Subgroups — #364×Dic₇₁ #31912772

Order 18176.

398 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985820	C₂	1	1	yes	maximal, characteristic
2	M₃₂×Dic₇₁	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#22586421	C₂	1	2	yes	maximal
5	C₂×#364×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
13	#17985820	C₄	1	4	yes	complemented
14	#17985774	—	2	2	no	—
15	C₁₆×Dic₇₁	—	2	2	no	—
16	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#364×C₇₁	C₄	1	4	yes	complemented
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	—	2	2	no	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	#14556068	—	2	2	no	—
25	#14556068	C₂×C₄	1	4	yes	—
26	C₂×C₇₁⋊C₁₆	—	2	2	no	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₈×Dic₇₁	—	2	1	no	—
29	C₈×Dic₇₁	—	2	1	no	—
30	C₈×Dic₇₁	C₈	1	4	yes	—
31	C₈×Dic₇₁	C₈	1	4	yes	—
32	C₈×Dic₇₁	D₄	1	1	yes	characteristic
33	C₈×Dic₇₁	D₄	1	1	yes	characteristic
34	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
35	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₇₁	—	2	2	no	—
40	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₇₁	—	2	2	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₇₁	—	2	1	no	—
45	C₂²×Dic₇₁	M₁₆	1	2	yes	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	2	1	no	—
48	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
51	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
53	C₄×Dic₇₁	—	2	8	no	—
54	C₄×Dic₇₁	—	2	1	no	—
55	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
56	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
57	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
58	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
59	C₇₁⋊C₁₆	—	2	4	no	—
60	C₂²×C₄×C₇₁	—	2	1	no	—
61	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₄²	1	2	yes	—
63	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
70	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
71	C₄²×C₇₁	—	2	1	no	—
72	C₄²×C₇₁	C₄²	1	2	yes	—
73	C₁₆×C₇₁	—	2	4	no	—
74	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₁	—	2	4	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₇₁	—	2	4	no	—
84	C₂×Dic₇₁	—	2	2	no	—
85	C₂×Dic₇₁	—	2	4	no	—
86	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
88	C₈×C₇₁	C₄×C₈	1	4	yes	—
89	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
90	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
91	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
92	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
93	C₇₁⋊C₈	—	2	2	no	—
94	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
95	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
96	C₂²×C₇₁	—	2	2	no	—
97	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
98	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₇₁	#431	1	1	yes	characteristic
100	C₄×C₇₁	—	2	4	no	—
101	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₇₁	#431	1	1	yes	characteristic
105	Dic₇₁	—	2	4	no	—
106	Dic₇₁	—	2	2	no	—
107	Dic₇₁	#364	1	4	yes	direct product factor
108	C₄×#364	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	—	2	1	no	—
111	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₇₁	C₂×#364	1	2	yes	—
113	C₂×#364	—	71	1	no	—
114	C₂×#364	D₇₁	1	1	yes	characteristic
115	C₄×M₃₂	—	71	2	no	—
116	C₄²×C₈	—	71	1	no	—
117	#1376	—	71	2	no	—
118	C₇₁	C₄×#364	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₃₂	—	71	2	no	—
120	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
121	C₂×C₄×C₈	—	71	1	no	—
122	C₂×C₄×C₈	—	71	1	no	—
123	C₂×C₄×C₈	—	71	1	no	—
124	C₂×C₄×C₈	—	71	1	no	—
125	C₂×C₄×C₈	—	71	1	no	—
126	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₄³	—	71	1	no	—
128	#364	—	71	4	no	—
129	#364	Dic₇₁	1	4	yes	direct product factor
130	#346	—	142	2	no	—
131	C₄×C₁₆	—	142	2	no	—
132	C₂²×C₈	—	71	1	no	—
133	C₂²×C₈	—	71	1	no	—
134	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₂×C₄²	—	71	1	no	—
137	C₂×C₄²	—	71	1	no	—
138	C₂×C₄²	—	142	2	no	—
139	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
140	M₃₂	—	2	2	no	—
141	M₃₂	—	71	4	no	—
142	M₃₂	—	142	2	no	—
143	M₃₂	C₂×Dic₇₁	1	4	yes	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	142	2	no	—
146	C₄×C₈	—	71	4	no	—
147	C₄×C₈	—	71	4	no	—
148	C₄×C₈	—	71	1	no	—
149	C₄×C₈	—	71	1	no	—
150	C₄×C₈	—	71	1	no	—
151	C₄×C₈	—	71	1	no	—
152	C₄×C₈	—	142	2	no	—
153	C₄×C₈	—	142	1	no	—
154	C₄×C₈	—	142	1	no	—
155	C₄×C₈	C₄×D₇₁	1	2	yes	—
156	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	71	2	no	—
159	C₂²×C₄	—	142	1	no	—
160	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
161	C₂×C₈	—	71	1	no	—
162	C₂×C₈	—	71	1	no	—
163	C₂×C₈	—	71	2	no	—
164	C₂×C₈	—	71	2	no	—
165	C₂×C₈	—	71	2	no	—
166	C₂×C₈	—	142	1	no	—
167	C₂×C₈	—	142	1	no	—
168	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
169	C₂×C₈	C₈×D₇₁	1	2	yes	—
170	C₂×C₈	C₈×D₇₁	1	2	yes	—
171	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
173	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
174	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
175	C₂×C₈	#11765864	1	1	yes	characteristic
176	C₂×C₈	#11765866	1	1	yes	characteristic
177	C₄²	—	2	1	no	—
178	C₄²	—	71	1	no	—
179	C₄²	—	71	1	no	—
180	C₄²	—	71	2	no	—
181	C₄²	—	71	2	no	—
182	C₄²	—	142	8	no	—
183	C₄²	—	142	1	no	—
184	C₄²	C₄×Dic₇₁	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₁₆	—	142	4	no	—
187	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	71	2	no	—
191	C₂×C₄	—	71	2	no	—
192	C₂×C₄	—	142	4	no	—
193	C₂×C₄	—	142	4	no	—
194	C₂×C₄	—	142	2	no	—
195	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
196	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
197	C₂×C₄	#14556086	1	1	yes	characteristic
198	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
200	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
201	C₈	—	71	2	no	—
202	C₈	—	71	2	no	—
203	C₈	—	142	2	no	—
204	C₈	C₈×Dic₇₁	1	4	yes	—
205	C₈	#14556090	1	1	yes	characteristic
206	C₈	D₄×Dic₇₁	1	1	yes	characteristic
207	C₈	#14556093	1	1	yes	characteristic
208	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	M₁₆×Dic₇₁	1	1	yes	characteristic
211	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
212	C₂²	#17986051	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	71	4	no	—
215	C₄	—	142	4	no	—
216	C₄	—	142	2	no	—
217	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
218	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
219	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
220	C₄	#17986051	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	C₄⋊C₈×Dic₇₁	1	1	yes	characteristic
224	C₂	#364×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#364×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#364	D₇₁
hypocenter	C₇₁	C₄×#364
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #17985820	C₄	semidirect
18. #364×C₇₁	C₄	semidirect
107. Dic₇₁	#364	direct (N × H)
118. C₇₁	C₄×#364	semidirect
129. #364	Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17985820	2
M₃₂×Dic₇₁	2
C₄×C₈×Dic₇₁	2
#22586421	2
C₂×#364×C₇₁	2
C₄×#364	71

Subgroups — #364×Dic71 #31912772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #364×Dic₇₁ #31912772