2	no	—
189	C₈	—	142	8	no	—
190	C₂²	#17985906	1	1	yes	characteristic
191	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
192	C₂²	#17985966	1	1	yes	characteristic
193	C₂²	#17985995	1	2	yes	—
194	C₂²	C₂×#14556055	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₄	Q₈×C₇₁⋊C₈	1	2	yes	—
201	C₄	#17986130	1	2	yes	—
202	C₄	#17986356	1	2	yes	—
203	C₄	#17986398	1	2	yes	—
204	C₂	#22586722	1	2	yes	—
205	C₂	#22586800	1	2	yes	—
206	C₂	#22589277	1	1	yes	characteristic
207	C₂	#22589306	1	1	yes	characteristic
208	C₂	#22589318	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#125×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#125	D₇₁
hypocenter	C₇₁	#9871
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
23. #125×C₇₁	C₈	semidirect
109. C₇₁	#9871	semidirect
124. #125	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588457	2
C₄²×C₇₁⋊C₈	2
#22586181	2
C₄×#125×C₇₁	2
#9871	71