Subgroups — #31916521

Order 18176.

414 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×#17986404	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985803	C₂	1	2	yes	maximal
3	#22586431	C₂	1	2	yes	maximal
4	#22586418	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	—	2	1	no	—
7	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
8	#17986404	—	2	4	no	—
9	C₂×#14556062	—	2	1	no	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	#17985803	—	2	4	no	—
12	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#391×C₇₁	—	2	1	no	—
17	C₂²×Dic₁₄₂	—	4	1	no	—
18	#14556128	—	2	4	no	—
19	#14556128	—	4	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	4	1	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	#14556062	—	2	4	no	—
26	#14556062	—	2	2	no	—
27	#14556062	D₄	1	4	yes	—
28	C₂²×Q₈×C₇₁	—	2	1	no	—
29	C₂×Q₁₆×C₇₁	—	2	2	no	—
30	C₂×Q₁₆×C₇₁	—	2	2	no	—
31	C₂×Q₁₆×C₇₁	—	2	1	no	—
32	C₂×Q₁₆×C₇₁	Q₈	1	2	yes	complemented
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	#107×C₇₁	—	2	2	no	—
38	#103×C₇₁	—	4	2	no	—
39	C₄×C₈×C₇₁	—	2	2	no	—
40	#95×C₇₁	—	4	1	no	—
41	#95×C₇₁	—	4	1	no	—
42	C₂×Dic₁₄₂	—	4	4	no	—
43	C₂×Dic₁₄₂	—	8	2	no	—
44	C₂×Dic₁₄₂	—	8	2	no	—
45	C₂²×Dic₇₁	—	4	2	no	—
46	C₄⋊Dic₇₁	—	2	4	no	—
47	C₄⋊Dic₇₁	—	4	4	no	—
48	C₄⋊Dic₇₁	—	4	2	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	2	2	no	—
51	C₂×Q₈×C₇₁	—	4	1	no	—
52	C₂×Q₈×C₇₁	—	4	1	no	—
53	C₂×Q₈×C₇₁	—	4	1	no	—
54	C₂×Q₈×C₇₁	—	4	1	no	—
55	C₂×Q₈×C₇₁	—	4	1	no	—
56	C₂²×C₄×C₇₁	—	4	1	no	—
57	C₂²×C₄×C₇₁	—	4	1	no	—
58	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₇₁	—	2	4	no	—
62	Q₁₆×C₇₁	—	4	2	no	—
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
66	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₇₁	#41	1	2	yes	—
69	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
70	C₄⋊C₄×C₇₁	—	4	1	no	—
71	C₄⋊C₄×C₇₁	—	4	1	no	—
72	C₄²×C₇₁	—	2	1	no	—
73	C₄²×C₇₁	—	2	1	no	—
74	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₇₁	—	4	2	no	—
76	Q₈×C₇₁	—	4	2	no	—
77	Q₈×C₇₁	—	8	1	no	—
78	Q₈×C₇₁	—	8	1	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	2	1	no	—
81	C₂×C₄×C₇₁	—	2	1	no	—
82	C₂×C₄×C₇₁	—	2	1	no	—
83	C₂×C₄×C₇₁	—	4	2	no	—
84	C₂×C₄×C₇₁	—	8	1	no	—
85	C₂×C₄×C₇₁	—	8	1	no	—
86	C₂×C₄×C₇₁	—	8	1	no	—
87	C₂×C₄×C₇₁	—	8	1	no	—
88	C₂×C₄×C₇₁	—	8	1	no	—
89	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
90	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
91	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
97	C₂×Dic₇₁	—	4	4	no	—
98	C₂×Dic₇₁	—	8	2	no	—
99	C₂×Dic₇₁	—	8	2	no	—
100	Dic₁₄₂	—	8	8	no	—
101	C₈×C₇₁	—	2	2	no	—
102	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₇₁	#124	1	2	yes	—
104	C₂²×C₇₁	#447	1	1	yes	characteristic
105	C₂²×C₇₁	#474	1	1	yes	characteristic
106	C₂²×C₇₁	#486	1	1	yes	characteristic
107	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
108	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
109	C₂²×C₇₁	#509	1	1	yes	characteristic
110	C₂²×C₇₁	#510	1	1	yes	characteristic
111	C₄×C₇₁	—	2	2	no	—
112	C₄×C₇₁	—	2	2	no	—
113	C₄×C₇₁	—	8	2	no	—
114	C₄×C₇₁	—	16	1	no	—
115	C₄×C₇₁	#393	1	1	yes	characteristic
116	C₄×C₇₁	#448	1	1	yes	characteristic
117	C₄×C₇₁	#476	1	1	yes	characteristic
118	C₄×C₇₁	#489	1	1	yes	characteristic
119	Dic₇₁	—	8	4	no	—
120	#13364	—	71	1	no	maximal, Sylow
121	C₂×C₇₁	#2023	1	1	yes	characteristic
122	C₂×C₇₁	#2187	1	1	yes	characteristic
123	C₂×C₇₁	#2188	1	1	yes	characteristic
124	C₂×C₇₁	#2225	1	1	yes	characteristic
125	C₂×C₇₁	#2227	1	1	yes	characteristic
126	C₂×C₇₁	#2241	1	1	yes	characteristic
127	C₂×C₇₁	#2243	1	1	yes	characteristic
128	C₂×#500	—	71	1	no	—
129	C₂×#358	—	71	2	no	—
130	#1969	D₇₁	1	1	yes	characteristic
131	#1380	—	71	2	no	—
132	#1374	—	71	1	no	—
133	C₇₁	#13364	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	—	142	1	no	—
136	C₂²×C₁₆	—	71	2	no	—
137	#500	—	142	4	no	—
138	C₂×#107	—	71	2	no	—
139	C₂×#107	—	142	1	no	—
140	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
141	C₂×#103	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
143	#391	—	2	1	no	—
144	#358	—	142	4	no	—
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	284	1	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	1	no	—
150	C₂×Q₁₆	Dic₁₄₂	1	2	yes	complemented
151	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
153	C₄⋊Q₈	—	284	2	no	—
154	C₄⋊Q₈	—	142	4	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	284	1	no	—
157	C₂×C₄⋊C₄	—	142	2	no	—
158	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
159	C₂×C₁₆	—	142	2	no	—
160	C₂×C₁₆	—	142	2	no	—
161	C₂×C₁₆	—	142	2	no	—
162	#107	—	2	2	no	—
163	#107	—	71	4	no	—
164	#107	—	142	4	no	—
165	#107	—	142	2	no	—
166	#103	—	4	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	284	4	no	—
177	C₂×Q₈	—	568	2	no	—
178	C₂×Q₈	—	568	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	284	2	no	—
182	C₂²×C₄	#11765850	1	1	yes	characteristic
183	C₂²×C₄	#11765850	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	#11765853	1	2	yes	—
190	C₂×C₈	D₄×D₇₁	1	2	yes	—
191	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
192	C₂×C₈	#11765864	1	2	yes	—
193	C₂×C₈	#11765865	1	1	yes	characteristic
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	284	4	no	—
197	C₄⋊C₄	—	284	2	no	—
198	C₄⋊C₄	—	142	4	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	284	2	no	—
202	C₂³	C₂×#11765850	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	568	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	284	4	no	—
219	C₂×C₄	—	568	2	no	—
220	C₂×C₄	—	568	2	no	—
221	C₂×C₄	#14556071	1	2	yes	—
222	C₂×C₄	#14556074	1	2	yes	—
223	C₂×C₄	#14556096	1	1	yes	characteristic
224	C₂×C₄	#14556107	1	1	yes	characteristic
225	C₂×C₄	#14556115	1	1	yes	characteristic
226	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
227	C₂×C₄	#14556153	1	1	yes	characteristic
228	C₂×C₄	#14556158	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#14556105	1	2	yes	—
231	C₈	#14556114	1	2	yes	—
232	C₂²	#17986132	1	1	yes	characteristic
233	C₂²	#17986279	1	1	yes	characteristic
234	C₂²	#17986345	1	1	yes	characteristic
235	C₂²	C₂×#14556071	1	1	yes	characteristic
236	C₂²	C₂×#14556074	1	1	yes	characteristic
237	C₂²	#17986434	1	1	yes	characteristic
238	C₂²	#17986439	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	568	4	no	—
244	C₄	#17985911	1	1	yes	characteristic
245	C₄	#17986137	1	1	yes	characteristic
246	C₄	#17986289	1	1	yes	characteristic
247	C₄	#17986358	1	1	yes	characteristic
248	C₂	#22589476	1	1	yes	characteristic
249	C₂	#22590268	1	1	yes	characteristic
250	C₂	#22590273	1	1	yes	characteristic
251	C₂	#22590482	1	1	yes	characteristic
252	C₂	#22590492	1	1	yes	characteristic
253	C₂	#22590567	1	1	yes	characteristic
254	C₂	#22590576	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1969	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#1969	D₇₁
hypocenter	C₇₁	#13364
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₇₁	Q₈	semidirect
133. C₇₁	#13364	semidirect
150. C₂×Q₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
C₂×#17985803	2
#22586431	2
#22586418	2
C₇₁×#1969	2
#13364	71