Subgroups — #32150647

Order 18176.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#22590166	C₂	1	8	yes	maximal, complemented
3	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589954	C₂	1	2	yes	maximal, complemented
5	C₂×#17985775	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
7	#17986416	—	2	8	no	—
8	C₂×#14556068	C₂²	1	2	yes	—
9	C₂×#14556068	C₂²	1	4	yes	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
12	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
14	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	8	yes	—
15	#17985775	C₂²	1	4	yes	—
16	C₂×#131×C₇₁	C₂²	1	2	yes	—
17	#532×C₇₁	C₄	1	1	yes	characteristic
18	#436×C₇₁	C₄	1	4	yes	—
19	C₈×D₄×C₇₁	C₂²	1	4	yes	—
20	#423×C₇₁	C₄	1	1	yes	characteristic
21	#408×C₇₁	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#14556068	—	2	8	no	—
24	#14556068	—	2	4	no	—
25	C₂×C₇₁⋊C₁₆	—	2	8	no	—
26	C₂×C₇₁⋊C₁₆	—	2	2	no	—
27	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	complemented
28	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
29	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
30	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
31	C₂×#41×C₇₁	C₂×C₄	1	2	yes	—
32	#131×C₇₁	—	2	4	no	—
33	#131×C₇₁	—	2	4	no	—
34	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₁	C₈	1	1	yes	characteristic
40	#127×C₇₁	C₈	1	1	yes	characteristic
41	#124×C₇₁	C₈	1	2	yes	—
42	#121×C₇₁	C₈	1	4	yes	—
43	C₄×D₄×C₇₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₁	C₂³	1	2	yes	—
49	C₇₁⋊C₁₆	—	2	4	no	—
50	C₇₁⋊C₁₆	—	2	2	no	—
51	C₇₁⋊C₁₆	C₂×D₄	1	4	yes	—
52	#41×C₇₁	—	2	4	no	—
53	#41×C₇₁	—	2	4	no	—
54	C₂×Q₈×C₇₁	C₂×C₈	1	2	yes	—
55	C₂×D₄×C₇₁	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
57	C₂×D₄×C₇₁	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₇₁	—	2	4	no	—
62	M₁₆×C₇₁	—	2	8	no	—
63	C₂×C₈×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	—	2	4	no	—
65	C₂×C₈×C₇₁	—	2	4	no	—
66	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₁	#41	1	2	yes	—
71	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
74	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
77	C₂³×C₇₁	C₂²×C₈	1	4	yes	—
78	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
79	Q₈×C₇₁	—	2	4	no	—
80	D₄×C₇₁	—	2	8	no	—
81	D₄×C₇₁	—	2	4	no	—
82	D₄×C₇₁	—	4	4	no	—
83	C₂×C₄×C₇₁	—	2	4	no	—
84	C₂×C₄×C₇₁	—	2	4	no	—
85	C₂×C₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#131	1	2	yes	—
92	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₇₁	—	2	4	no	—
94	C₈×C₇₁	—	2	2	no	—
95	C₈×C₇₁	C₄×D₄	1	2	yes	—
96	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
98	C₂²×C₇₁	—	2	2	no	—
99	C₂²×C₇₁	—	2	8	no	—
100	C₂²×C₇₁	—	2	4	no	—
101	C₂²×C₇₁	C₈×D₄	1	2	yes	—
102	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₇₁	—	2	2	no	—
104	C₄×C₇₁	—	2	2	no	—
105	C₄×C₇₁	—	2	4	no	—
106	C₄×C₇₁	C₈×D₄	1	2	yes	—
107	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
109	#33435	—	71	1	no	maximal, Sylow
110	C₂×C₇₁	—	2	2	no	—
111	C₂×C₇₁	—	4	4	no	—
112	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
113	C₂×C₇₁	#3407	1	2	yes	—
114	C₂×#504	—	71	2	no	—
115	#2931	D₇₁	1	1	yes	characteristic
116	#2168	—	71	8	no	—
117	#2150	—	71	1	no	—
118	#2113	—	71	2	no	—
119	C₂×#346	—	71	1	no	—
120	C₇₁	#33435	1	1	yes	characteristic, complemented, Sylow
121	C₂×#131	C₂×D₇₁	1	2	yes	—
122	#532	Dic₇₁	1	1	yes	characteristic
123	#504	—	142	8	no	—
124	C₂×M₃₂	—	71	2	no	—
125	C₂×M₃₂	—	71	4	no	—
126	C₂²×C₁₆	—	71	1	no	—
127	C₂²×C₁₆	—	71	1	no	—
128	C₂²×C₁₆	—	71	4	no	—
129	#436	Dic₇₁	1	4	yes	—
130	C₈×D₄	C₂×D₇₁	1	4	yes	—
131	#423	Dic₇₁	1	1	yes	characteristic
132	#408	Dic₇₁	1	2	yes	—
133	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
134	C₄⋊C₁₆	—	71	4	no	—
135	C₂²⋊C₁₆	—	71	8	no	—
136	#346	—	71	4	no	—
137	C₂×#41	C₂×Dic₇₁	1	2	yes	—
138	#131	—	2	4	no	—
139	#131	—	2	4	no	—
140	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
141	C₂×M₁₆	C₂×Dic₇₁	1	4	yes	—
142	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₇₁	1	4	yes	—
144	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
145	C₄⋊Q₈	C₇₁⋊C₈	1	1	yes	characteristic
146	#127	C₇₁⋊C₈	1	1	yes	characteristic
147	#124	C₇₁⋊C₈	1	2	yes	—
148	#121	C₇₁⋊C₈	1	4	yes	—
149	C₄×D₄	C₂×Dic₇₁	1	4	yes	—
150	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
151	M₃₂	—	142	8	no	—
152	M₃₂	—	142	4	no	—
153	C₂×C₁₆	—	71	4	no	—
154	C₂×C₁₆	—	71	2	no	—
155	C₂×C₁₆	—	71	2	no	—
156	C₂×C₁₆	—	71	4	no	—
157	C₂×C₁₆	—	142	8	no	—
158	C₂×C₁₆	—	142	2	no	—
159	C₄⋊C₈	C₂×Dic₇₁	1	4	yes	—
160	C₂²⋊C₈	C₂×Dic₇₁	1	8	yes	—
161	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
162	C₄×C₈	C₂²×D₇₁	1	2	yes	—
163	#41	—	2	4	no	—
164	#41	—	2	4	no	—
165	C₂×Q₈	C₂×C₇₁⋊C₈	1	2	yes	—
166	C₂×D₄	C₂×C₇₁⋊C₈	1	4	yes	—
167	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
168	C₂×D₄	C₂×C₇₁⋊C₈	1	4	yes	—
169	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₇₁	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
172	M₁₆	—	2	4	no	—
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
178	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
179	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
180	C₂×C₈	D₄×D₇₁	1	2	yes	—
181	C₂×C₈	#11765864	1	2	yes	—
182	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
183	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
184	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
185	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
186	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
187	C₄²	C₂²×Dic₇₁	1	2	yes	—
188	C₁₆	—	71	4	no	—
189	C₁₆	—	142	4	no	—
190	C₁₆	—	142	2	no	—
191	C₂³	C₂²×C₇₁⋊C₈	1	4	yes	—
192	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
201	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
202	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
203	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
204	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
205	C₂×C₄	#14556135	1	2	yes	—
206	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	2	2	no	—
209	C₈	D₄×Dic₇₁	1	2	yes	—
210	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
211	C₈	C₂×#11765864	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
216	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
221	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
222	C₄	C₂×#14556135	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	4	4	no	—
225	C₂	C₂×D₄×C₇₁⋊C₈	1	1	yes	characteristic
226	C₂	#22601443	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2931	C₂
prime radical	#532×C₇₁	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2931	D₇₁
hypocenter	C₇₁	#33435
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#17986416	C₂	semidirect
2. #22590166	C₂	semidirect
3. #22590096	C₂	semidirect
4. #22589954	C₂	semidirect
5. C₂×#17985775	C₂	semidirect
27. C₂×C₇₁⋊C₁₆	D₄	semidirect
120. C₇₁	#33435	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
#22590166	2
#22590096	2
#22589954	2
C₂×#17985775	2
C₇₁×#2931	2
#33435	71