Subgroups — #31891195

Order 18176.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 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	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	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#17985754	—	2	2	no	—
14	#17985751	—	2	4	no	—
15	#17985749	—	2	4	no	—
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
17	C₄³×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	—	2	2	no	—
19	C₂²×C₇₁⋊C₈	—	2	2	no	—
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₄×C₇₁⋊C₈	—	2	2	no	—
29	C₄×C₇₁⋊C₈	C₈	1	4	yes	complemented
30	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
31	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
32	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
33	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
34	C₂²×C₈×C₇₁	—	2	2	no	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	D₄	1	2	yes	—
38	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₁	—	2	2	no	—
41	C₄⋊C₈×C₇₁	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	—	4	4	no	—
48	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
52	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
53	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
54	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
55	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₁	—	4	2	no	—
62	C₂×C₈×C₇₁	—	4	4	no	—
63	C₄²×C₇₁	—	2	4	no	—
64	C₄²×C₇₁	—	2	1	no	—
65	C₄²×C₇₁	—	2	1	no	—
66	C₄²×C₇₁	—	2	4	no	—
67	C₄²×C₇₁	C₂×C₈	1	2	yes	—
68	C₄²×C₇₁	M₁₆	1	2	yes	—
69	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₇₁	#108	1	4	yes	—
81	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#135	1	4	yes	—
92	C₈×C₇₁	—	8	2	no	—
93	C₇₁⋊C₈	—	2	2	no	—
94	C₇₁⋊C₈	—	2	2	no	—
95	C₇₁⋊C₈	—	8	2	no	—
96	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
97	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
98	C₂²×C₇₁	#384	1	1	yes	characteristic
99	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₇₁	#423	1	1	yes	characteristic
101	C₂²×C₇₁	#427	1	2	yes	—
102	C₂²×C₇₁	C₂×#108	1	2	yes	—
103	C₄×C₇₁	—	2	4	no	—
104	C₄×C₇₁	—	2	4	no	—
105	C₄×C₇₁	—	2	1	no	—
106	C₄×C₇₁	—	2	1	no	—
107	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₇₁	#423	1	1	yes	characteristic
110	C₄×C₇₁	#423	1	1	yes	characteristic
111	C₄×C₇₁	#495	1	2	yes	—
112	C₄×C₇₁	#499	1	2	yes	—
113	#8807	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	#1564	1	2	yes	—
115	C₂×C₇₁	#1568	1	2	yes	—
116	C₂×C₇₁	#1842	1	1	yes	characteristic
117	C₂×C₇₁	#1846	1	1	yes	characteristic
118	C₂×C₇₁	#1848	1	1	yes	characteristic
119	#1831	—	71	1	no	—
120	#1831	D₇₁	1	1	yes	characteristic
121	C₄²×C₈	—	71	1	no	—
122	C₂×#335	—	71	2	no	—
123	C₂×#334	—	71	2	no	—
124	C₇₁	#8807	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊C₈	—	71	2	no	—
126	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
127	C₂×C₄×C₈	—	71	1	no	—
128	C₂×C₄×C₈	—	71	1	no	—
129	C₂×C₄×C₈	—	71	1	no	—
130	C₂×C₄×C₈	—	71	1	no	—
131	C₂×C₄×C₈	—	71	1	no	—
132	C₂×C₄×C₈	—	71	1	no	—
133	C₄³	C₂×D₇₁	1	1	yes	characteristic
134	#336	—	2	2	no	—
135	#336	—	142	2	no	—
136	#335	—	142	4	no	—
137	#334	—	142	4	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	71	1	no	—
140	C₂²×C₈	—	71	1	no	—
141	C₂²×C₈	—	71	1	no	—
142	C₂²×C₈	—	71	1	no	—
143	C₂²×C₈	—	142	2	no	—
144	C₂×C₄²	Dic₁₄₂	1	2	yes	—
145	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
148	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	142	2	no	—
152	C₄⋊C₈	—	142	2	no	—
153	C₄×C₈	—	71	4	no	—
154	C₄×C₈	—	71	2	no	—
155	C₄×C₈	—	71	2	no	—
156	C₄×C₈	—	71	2	no	—
157	C₄×C₈	—	71	2	no	—
158	C₄×C₈	—	142	2	no	—
159	C₄×C₈	—	142	2	no	—
160	C₄×C₈	—	142	2	no	—
161	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
162	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
163	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	284	2	no	—
169	C₂×C₈	—	284	4	no	—
170	C₂×C₈	—	71	2	no	—
171	C₂×C₈	—	71	2	no	—
172	C₂×C₈	—	71	2	no	—
173	C₂×C₈	—	71	2	no	—
174	C₂×C₈	—	71	2	no	—
175	C₂×C₈	—	71	2	no	—
176	C₂×C₈	—	71	2	no	—
177	C₂×C₈	—	71	2	no	—
178	C₂×C₈	—	142	1	no	—
179	C₂×C₈	—	142	1	no	—
180	C₂×C₈	—	142	1	no	—
181	C₂×C₈	—	142	1	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	4	no	—
186	C₄²	C₈×D₇₁	1	2	yes	—
187	C₄²	#11765848	1	2	yes	—
188	C₄²	D₄×D₇₁	1	1	yes	characteristic
189	C₄²	D₄×D₇₁	1	1	yes	characteristic
190	C₄²	Q₈×D₇₁	1	1	yes	characteristic
191	C₄²	Q₈×D₇₁	1	1	yes	characteristic
192	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	Dic₇₁⋊C₈	1	4	yes	—
199	C₂×C₄	#14556063	1	4	yes	—
200	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
201	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
202	C₂×C₄	#14556124	1	1	yes	characteristic
203	C₂×C₄	#14556126	1	1	yes	characteristic
204	C₂×C₄	#14556127	1	1	yes	characteristic
205	C₂×C₄	#14556127	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
209	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
210	C₂×C₄	#14556148	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	568	2	no	—
213	C₈	—	71	4	no	—
214	C₈	—	71	4	no	—
215	C₈	—	142	2	no	—
216	C₈	—	142	2	no	—
217	C₂²	#17985868	1	1	yes	characteristic
218	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
219	C₂²	#17986022	1	1	yes	characteristic
220	C₂²	#17986035	1	2	yes	—
221	C₂²	C₂×#14556063	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
228	C₄	#17986021	1	1	yes	characteristic
229	C₄	#17986021	1	1	yes	characteristic
230	C₄	#17986381	1	2	yes	—
231	C₄	#17986398	1	2	yes	—
232	C₂	#22587086	1	2	yes	—
233	C₂	#22587101	1	2	yes	—
234	C₂	#22588491	1	1	yes	characteristic
235	C₂	#22588502	1	1	yes	characteristic
236	C₂	#22588511	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	C₇₁×#1831	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1831	D₇₁
hypocenter	C₇₁	#8807
LCS	G ▷ C₂×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
29. C₄×C₇₁⋊C₈	C₈	semidirect
124. C₇₁	#8807	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
C₂×#17985751	2
C₂×#17985749	2
C₇₁×#1831	2
#8807	71