Subgroups — #31913377

Order 18176.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal
3	#22586418	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
6	C₄⋊(C₇₁⋊C₁₆)	—	2	4	no	—
7	C₂²⋊(C₇₁⋊C₁₆)	—	2	4	no	—
8	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₇₁	C₄	1	1	yes	characteristic
11	#405×C₇₁	—	2	4	no	—
12	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
13	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂×C₇₁⋊C₁₆	—	2	4	no	—
18	C₂×C₇₁⋊C₁₆	—	2	4	no	—
19	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	—	2	2	no	—
21	C₂×M₁₆×C₇₁	—	2	2	no	—
22	C₂×M₁₆×C₇₁	—	2	2	no	—
23	C₂²×C₈×C₇₁	—	2	2	no	—
24	C₂²×C₈×C₇₁	—	2	2	no	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	D₄	1	2	yes	—
29	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	#117×C₇₁	—	2	2	no	—
32	#117×C₇₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₁	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	#97×C₇₁	—	2	2	no	—
38	#97×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	—	2	2	no	—
40	C₄×C₈×C₇₁	D₄	1	4	yes	—
41	C₇₁⋊C₁₆	—	4	4	no	—
42	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	—	2	1	no	—
44	C₂²×C₄×C₇₁	—	2	1	no	—
45	C₂²×C₄×C₇₁	—	2	1	no	—
46	C₂²×C₄×C₇₁	—	2	1	no	—
47	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₇₁	—	2	8	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	2	1	no	—
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	—	2	1	no	—
60	C₂×C₈×C₇₁	—	2	4	no	—
61	C₂×C₈×C₇₁	—	2	4	no	—
62	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
65	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₇₁	#41	1	2	yes	—
69	C₄⋊C₄×C₇₁	—	2	4	no	—
70	C₂²⋊C₄×C₇₁	—	2	4	no	—
71	C₄²×C₇₁	—	2	1	no	—
72	C₄²×C₇₁	—	2	1	no	—
73	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₇₁	M₁₆	1	2	yes	—
75	C₂³×C₇₁	—	2	1	no	—
76	C₂³×C₇₁	—	2	1	no	—
77	C₂³×C₇₁	—	2	4	no	—
78	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
79	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	—	2	1	no	—
81	C₂×C₄×C₇₁	—	2	1	no	—
82	C₂×C₄×C₇₁	—	2	4	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	2	2	no	—
85	C₂×C₄×C₇₁	—	2	2	no	—
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	2	4	no	—
89	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
97	C₈×C₇₁	—	2	2	no	—
98	C₈×C₇₁	—	2	2	no	—
99	C₈×C₇₁	—	2	2	no	—
100	C₈×C₇₁	#121	1	2	yes	—
101	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
102	C₂²×C₇₁	—	2	2	no	—
103	C₂²×C₇₁	—	2	4	no	—
104	C₂²×C₇₁	—	4	2	no	—
105	C₂²×C₇₁	—	4	2	no	—
106	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₇₁	#431	1	1	yes	characteristic
109	C₂²×C₇₁	#504	1	4	yes	—
110	C₄×C₇₁	—	2	2	no	—
111	C₄×C₇₁	—	2	4	no	—
112	C₄×C₇₁	—	4	2	no	—
113	C₄×C₇₁	#380	1	1	yes	characteristic
114	C₄×C₇₁	#407	1	1	yes	characteristic
115	C₄×C₇₁	#423	1	1	yes	characteristic
116	C₄×C₇₁	#432	1	1	yes	characteristic
117	#12784	—	71	1	no	maximal, Sylow
118	C₂×C₇₁	—	4	2	no	—
119	C₂×C₇₁	#1797	1	1	yes	characteristic
120	C₂×C₇₁	#2113	1	2	yes	—
121	C₂×C₇₁	#2151	1	2	yes	—
122	C₂×C₇₁	#2164	1	2	yes	—
123	C₂×#405	D₇₁	1	1	yes	characteristic
124	C₂×C₄⋊C₁₆	—	71	2	no	—
125	C₂×C₂²⋊C₁₆	—	71	2	no	—
126	#1374	—	71	2	no	—
127	C₇₁	#12784	1	1	yes	characteristic, complemented, Sylow
128	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
129	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
130	C₂×#117	Dic₇₁	1	1	yes	characteristic
131	C₂²×C₁₆	—	71	4	no	—
132	#405	—	2	4	no	—
133	C₂×#97	Dic₇₁	1	1	yes	characteristic
134	C₂×#97	Dic₇₁	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
137	C₄⋊C₁₆	—	142	4	no	—
138	C₂²⋊C₁₆	—	142	4	no	—
139	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	Dic₁₄₂	1	2	yes	—
146	C₂²×C₈	D₂₈₄	1	2	yes	—
147	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
149	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
151	#117	—	2	2	no	—
152	#117	—	2	2	no	—
153	C₂×C₄⋊C₄	C₇₁⋊C₈	1	2	yes	—
154	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
155	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
157	C₂×C₁₆	—	142	4	no	—
158	C₂×C₁₆	—	142	4	no	—
159	C₂×C₁₆	—	142	4	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	C₂²⋊D₇₁	1	4	yes	—
164	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
170	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
171	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
172	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	#11765849	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
185	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
186	C₂×C₈	#11765849	1	2	yes	—
187	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
188	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
189	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
190	C₂×C₈	#11765865	1	2	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
196	C₄²	#11765849	1	2	yes	—
197	C₁₆	—	284	4	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
202	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	4	yes	—
213	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
216	C₂×C₄	#14556087	1	1	yes	characteristic
217	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
218	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
219	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	#14556102	1	2	yes	—
224	C₈	#14556107	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
231	C₂²	#17986052	1	1	yes	characteristic
232	C₂²	#17986416	1	4	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#17985859	1	1	yes	characteristic
237	C₄	#17985963	1	1	yes	characteristic
238	C₄	#17986023	1	1	yes	characteristic
239	C₄	#17986056	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#22588320	1	1	yes	characteristic
242	C₂	#22589954	1	2	yes	—
243	C₂	#22590099	1	2	yes	—
244	C₂	#22590147	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator, socle	C₂³×C₇₁	C₂²×C₈
Fitting	C₂×#405×C₇₁	C₂
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#12784
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
3. #22586418	C₂	semidirect
127. C₇₁	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×C₂²⋊(C₇₁⋊C₁₆)	2
#22586418	2
C₂×#405×C₇₁	2
#12784	71