Subgroups — #31891188

Order 18176.

462 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588491	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985749	C₂	1	2	yes	maximal
5	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₇₁⋊C₈	—	2	4	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
13	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
14	#17985750	—	2	2	no	—
15	#17985749	—	2	4	no	—
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#334×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	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₁	D₄	1	2	yes	—
25	C₂×C₄×Dic₇₁	Q₈	1	2	yes	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	Dic₇₁⋊C₈	—	4	8	no	—
29	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
30	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
31	C₈×Dic₇₁	—	2	2	no	—
32	C₈×Dic₇₁	—	2	2	no	—
33	C₈×Dic₇₁	—	2	2	no	—
34	C₈×Dic₇₁	C₈	1	4	yes	complemented
35	C₂²×C₈×C₇₁	—	2	2	no	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	—	2	2	no	—
40	C₄⋊C₈×C₇₁	—	2	2	no	—
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	C₄×C₈×C₇₁	Q₈	1	2	yes	—
43	C₂²×Dic₇₁	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₇₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	4	4	no	—
49	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
53	C₄×Dic₇₁	—	2	1	no	—
54	C₄×Dic₇₁	—	2	1	no	—
55	C₄×Dic₇₁	—	2	4	no	—
56	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
57	C₄×Dic₇₁	M₁₆	1	2	yes	—
58	C₄×Dic₇₁	SD₁₆	1	8	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	4	2	no	—
65	C₂×C₈×C₇₁	—	4	4	no	—
66	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
70	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
87	C₂×Dic₇₁	—	2	2	no	—
88	C₂×Dic₇₁	—	2	2	no	—
89	C₂×Dic₇₁	C₄⋊C₈	1	4	yes	—
90	C₂×Dic₇₁	#106	1	4	yes	—
91	C₂×Dic₇₁	#106	1	4	yes	—
92	C₂×Dic₇₁	#108	1	4	yes	—
93	C₈×C₇₁	—	2	2	no	—
94	C₈×C₇₁	—	8	2	no	—
95	C₈×C₇₁	C₄⋊C₈	1	4	yes	complemented
96	C₇₁⋊C₈	—	2	2	no	—
97	C₇₁⋊C₈	—	8	2	no	—
98	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	complemented
99	C₂²×C₇₁	#384	1	1	yes	characteristic
100	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₇₁	#423	1	1	yes	characteristic
102	C₂²×C₇₁	C₂×#106	1	2	yes	—
103	C₂²×C₇₁	C₂×#108	1	2	yes	—
104	C₄×C₇₁	—	2	1	no	—
105	C₄×C₇₁	—	2	1	no	—
106	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₇₁	#423	1	1	yes	characteristic
109	C₄×C₇₁	#423	1	1	yes	characteristic
110	C₄×C₇₁	#492	1	2	yes	—
111	C₄×C₇₁	#498	1	2	yes	—
112	Dic₇₁	—	2	4	no	—
113	Dic₇₁	#334	1	8	yes	complemented
114	#8805	—	71	1	no	maximal, Sylow
115	C₂×C₇₁	C₂×#334	1	2	yes	—
116	C₂×C₇₁	#1566	1	2	yes	—
117	C₂×C₇₁	#1842	1	1	yes	characteristic
118	C₂×C₇₁	#1844	1	1	yes	characteristic
119	C₂×C₇₁	#1848	1	1	yes	characteristic
120	#1842	—	71	2	no	—
121	C₄²×C₈	—	71	1	no	—
122	C₂×#334	—	71	2	no	—
123	C₂×#334	—	71	1	no	—
124	C₂×#334	D₇₁	1	1	yes	characteristic
125	C₇₁	#8805	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄⋊C₈	—	71	2	no	—
127	C₂×C₄⋊C₈	—	142	4	no	—
128	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
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	C₂×C₄×C₈	—	71	1	no	—
134	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
135	C₄³	—	71	1	no	—
136	#334	—	2	2	no	—
137	#334	—	142	4	no	—
138	#334	—	142	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	71	1	no	—
141	C₂²×C₈	—	71	1	no	—
142	C₂²×C₈	—	142	2	no	—
143	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
144	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
145	C₂×C₄²	—	71	1	no	—
146	C₂×C₄²	—	71	2	no	—
147	C₂×C₄²	—	71	1	no	—
148	C₂×C₄²	—	71	2	no	—
149	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	284	8	no	—
153	C₄⋊C₈	—	142	2	no	—
154	C₄⋊C₈	—	142	2	no	—
155	C₄×C₈	—	71	4	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	C₄×C₈	Dic₁₄₂	1	2	yes	—
162	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
163	C₂²×C₄	—	71	2	no	—
164	C₂²×C₄	—	71	2	no	—
165	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
166	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
167	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	284	2	no	—
173	C₂×C₈	—	284	4	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	Dic₇₁⋊C₄	1	2	yes	—
181	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
182	C₂×C₈	C₈×D₇₁	1	2	yes	—
183	C₂×C₈	#11765848	1	2	yes	—
184	C₄²	—	71	2	no	—
185	C₄²	—	71	2	no	—
186	C₄²	—	71	8	no	—
187	C₄²	—	142	4	no	—
188	C₄²	—	142	1	no	—
189	C₄²	—	142	1	no	—
190	C₄²	D₄×D₇₁	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
192	C₄²	Q₈×D₇₁	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
194	C₂³	C₄⋊C₄×D₇₁	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	71	4	no	—
197	C₂×C₄	—	71	4	no	—
198	C₂×C₄	—	71	4	no	—
199	C₂×C₄	—	71	4	no	—
200	C₂×C₄	—	142	2	no	—
201	C₂×C₄	—	142	2	no	—
202	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
203	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#14556124	1	1	yes	characteristic
205	C₂×C₄	#14556126	1	1	yes	characteristic
206	C₂×C₄	#14556127	1	1	yes	characteristic
207	C₂×C₄	#14556127	1	1	yes	characteristic
208	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
209	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
211	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
212	C₂×C₄	SD₁₆×D₇₁	1	4	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	8	2	no	—
215	C₈	—	568	2	no	—
216	C₈	—	71	4	no	—
217	C₈	—	142	2	no	—
218	C₈	Dic₇₁⋊C₈	1	4	yes	complemented
219	C₂²	#17985868	1	1	yes	characteristic
220	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
221	C₂²	#17986021	1	1	yes	characteristic
222	C₂²	#106×D₇₁	1	2	yes	—
223	C₂²	#108×D₇₁	1	2	yes	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	71	8	no	—
227	C₄	—	142	4	no	—
228	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
229	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
230	C₄	#17986021	1	1	yes	characteristic
231	C₄	#17986022	1	1	yes	characteristic
232	C₄	#17986370	1	2	yes	—
233	C₄	#17986396	1	2	yes	—
234	C₂	#334×D₇₁	1	2	yes	—
235	C₂	#22587094	1	2	yes	—
236	C₂	#22588491	1	1	yes	characteristic
237	C₂	#22588495	1	1	yes	characteristic
238	C₂	#22588510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#8805
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₇₁	C₈	semidirect
95. C₈×C₇₁	C₄⋊C₈	semidirect
98. C₇₁⋊C₈	C₄⋊C₈	semidirect
113. Dic₇₁	#334	semidirect
125. C₇₁	#8805	semidirect
218. C₈	Dic₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588491	2
C₄×C₈×Dic₇₁	2
C₂×#17985750	2
C₂×#17985749	2
C₂×#334×C₇₁	2
#8805	71