Subgroups — #31915628

Order 18176.

466 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985781	C₂	1	4	yes	maximal, complemented
3	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	#17985781	—	2	8	no	—
10	#17985775	—	2	1	no	—
11	C₄×C₇₁⋊C₁₆	—	2	1	no	—
12	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₇₁	C₄	1	1	yes	characteristic
14	#436×C₇₁	—	2	1	no	—
15	#436×C₇₁	—	2	1	no	—
16	#436×C₇₁	—	2	1	no	—
17	#436×C₇₁	—	2	1	no	—
18	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	#14556068	—	2	4	no	—
22	#14556068	—	2	2	no	—
23	#14556068	D₄	1	4	yes	complemented
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
26	C₂²×D₄×C₇₁	C₈	1	1	yes	characteristic
27	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂×M₁₆×C₇₁	—	2	4	no	—
31	C₂×M₁₆×C₇₁	—	2	4	no	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₇₁	—	2	1	no	—
36	C₄×D₄×C₇₁	—	2	1	no	—
37	C₄×D₄×C₇₁	—	2	1	no	—
38	C₄×D₄×C₇₁	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
41	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	—	2	1	no	—
43	C₄⋊C₈×C₇₁	—	2	1	no	—
44	C₂²⋊C₈×C₇₁	—	2	2	no	—
45	C₂²⋊C₈×C₇₁	—	2	2	no	—
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₇₁⋊C₁₆	—	2	4	no	—
51	C₇₁⋊C₁₆	—	4	2	no	—
52	C₂⁴×C₇₁	C₂×C₈	1	2	yes	—
53	C₂×D₄×C₇₁	—	2	1	no	—
54	C₂×D₄×C₇₁	—	2	1	no	—
55	C₂×D₄×C₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	—	2	4	no	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	4	no	—
60	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₇₁	—	4	4	no	—
65	M₁₆×C₇₁	—	4	2	no	—
66	M₁₆×C₇₁	—	4	2	no	—
67	C₂×C₈×C₇₁	—	2	2	no	—
68	C₂×C₈×C₇₁	—	2	2	no	—
69	C₂×C₈×C₇₁	—	2	4	no	—
70	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₁	#41	1	2	yes	—
76	C₄⋊C₄×C₇₁	—	2	1	no	—
77	C₄⋊C₄×C₇₁	—	2	1	no	—
78	C₂²⋊C₄×C₇₁	—	2	2	no	—
79	C₂²⋊C₄×C₇₁	—	2	2	no	—
80	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
83	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
84	C₂³×C₇₁	—	2	2	no	—
85	C₂³×C₇₁	—	2	4	no	—
86	C₂³×C₇₁	—	4	2	no	—
87	C₂³×C₇₁	—	4	2	no	—
88	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
89	D₄×C₇₁	—	4	2	no	—
90	D₄×C₇₁	—	4	2	no	—
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	2	no	—
93	C₂×C₄×C₇₁	—	2	4	no	—
94	C₂×C₄×C₇₁	—	4	1	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	—	4	4	no	—
98	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#131	1	2	yes	—
106	C₈×C₇₁	—	2	1	no	—
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	—	4	2	no	—
109	C₈×C₇₁	#121	1	2	yes	—
110	C₈×C₇₁	#124	1	1	yes	characteristic
111	C₈×C₇₁	#127	1	1	yes	characteristic
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	1	no	—
114	C₂²×C₇₁	—	4	2	no	—
115	C₂²×C₇₁	—	4	2	no	—
116	C₂²×C₇₁	—	4	2	no	—
117	C₂²×C₇₁	—	8	4	no	—
118	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₇₁	C₈×D₄	1	2	yes	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	4	2	no	—
124	C₄×C₇₁	#349	1	2	yes	complemented
125	C₄×C₇₁	#390	1	1	yes	characteristic
126	C₄×C₇₁	#408	1	1	yes	characteristic
127	C₄×C₇₁	#436	1	2	yes	—
128	#13194	—	71	1	no	maximal, Sylow
129	C₂×C₇₁	—	2	2	no	—
130	C₂×C₇₁	—	8	2	no	—
131	C₂×C₇₁	#1954	1	1	yes	characteristic
132	C₂×C₇₁	C₂×#349	1	1	yes	characteristic
133	C₂×C₇₁	#2115	1	1	yes	characteristic
134	C₂×#436	D₇₁	1	1	yes	characteristic
135	#2150	—	71	1	no	—
136	C₂×#349	—	71	4	no	—
137	C₄×M₃₂	—	71	1	no	—
138	C₇₁	#13194	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	C₂×D₇₁	1	2	yes	—
140	C₂×C₄×D₄	Dic₇₁	1	1	yes	characteristic
141	C₂×M₃₂	—	71	2	no	—
142	C₂×M₃₂	—	71	2	no	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	#436	—	2	1	no	—
147	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₈	Dic₇₁	1	2	yes	—
149	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
150	C₄⋊C₁₆	—	142	1	no	—
151	C₄⋊C₁₆	—	142	1	no	—
152	#349	—	142	8	no	—
153	#346	—	142	1	no	—
154	C₄×C₁₆	—	142	1	no	—
155	C₂²×D₄	C₇₁⋊C₈	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₇₁	1	2	yes	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
162	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
170	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
171	M₃₂	—	71	4	no	—
172	M₃₂	—	142	4	no	—
173	M₃₂	—	142	2	no	—
174	C₂×C₁₆	—	142	2	no	—
175	C₂×C₁₆	—	142	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
184	C₂⁴	C₂×C₇₁⋊C₈	1	2	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
193	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
204	C₂×C₈	D₄×D₇₁	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
207	C₂×C₈	#11765864	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
213	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
214	C₄²	#11765849	1	1	yes	characteristic
215	C₄²	#11765849	1	1	yes	characteristic
216	C₁₆	—	284	2	no	—
217	C₁₆	—	142	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
238	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
239	C₂×C₄	#14556135	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	#14556099	1	2	yes	—
244	C₈	#14556115	1	1	yes	characteristic
245	C₈	#14556124	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
253	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#17985781	1	2	yes	complemented
259	C₄	#17985902	1	1	yes	characteristic
260	C₄	#17985966	1	1	yes	characteristic
261	C₄	#17986077	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#22589135	1	1	yes	characteristic
265	C₂	C₂×#17985781	1	1	yes	characteristic
266	C₂	#22589961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₁
commutator	C₂³×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#436×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₇₁	C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#436	D₇₁
hypocenter	C₇₁	#13194
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22590096	C₂	semidirect
2. C₂×#17985781	C₂	semidirect
3. C₄×#14556068	C₂	semidirect
23. #14556068	D₄	semidirect
124. C₄×C₇₁	#349	semidirect
138. C₇₁	#13194	semidirect
258. C₄	#17985781	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#17985781	2
C₄×#14556068	2
C₂×#436×C₇₁	2
#13194	71