Subgroups — #32150693

Order 18176.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986416	C₂	1	2	yes	maximal, complemented
2	#22590182	C₂	1	4	yes	maximal, complemented
3	#22590177	C₂	1	4	yes	maximal, complemented
4	#22590106	C₂	1	1	yes	maximal, characteristic, complemented
5	#22589968	C₂	1	2	yes	maximal, complemented
6	#22589937	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2955	C₂	1	1	yes	maximal, characteristic
8	#17986416	—	2	2	no	—
9	#17986416	—	2	2	no	—
10	#17986416	C₂²	1	4	yes	complemented
11	#17986416	C₂²	1	4	yes	complemented
12	C₂×#14556068	—	2	2	no	—
13	C₂×#14556068	C₂²	1	2	yes	—
14	C₂×#14556068	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
16	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
17	#17985820	C₂²	1	4	yes	complemented
18	#17985783	C₂²	1	8	yes	complemented
19	#17985775	C₂²	1	2	yes	complemented
20	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
21	#575×C₇₁	C₄	1	1	yes	characteristic
22	C₂×#131×C₇₁	C₂²	1	2	yes	—
23	#444×C₇₁	C₄	1	4	yes	—
24	#443×C₇₁	C₂²	1	4	yes	—
25	#430×C₇₁	C₄	1	1	yes	characteristic
26	#421×C₇₁	C₄	1	2	yes	—
27	#405×C₇₁	C₂²	1	1	yes	characteristic
28	#14556068	—	2	4	no	—
29	#14556068	—	2	4	no	—
30	#14556068	—	2	2	no	—
31	#14556068	—	2	2	no	—
32	#14556068	C₂³	1	4	yes	—
33	C₂×C₇₁⋊C₁₆	—	2	4	no	—
34	C₂×C₇₁⋊C₁₆	—	2	4	no	—
35	C₂×C₇₁⋊C₁₆	—	2	2	no	—
36	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
37	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
38	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
39	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₂×#41×C₇₁	C₂×C₄	1	2	yes	—
42	#137×C₇₁	C₈	1	4	yes	—
43	AGL(1,ℤ/8)×C₇₁	C₈	1	4	yes	—
44	#135×C₇₁	C₂×C₄	1	4	yes	—
45	#131×C₇₁	—	2	2	no	—
46	#131×C₇₁	—	2	2	no	—
47	#131×C₇₁	C₂×C₄	1	4	yes	—
48	#131×C₇₁	C₂³	1	4	yes	—
49	C₂×M₁₆×C₇₁	—	2	2	no	—
50	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
51	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	—	2	2	no	—
53	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
55	#108×C₇₁	C₂×C₄	1	4	yes	—
56	#104×C₇₁	C₂×C₄	1	8	yes	—
57	#97×C₇₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₁	C₂³	1	2	yes	—
59	C₇₁⋊C₁₆	—	2	4	no	—
60	C₇₁⋊C₁₆	—	2	2	no	—
61	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
62	C₇₁⋊C₁₆	C₂×D₄	1	2	yes	—
63	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
64	#41×C₇₁	—	2	2	no	—
65	#41×C₇₁	—	2	2	no	—
66	#41×C₇₁	C₂×C₈	1	4	yes	—
67	#41×C₇₁	C₂²×C₄	1	4	yes	—
68	C₂×Q₈×C₇₁	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₇₁	—	2	2	no	—
70	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₇₁	—	2	2	no	—
72	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
73	Q₁₆×C₇₁	C₂×C₈	1	4	yes	—
74	SD₁₆×C₇₁	C₂×C₈	1	8	yes	—
75	D₈×C₇₁	C₂×C₈	1	4	yes	—
76	M₁₆×C₇₁	—	2	2	no	—
77	M₁₆×C₇₁	—	2	4	no	—
78	M₁₆×C₇₁	—	2	4	no	—
79	M₁₆×C₇₁	C₂×C₈	1	4	yes	—
80	M₁₆×C₇₁	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	—	2	4	no	—
83	C₂×C₈×C₇₁	—	2	4	no	—
84	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₇₁	#41	1	2	yes	—
88	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₇₁	—	2	1	no	—
90	C₂²⋊C₄×C₇₁	—	2	1	no	—
91	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
92	C₂³×C₇₁	—	2	2	no	—
93	C₂³×C₇₁	#131	1	1	yes	characteristic
94	Q₈×C₇₁	—	2	2	no	—
95	Q₈×C₇₁	C₂²×C₈	1	4	yes	—
96	D₄×C₇₁	—	2	4	no	—
97	D₄×C₇₁	—	2	4	no	—
98	D₄×C₇₁	—	2	2	no	—
99	D₄×C₇₁	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	2	4	no	—
102	C₂×C₄×C₇₁	—	2	4	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₇₁	—	2	4	no	—
109	C₈×C₇₁	C₄×D₄	1	2	yes	—
110	C₈×C₇₁	C₂²×C₈	1	4	yes	—
111	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
113	C₂²×C₇₁	—	2	4	no	—
114	C₂²×C₇₁	—	4	1	no	—
115	C₂²×C₇₁	—	4	4	no	—
116	C₂²×C₇₁	C₈×D₄	1	2	yes	—
117	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
118	C₄×C₇₁	—	2	4	no	—
119	C₄×C₇₁	—	4	2	no	—
120	C₄×C₇₁	C₈×D₄	1	2	yes	—
121	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
123	#33441	—	71	1	no	maximal, Sylow
124	C₂×C₇₁	—	2	1	no	—
125	C₂×C₇₁	—	2	2	no	—
126	C₂×C₇₁	—	4	4	no	—
127	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
128	C₂×#504	—	71	2	no	—
129	#2955	D₇₁	1	1	yes	characteristic
130	#2171	—	71	4	no	—
131	#2170	—	71	4	no	—
132	#2153	—	71	1	no	—
133	#2117	—	71	2	no	—
134	#2108	—	71	1	no	—
135	C₇₁	#33441	1	1	yes	characteristic, complemented, Sylow
136	#575	Dic₇₁	1	1	yes	characteristic
137	C₂×#131	C₂×D₇₁	1	2	yes	—
138	#504	—	71	4	no	—
139	#504	—	71	4	no	—
140	#504	—	142	2	no	—
141	#504	—	142	2	no	—
142	C₂×M₃₂	—	71	1	no	—
143	C₂×M₃₂	—	71	2	no	—
144	C₂×M₃₂	—	142	2	no	—
145	C₂²×C₁₆	—	71	1	no	—
146	C₂²×C₁₆	—	142	2	no	—
147	#444	Dic₇₁	1	4	yes	—
148	#443	C₂×D₇₁	1	4	yes	—
149	#430	Dic₇₁	1	1	yes	characteristic
150	#421	Dic₇₁	1	2	yes	—
151	#405	C₂×D₇₁	1	1	yes	characteristic
152	#364	—	71	4	no	—
153	#350	—	71	8	no	—
154	#346	—	71	2	no	—
155	C₄×C₁₆	—	71	2	no	—
156	C₂×#41	C₂×Dic₇₁	1	2	yes	—
157	#137	C₇₁⋊C₈	1	4	yes	—
158	AGL(1,ℤ/8)	C₇₁⋊C₈	1	4	yes	—
159	#135	C₂×Dic₇₁	1	4	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×Dic₇₁	1	4	yes	—
163	#131	C₂²×D₇₁	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
166	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
169	#117	C₂×Dic₇₁	1	1	yes	characteristic
170	M₃₂	—	71	4	no	—
171	M₃₂	—	142	4	no	—
172	M₃₂	—	142	4	no	—
173	M₃₂	—	142	2	no	—
174	M₃₂	—	142	2	no	—
175	C₂×C₁₆	—	71	2	no	—
176	C₂×C₁₆	—	71	2	no	—
177	C₂×C₁₆	—	71	1	no	—
178	C₂×C₁₆	—	71	1	no	—
179	C₂×C₁₆	—	71	2	no	—
180	C₂×C₁₆	—	142	4	no	—
181	C₂×C₁₆	—	142	4	no	—
182	C₂×C₁₆	—	142	2	no	—
183	#108	C₂×Dic₇₁	1	4	yes	—
184	#104	C₂×Dic₇₁	1	8	yes	—
185	#97	C₂×Dic₇₁	1	2	yes	—
186	C₄×C₈	C₂²×D₇₁	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×C₇₁⋊C₈	1	4	yes	—
190	#41	C₂²×Dic₇₁	1	4	yes	—
191	C₂×Q₈	C₂×C₇₁⋊C₈	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
196	Q₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
197	SD₁₆	C₂×C₇₁⋊C₈	1	8	yes	—
198	D₈	C₂×C₇₁⋊C₈	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
203	M₁₆	C₂²×Dic₇₁	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
209	C₂×C₈	D₄×D₇₁	1	2	yes	—
210	C₂×C₈	#11765864	1	2	yes	—
211	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂²×Dic₇₁	1	2	yes	—
215	C₁₆	—	71	1	no	—
216	C₁₆	—	71	2	no	—
217	C₁₆	—	71	1	no	—
218	C₁₆	—	142	4	no	—
219	C₁₆	—	142	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#14556135	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×C₇₁⋊C₈	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×C₇₁⋊C₈	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
233	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
234	C₂×C₄	#14556135	1	1	yes	characteristic
235	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	D₄×Dic₇₁	1	2	yes	—
238	C₈	C₂²×C₇₁⋊C₈	1	4	yes	—
239	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
240	C₈	C₂×#11765864	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
245	C₂²	C₂×#14556135	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
249	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
250	C₄	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×D₄×C₇₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₇₁
commutator	C₄×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2955	C₂
socle	C₂×C₇₁	C₂×C₈×D₄
prime radical	#575×C₇₁	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2955	D₇₁
hypocenter	C₇₁	#33441
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. C₂×#17986416	C₂	semidirect
2. #22590182	C₂	semidirect
3. #22590177	C₂	semidirect
4. #22590106	C₂	semidirect
5. #22589968	C₂	semidirect
6. #22589937	C₂	semidirect
10. #17986416	C₂²	semidirect
11. #17986416	C₂²	semidirect
17. #17985820	C₂²	semidirect
18. #17985783	C₂²	semidirect
19. #17985775	C₂²	semidirect
20. C₄×C₇₁⋊C₁₆	C₂²	semidirect
135. C₇₁	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986416	2
#22590182	2
#22590177	2
#22590106	2
#22589968	2
#22589937	2
C₇₁×#2955	2
#33441	71