Subgroups — #31913530

Order 18176.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	#22590096	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985814	C₂	1	2	yes	maximal
4	#22586450	C₂	1	2	yes	maximal
5	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₂	—	2	1	no	—
8	C₂×#14556068	C₂²	1	2	yes	—
9	C₄×Dic₂₈₄	—	2	1	no	—
10	C₄×Dic₂₈₄	—	2	1	no	—
11	C₄×Dic₂₈₄	C₄	1	2	yes	—
12	C₄×Dic₂₈₄	C₄	1	2	yes	—
13	C₂×#14556062	C₂²	1	1	yes	characteristic
14	C₂×#14556053	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
17	#17985814	—	2	4	no	—
18	C₂×#108×C₇₁	—	2	1	no	—
19	C₂×#108×C₇₁	C₂²	1	2	yes	—
20	#423×C₇₁	—	2	1	no	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₂₈₄	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	1	no	—
25	C₂×Dic₂₈₄	—	2	1	no	—
26	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₈₄	D₄	1	2	yes	—
28	C₂×Dic₂₈₄	Q₈	1	2	yes	—
29	C₄×Dic₁₄₂	—	2	2	no	—
30	C₄×Dic₁₄₂	—	2	2	no	—
31	C₄×Dic₁₄₂	—	4	1	no	—
32	C₄×Dic₁₄₂	—	4	1	no	—
33	C₂×C₄⋊Dic₇₁	—	2	1	no	—
34	C₂×Dic₇₁⋊C₄	—	4	1	no	—
35	C₂×C₄×Dic₇₁	—	4	1	no	—
36	#14556068	—	2	2	no	—
37	#14556068	—	2	2	no	—
38	C₂×C₇₁⋊C₁₆	—	2	2	no	—
39	#14556062	—	2	1	no	—
40	#14556062	C₂×C₄	1	2	yes	—
41	#14556053	—	2	2	no	—
42	#14556053	—	2	2	no	—
43	C₂×M₁₆×C₇₁	—	2	2	no	—
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
47	#108×C₇₁	—	2	2	no	—
48	#108×C₇₁	—	2	2	no	—
49	#108×C₇₁	—	4	2	no	—
50	C₄⋊C₈×C₇₁	—	4	1	no	—
51	C₄⋊C₈×C₇₁	—	4	1	no	—
52	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
56	C₂×Dic₁₄₂	—	2	2	no	—
57	C₂×Dic₁₄₂	—	2	2	no	—
58	C₂×Dic₁₄₂	—	2	2	no	—
59	C₂×Dic₁₄₂	—	4	1	no	—
60	C₂×Dic₁₄₂	—	4	1	no	—
61	C₂×Dic₁₄₂	—	4	1	no	—
62	C₂²×Dic₇₁	—	4	1	no	—
63	C₂²×Dic₇₁	—	4	1	no	—
64	Dic₂₈₄	—	2	2	no	—
65	Dic₂₈₄	—	2	2	no	—
66	Dic₂₈₄	—	2	2	no	—
67	Dic₂₈₄	C₄⋊C₄	1	4	yes	—
68	C₄⋊Dic₇₁	—	2	2	no	—
69	C₄⋊Dic₇₁	—	4	1	no	—
70	Dic₇₁⋊C₄	—	4	2	no	—
71	Dic₇₁⋊C₄	—	4	2	no	—
72	C₄×Dic₇₁	—	4	2	no	—
73	C₄×Dic₇₁	—	4	2	no	—
74	C₇₁⋊C₁₆	—	4	2	no	—
75	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₇₁	—	4	2	no	—
79	M₁₆×C₇₁	—	4	2	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	—	2	1	no	—
82	C₂×C₈×C₇₁	—	4	2	no	—
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
90	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
91	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₇₁	D₈	1	1	yes	characteristic
94	C₄²×C₇₁	D₈	1	1	yes	characteristic
95	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₁	—	2	2	no	—
97	C₂×C₄×C₇₁	—	4	1	no	—
98	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
101	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
108	C₂×Dic₇₁	—	4	2	no	—
109	C₂×Dic₇₁	—	4	2	no	—
110	C₂×Dic₇₁	—	4	2	no	—
111	C₂×Dic₇₁	—	4	2	no	—
112	C₂×Dic₇₁	—	8	2	no	—
113	Dic₁₄₂	—	2	4	no	—
114	Dic₁₄₂	—	4	2	no	—
115	Dic₁₄₂	—	4	2	no	—
116	Dic₁₄₂	—	4	2	no	—
117	C₈×C₇₁	—	2	1	no	—
118	C₈×C₇₁	—	2	1	no	—
119	C₈×C₇₁	—	8	2	no	—
120	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₇₁	#117	1	1	yes	characteristic
122	C₈×C₇₁	#121	1	1	yes	characteristic
123	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂²×C₇₁	—	2	1	no	—
125	C₂²×C₇₁	—	2	1	no	—
126	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇₁	#461	1	1	yes	characteristic
128	C₂²×C₇₁	#476	1	1	yes	characteristic
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	—	2	1	no	—
131	C₄×C₇₁	—	2	1	no	—
132	C₄×C₇₁	#362	1	2	yes	complemented
133	C₄×C₇₁	#380	1	1	yes	characteristic
134	C₄×C₇₁	#417	1	1	yes	characteristic
135	C₄×C₇₁	#459	1	1	yes	characteristic
136	C₄×C₇₁	#474	1	1	yes	characteristic
137	Dic₇₁	—	4	4	no	—
138	Dic₇₁	—	8	2	no	—
139	#12819	—	71	1	no	maximal, Sylow
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	—	2	1	no	—
142	C₂×C₇₁	#1802	1	1	yes	characteristic
143	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
144	C₂×C₇₁	#2148	1	1	yes	characteristic
145	C₂×C₄×Q₁₆	—	71	1	no	—
146	#2150	—	71	1	no	—
147	C₂×#362	—	71	2	no	—
148	#1848	D₇₁	1	1	yes	characteristic
149	#1387	—	71	2	no	—
150	C₇₁	#12819	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	71	1	no	—
152	C₂×C₄×Q₈	—	142	1	no	—
153	C₂×M₃₂	—	71	2	no	—
154	C₄×Q₁₆	—	71	2	no	—
155	C₄×Q₁₆	—	71	2	no	—
156	C₄×Q₁₆	—	142	1	no	—
157	C₄×Q₁₆	—	142	1	no	—
158	C₂×#108	—	2	1	no	—
159	C₂×#108	C₂×D₇₁	1	2	yes	—
160	C₂×#107	—	71	1	no	—
161	#423	—	2	1	no	—
162	C₂×#103	—	142	1	no	—
163	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
164	C₄⋊C₁₆	—	142	1	no	—
165	C₄⋊C₁₆	—	142	1	no	—
166	#362	—	142	4	no	—
167	C₂²×Q₈	—	142	1	no	—
168	C₂×Q₁₆	—	71	2	no	—
169	C₂×Q₁₆	—	71	2	no	—
170	C₂×Q₁₆	—	71	2	no	—
171	C₂×Q₁₆	—	142	1	no	—
172	C₂×Q₁₆	—	142	1	no	—
173	C₂×Q₁₆	—	142	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
177	C₄×Q₈	—	284	1	no	—
178	C₄×Q₈	—	284	1	no	—
179	C₄×Q₈	—	142	2	no	—
180	C₄×Q₈	—	142	2	no	—
181	C₂×C₄⋊C₄	—	284	1	no	—
182	C₂×C₄⋊C₄	—	142	1	no	—
183	C₂×C₄²	—	284	1	no	—
184	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
185	M₃₂	—	142	2	no	—
186	M₃₂	—	142	2	no	—
187	C₂×C₁₆	—	142	2	no	—
188	#108	—	2	2	no	—
189	#108	—	2	2	no	—
190	#108	—	4	2	no	—
191	#107	—	71	2	no	—
192	#107	—	142	1	no	—
193	C₄⋊C₈	—	4	1	no	—
194	C₄⋊C₈	—	4	1	no	—
195	#103	—	142	2	no	—
196	#103	—	142	2	no	—
197	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
198	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
199	C₄×C₈	D₂₈₄	1	1	yes	characteristic
200	C₄×C₈	D₂₈₄	1	1	yes	characteristic
201	C₂×Q₈	—	284	1	no	—
202	C₂×Q₈	—	284	1	no	—
203	C₂×Q₈	—	284	1	no	—
204	C₂×Q₈	—	142	2	no	—
205	C₂×Q₈	—	142	2	no	—
206	C₂×Q₈	—	142	2	no	—
207	C₂²×C₄	—	284	1	no	—
208	C₂²×C₄	—	284	1	no	—
209	C₂²×C₄	#11765852	1	1	yes	characteristic
210	C₂²×C₄	#11765852	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
212	Q₁₆	—	71	4	no	—
213	Q₁₆	—	142	2	no	—
214	Q₁₆	—	142	2	no	—
215	Q₁₆	—	142	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
222	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
224	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
227	C₂×C₈	#11765864	1	1	yes	characteristic
228	C₂×C₈	#11765866	1	1	yes	characteristic
229	C₄⋊C₄	—	284	2	no	—
230	C₄⋊C₄	—	284	2	no	—
231	C₄⋊C₄	—	284	1	no	—
232	C₄⋊C₄	—	142	2	no	—
233	C₄²	—	284	2	no	—
234	C₄²	—	284	2	no	—
235	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
236	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
237	C₄²	#11765850	1	1	yes	characteristic
238	C₄²	#11765850	1	1	yes	characteristic
239	C₁₆	—	284	2	no	—
240	C₂³	C₂×#11765852	1	1	yes	characteristic
241	Q₈	—	284	2	no	—
242	Q₈	—	284	2	no	—
243	Q₈	—	284	2	no	—
244	Q₈	—	142	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	284	2	no	—
248	C₂×C₄	—	284	2	no	—
249	C₂×C₄	—	284	2	no	—
250	C₂×C₄	—	284	2	no	—
251	C₂×C₄	—	568	2	no	—
252	C₂×C₄	#14556048	1	1	yes	characteristic
253	C₂×C₄	#14556048	1	1	yes	characteristic
254	C₂×C₄	#14556048	1	1	yes	characteristic
255	C₂×C₄	#14556048	1	1	yes	characteristic
256	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	#14556099	1	1	yes	characteristic
258	C₂×C₄	#14556104	1	1	yes	characteristic
259	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
260	C₂×C₄	#14556154	1	1	yes	characteristic
261	C₂×C₄	#14556159	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	8	2	no	—
265	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
266	C₈	#14556086	1	1	yes	characteristic
267	C₈	#14556103	1	1	yes	characteristic
268	C₈	#14556108	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#14556048	1	1	yes	characteristic
272	C₂²	#17986201	1	1	yes	characteristic
273	C₂²	#17986286	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	284	4	no	—
278	C₄	—	568	2	no	—
279	C₄	#17985814	1	2	yes	complemented
280	C₄	#17985856	1	1	yes	characteristic
281	C₄	#17985998	1	1	yes	characteristic
282	C₄	#17986191	1	1	yes	characteristic
283	C₄	#17986276	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#22588337	1	1	yes	characteristic
287	C₂	C₂×#17985814	1	1	yes	characteristic
288	C₂	#22590087	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556048
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#12819
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
132. C₄×C₇₁	#362	semidirect
150. C₇₁	#12819	semidirect
279. C₄	#17985814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₈₄	2
#22590096	2
C₂×#17985814	2
#22586450	2
C₇₁×#1848	2
#12819	71