Subgroups — #31890538

Order 18176.

434 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986129	C₂	1	1	yes	maximal, characteristic
2	#22588342	C₂	1	2	yes	maximal
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	#22586223	C₂	1	2	yes	maximal
5	C₇₁×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
7	#17986129	—	2	2	no	—
8	#17986129	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₁⋊C₈	—	2	1	no	—
12	C₂×#14556051	—	2	2	no	—
13	C₂×#14556038	—	2	1	no	—
14	#17985761	—	2	2	no	—
15	#17985752	—	2	2	no	—
16	C₂×#106×C₇₁	—	2	1	no	—
17	C₂×#106×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	#384×C₇₁	C₂²	1	2	yes	—
20	C₂²×Dic₁₄₂	D₄	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	—	2	1	no	—
22	C₂²×C₇₁⋊C₈	—	2	2	no	—
23	C₄×Dic₁₄₂	—	2	1	no	—
24	C₄×Dic₁₄₂	—	2	1	no	—
25	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₄₂	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
28	C₂×Dic₇₁⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₇₁	—	2	1	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₇₁⋊C₈	—	2	4	no	—
35	#14556051	—	4	4	no	—
36	#14556038	—	2	4	no	—
37	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
42	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	#106×C₇₁	—	4	2	no	—
44	#106×C₇₁	—	4	2	no	—
45	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₁	D₄	1	2	yes	—
47	C₂×Dic₁₄₂	—	2	2	no	—
48	C₂×Dic₁₄₂	—	2	2	no	—
49	C₂×Dic₁₄₂	—	2	1	no	—
50	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₇₁	—	2	1	no	—
52	C₂²×Dic₇₁	—	2	1	no	—
53	C₂×C₇₁⋊C₈	—	2	2	no	—
54	C₂×C₇₁⋊C₈	—	4	1	no	—
55	C₂×C₇₁⋊C₈	—	4	1	no	—
56	C₂×C₇₁⋊C₈	—	4	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₂×C₇₁⋊C₈	—	4	2	no	—
59	C₄⋊Dic₇₁	—	2	1	no	—
60	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
61	Dic₇₁⋊C₄	—	2	2	no	—
62	Dic₇₁⋊C₄	—	2	2	no	—
63	C₄×Dic₇₁	—	2	2	no	—
64	C₄×Dic₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	—	4	2	no	—
66	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₁	—	2	1	no	—
70	C₂×C₈×C₇₁	—	2	1	no	—
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
74	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
75	C₄⋊C₄×C₇₁	—	4	2	no	—
76	C₄⋊C₄×C₇₁	—	4	2	no	—
77	C₄⋊C₄×C₇₁	—	4	2	no	—
78	C₄⋊C₄×C₇₁	—	4	2	no	—
79	C₄⋊C₄×C₇₁	—	8	2	no	—
80	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₇₁	#41	1	1	yes	characteristic
84	C₂³×C₇₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₁	—	2	2	no	—
86	C₂×C₄×C₇₁	—	8	2	no	—
87	C₂×C₄×C₇₁	—	8	2	no	—
88	C₂×C₄×C₇₁	—	8	2	no	—
89	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₇₁	#137	1	2	yes	—
100	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
101	C₂×Dic₇₁	—	2	2	no	—
102	C₂×Dic₇₁	—	2	2	no	—
103	C₂×Dic₇₁	—	4	2	no	—
104	C₂×Dic₇₁	—	4	2	no	—
105	Dic₁₄₂	—	2	4	no	—
106	Dic₁₄₂	—	4	1	no	—
107	Dic₁₄₂	—	4	1	no	—
108	C₈×C₇₁	—	2	2	no	—
109	C₈×C₇₁	#103	1	4	yes	complemented
110	C₇₁⋊C₈	—	4	2	no	—
111	C₇₁⋊C₈	—	8	2	no	—
112	C₂²×C₇₁	#380	1	1	yes	characteristic
113	C₂²×C₇₁	#460	1	1	yes	characteristic
114	C₂²×C₇₁	#462	1	1	yes	characteristic
115	C₂²×C₇₁	#468	1	2	yes	—
116	C₂²×C₇₁	#471	1	2	yes	—
117	C₄×C₇₁	—	2	1	no	—
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	16	2	no	—
120	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
121	C₄×C₇₁	#417	1	1	yes	characteristic
122	C₄×C₇₁	#428	1	2	yes	—
123	C₄×C₇₁	#469	1	1	yes	characteristic
124	C₄×C₇₁	#470	1	1	yes	characteristic
125	C₄×C₇₁	#475	1	1	yes	characteristic
126	C₄×C₇₁	#477	1	1	yes	characteristic
127	Dic₇₁	—	4	2	no	—
128	Dic₇₁	—	4	2	no	—
129	#8651	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	#1686	1	2	yes	—
131	C₂×C₇₁	#1692	1	2	yes	—
132	C₂×C₇₁	#1803	1	1	yes	characteristic
133	C₂×C₇₁	#1809	1	1	yes	characteristic
134	C₂×C₇₁	#1815	1	1	yes	characteristic
135	C₂×#446	—	71	1	no	—
136	#1844	D₇₁	1	1	yes	characteristic
137	#1803	—	71	2	no	—
138	C₂×#335	—	71	1	no	—
139	#1296	—	71	2	no	—
140	C₇₁	#8651	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	—	71	1	no	—
142	#446	—	71	4	no	—
143	#446	—	142	2	no	—
144	C₂×#106	—	2	1	no	—
145	C₂×#106	—	2	1	no	—
146	C₂×C₄⋊C₈	—	71	1	no	—
147	C₂×C₄⋊C₈	—	71	2	no	—
148	C₂×C₄⋊C₈	—	142	1	no	—
149	C₂×#103	—	142	2	no	—
150	C₂×#97	—	142	1	no	—
151	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
152	#384	C₂×D₇₁	1	2	yes	—
153	#340	—	142	2	no	—
154	#335	—	142	2	no	—
155	C₂²×Q₈	—	71	1	no	—
156	C₂²×C₈	—	142	2	no	—
157	C₂²×C₈	—	142	1	no	—
158	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
160	C₄×Q₈	—	71	2	no	—
161	C₄×Q₈	—	71	2	no	—
162	C₄×Q₈	—	142	1	no	—
163	C₄×Q₈	—	142	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	71	1	no	—
168	C₂×C₄⋊C₄	—	142	1	no	—
169	C₂×C₄²	—	142	1	no	—
170	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
171	#106	—	4	2	no	—
172	#106	—	4	2	no	—
173	C₄⋊C₈	—	71	2	no	—
174	C₄⋊C₈	—	142	4	no	—
175	C₄⋊C₈	—	142	2	no	—
176	C₄⋊C₈	—	142	1	no	—
177	C₄⋊C₈	—	142	2	no	—
178	#103	—	284	4	no	—
179	#97	—	142	4	no	—
180	C₄×C₈	C₄×D₇₁	1	2	yes	—
181	C₄×C₈	D₂₈₄	1	2	yes	—
182	C₂×Q₈	—	71	2	no	—
183	C₂×Q₈	—	142	2	no	—
184	C₂×Q₈	—	142	2	no	—
185	C₂×Q₈	—	142	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	142	1	no	—
188	C₂²×C₄	—	142	1	no	—
189	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
191	C₂²×C₄	#11765864	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	284	1	no	—
195	C₂×C₈	—	284	1	no	—
196	C₂×C₈	—	284	2	no	—
197	C₂×C₈	—	284	2	no	—
198	C₂×C₈	—	284	2	no	—
199	C₂×C₈	—	142	2	no	—
200	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
202	C₂×C₈	#11765853	1	2	yes	—
203	C₂×C₈	#11765855	1	2	yes	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	—	71	2	no	—
210	C₄⋊C₄	—	142	2	no	—
211	C₄⋊C₄	—	142	1	no	—
212	C₄⋊C₄	—	142	2	no	—
213	C₄²	—	142	2	no	—
214	C₄²	—	142	2	no	—
215	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
216	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
217	C₄²	Q₈×D₇₁	1	1	yes	characteristic
218	C₄²	#11765866	1	1	yes	characteristic
219	C₂³	#14556099	1	1	yes	characteristic
220	Q₈	—	284	1	no	—
221	Q₈	—	284	1	no	—
222	Q₈	—	142	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	284	2	no	—
228	C₂×C₄	—	284	2	no	—
229	C₂×C₄	—	142	2	no	—
230	C₂×C₄	—	142	2	no	—
231	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
233	C₂×C₄	#14556086	1	1	yes	characteristic
234	C₂×C₄	#14556103	1	1	yes	characteristic
235	C₂×C₄	#14556104	1	1	yes	characteristic
236	C₂×C₄	#14556108	1	1	yes	characteristic
237	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
238	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
239	C₂×C₄	#14556154	1	1	yes	characteristic
240	C₂×C₄	#14556155	1	2	yes	—
241	C₂×C₄	#14556157	1	2	yes	—
242	C₂×C₄	#14556159	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	284	2	no	—
245	C₈	—	568	2	no	—
246	C₈	#14556051	1	4	yes	complemented
247	C₂²	#17985856	1	1	yes	characteristic
248	C₂²	#17986196	1	1	yes	characteristic
249	C₂²	#17986206	1	1	yes	characteristic
250	C₂²	#17986242	1	2	yes	—
251	C₂²	#17986259	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	16	2	no	—
255	C₄	—	284	2	no	—
256	C₄	—	284	2	no	—
257	C₄	C₂×#14556051	1	1	yes	characteristic
258	C₄	#17985998	1	1	yes	characteristic
259	C₄	#17986038	1	2	yes	—
260	C₄	#17986253	1	1	yes	characteristic
261	C₄	#17986258	1	1	yes	characteristic
262	C₄	#17986281	1	1	yes	characteristic
263	C₄	#17986291	1	1	yes	characteristic
264	C₂	#22587809	1	2	yes	—
265	C₂	#22587843	1	2	yes	—
266	C₂	#22588342	1	1	yes	characteristic
267	C₂	#22588370	1	1	yes	characteristic
268	C₂	#22588400	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556099
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1844	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1844	D₇₁
hypocenter	C₇₁	#8651
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
8. #17986129	C₄	semidirect
109. C₈×C₇₁	#103	semidirect
140. C₇₁	#8651	semidirect
246. C₈	#14556051	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986129	2
#22588342	2
C₂×#17985752	2
#22586223	2
C₇₁×#1844	2
#8651	71