Subgroups — #31969270

Order 18176.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598926	C₂	1	1	yes	maximal, characteristic
2	#22597287	C₂	1	2	yes	maximal
3	#22587250	C₂	1	2	yes	maximal, complemented
4	#22587213	C₂	1	4	yes	maximal, complemented
5	#22587160	C₂	1	2	yes	maximal, complemented
6	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
7	#22586744	C₂	1	2	yes	maximal
8	C₇₁×#1451	C₂	1	1	yes	maximal, characteristic
9	#17986556	C₂²	1	2	yes	—
10	#17986404	C₄	1	1	yes	characteristic
11	#17986379	C₄	1	1	yes	characteristic
12	#17986350	C₄	1	2	yes	—
13	#17986240	C₄	1	2	yes	—
14	#17986240	C₄	1	2	yes	—
15	#17986129	C₂²	1	4	yes	—
16	#17986127	—	2	4	no	—
17	C₈×Dic₁₄₂	C₂²	1	4	yes	—
18	C₄×Dic₂₈₄	C₂²	1	2	yes	—
19	C₄×Dic₂₈₄	C₂²	1	2	yes	—
20	#17986027	—	2	2	no	—
21	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
22	#17985957	—	2	2	no	—
23	#17985750	C₂²	1	1	yes	characteristic
24	#17985750	C₂²	1	2	yes	—
25	#17985750	C₂²	1	1	yes	characteristic
26	#17985731	C₂²	1	4	yes	—
27	#17985731	C₂²	1	4	yes	—
28	#431×C₇₁	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
31	#322×C₇₁	C₂²	1	2	yes	—
32	#322×C₇₁	C₂²	1	2	yes	—
33	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
35	C₂²×C₇₁⋊C₈	—	2	2	no	—
36	#14556128	C₂×C₄	1	2	yes	—
37	#14556128	C₂×C₄	1	2	yes	—
38	#14556120	—	2	2	no	—
39	#14556107	—	2	2	no	—
40	#14556107	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₄₂	—	2	2	no	—
42	C₄×Dic₁₄₂	C₂³	1	4	yes	—
43	#14556087	—	2	2	no	—
44	#14556062	C₂×C₄	1	2	yes	—
45	#14556062	C₂×C₄	1	2	yes	—
46	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
49	Dic₇₁⋊C₈	—	2	4	no	—
50	Dic₇₁⋊C₈	—	2	4	no	—
51	#14556053	C₂×C₄	1	4	yes	—
52	#14556053	C₂×C₄	1	4	yes	—
53	#14556038	—	2	4	no	—
54	C₈×Dic₇₁	—	2	4	no	—
55	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
56	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₁	—	2	1	no	—
60	C₂²⋊C₈×C₇₁	—	2	1	no	—
61	#97×C₇₁	D₄	1	2	yes	—
62	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₁	C₂³	1	2	yes	—
65	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₁	C₂³	1	2	yes	—
68	C₂×Dic₁₄₂	—	2	2	no	—
69	C₂×Dic₁₄₂	C₂²×C₄	1	4	yes	—
70	Dic₂₈₄	—	2	4	no	—
71	Dic₂₈₄	—	4	2	no	—
72	C₂×C₇₁⋊C₈	—	2	2	no	—
73	C₂×C₇₁⋊C₈	—	2	2	no	—
74	C₂×C₇₁⋊C₈	—	4	4	no	—
75	C₄⋊Dic₇₁	—	2	2	no	—
76	C₄⋊Dic₇₁	—	2	2	no	—
77	C₄⋊Dic₇₁	—	2	2	no	—
78	C₄⋊Dic₇₁	C₂²×C₄	1	4	yes	—
79	Dic₇₁⋊C₄	—	2	4	no	—
80	Dic₇₁⋊C₄	—	2	4	no	—
81	C₂²⋊Dic₇₁	—	2	4	no	—
82	C₄×Dic₇₁	—	2	4	no	—
83	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
86	M₁₆×C₇₁	—	2	4	no	—
87	C₂×C₈×C₇₁	—	2	2	no	—
88	C₂×C₈×C₇₁	—	2	2	no	—
89	C₂×C₈×C₇₁	—	4	1	no	—
90	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
95	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₁	#41	1	1	yes	characteristic
98	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
99	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#137	1	4	yes	—
107	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₇₁	—	2	4	no	—
111	C₂×Dic₇₁	—	2	4	no	—
112	Dic₁₄₂	—	2	4	no	—
113	Dic₁₄₂	—	2	4	no	—
114	Dic₁₄₂	—	8	2	no	—
115	C₈×C₇₁	—	2	1	no	—
116	C₈×C₇₁	—	2	1	no	—
117	C₈×C₇₁	—	2	1	no	—
118	C₈×C₇₁	—	2	4	no	—
119	C₈×C₇₁	—	4	2	no	—
120	C₈×C₇₁	#131	1	2	yes	—
121	C₇₁⋊C₈	—	4	2	no	—
122	C₇₁⋊C₈	—	4	2	no	—
123	C₂²×C₇₁	—	2	1	no	—
124	C₂²×C₇₁	—	2	1	no	—
125	C₂²×C₇₁	#441	1	2	yes	—
126	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₇₁	—	2	1	no	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	#441	1	2	yes	—
131	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
132	C₄×C₇₁	#568	1	1	yes	characteristic
133	C₄×C₇₁	C₂×#137	1	2	yes	—
134	Dic₇₁	—	4	4	no	—
135	Dic₇₁	—	4	4	no	—
136	#18072	—	71	1	no	maximal, Sylow
137	C₂×C₇₁	—	2	2	no	—
138	C₂×C₇₁	#2935	1	1	yes	characteristic
139	C₂×C₇₁	C₂×#441	1	1	yes	characteristic
140	C₂×C₇₁	#2957	1	1	yes	characteristic
141	#3170	—	71	1	no	—
142	#2962	—	71	2	no	—
143	#1593	—	71	2	no	—
144	#1587	—	71	4	no	—
145	#1579	—	71	2	no	—
146	C₂×#334	—	71	1	no	—
147	#1481	—	71	2	no	—
148	#1451	D₇₁	1	1	yes	characteristic
149	C₇₁	#18072	1	1	yes	characteristic, complemented, Sylow
150	#533	—	71	2	no	—
151	#500	—	71	1	no	—
152	#494	—	71	1	no	—
153	#487	—	71	2	no	—
154	#467	—	71	2	no	—
155	#467	—	71	2	no	—
156	#446	—	71	4	no	—
157	#446	—	142	4	no	—
158	C₈×Q₈	—	71	4	no	—
159	C₄×Q₁₆	—	71	2	no	—
160	C₄×Q₁₆	—	71	2	no	—
161	#431	C₂×D₇₁	1	1	yes	characteristic
162	#424	—	142	2	no	—
163	C₂×C₄⋊C₈	—	71	2	no	—
164	#405	—	142	2	no	—
165	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
167	#334	—	71	1	no	—
168	#334	—	71	2	no	—
169	#334	—	71	1	no	—
170	Q₈⋊C₈	—	71	4	no	—
171	Q₈⋊C₈	—	71	4	no	—
172	#322	C₂×D₇₁	1	2	yes	—
173	#322	C₂×D₇₁	1	2	yes	—
174	C₂×Q₁₆	—	71	2	no	—
175	C₂×Q₁₆	—	71	2	no	—
176	C₂×M₁₆	D₂₈₄	1	2	yes	—
177	C₂²×C₈	—	142	2	no	—
178	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
180	C₄⋊Q₈	—	71	2	no	—
181	C₄⋊Q₈	—	71	2	no	—
182	#125	—	142	2	no	—
183	C₂²⋊Q₈	—	71	4	no	—
184	C₂²⋊Q₈	—	142	2	no	—
185	C₄×Q₈	—	71	4	no	—
186	C₄×Q₈	—	142	2	no	—
187	#117	—	142	2	no	—
188	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
189	#107	—	71	2	no	—
190	#107	—	71	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	71	2	no	—
193	C₄⋊C₈	—	71	2	no	—
194	C₄⋊C₈	—	142	4	no	—
195	C₄⋊C₈	—	142	4	no	—
196	C₄⋊C₈	—	142	2	no	—
197	#103	—	71	4	no	—
198	#103	—	71	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	142	4	no	—
201	#97	D₂₈₄	1	2	yes	—
202	C₄×C₈	—	142	4	no	—
203	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
204	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₇₁	1	2	yes	—
206	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₁	1	2	yes	—
209	C₂×Q₈	—	71	4	no	—
210	C₂×Q₈	—	142	2	no	—
211	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
213	C₂²×C₄	#11765865	1	1	yes	characteristic
214	Q₁₆	—	284	2	no	—
215	Q₁₆	—	142	4	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	284	4	no	—
221	C₂×C₈	—	142	2	no	—
222	C₂×C₈	—	142	2	no	—
223	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₈₄	1	4	yes	—
228	C₄⋊C₄	—	71	4	no	—
229	C₄⋊C₄	—	142	4	no	—
230	C₄⋊C₄	—	142	4	no	—
231	C₄⋊C₄	—	142	2	no	—
232	C₄⋊C₄	—	142	2	no	—
233	C₄⋊C₄	—	142	2	no	—
234	C₂²⋊C₄	—	142	4	no	—
235	C₄²	—	142	4	no	—
236	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
237	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
238	C₄²	#11765865	1	1	yes	characteristic
239	C₄²	C₂³×D₇₁	1	1	yes	characteristic
240	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
241	Q₈	—	568	2	no	—
242	Q₈	—	142	4	no	—
243	Q₈	—	142	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	142	4	no	—
248	C₂×C₄	—	142	4	no	—
249	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
250	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
252	C₂×C₄	#14556161	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
255	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	284	2	no	—
262	C₈	—	284	2	no	—
263	C₈	#14556136	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#17986107	1	2	yes	—
267	C₂²	C₂×C₄×D₂₈₄	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	284	4	no	—
272	C₄	—	284	4	no	—
273	C₄	#17986107	1	2	yes	—
274	C₄	C₂×#14556136	1	1	yes	characteristic
275	C₄	#17986765	1	1	yes	characteristic
276	C₄	C₂×#14556161	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#22597086	1	1	yes	characteristic
279	C₂	C₂×#17986107	1	1	yes	characteristic
280	C₂	#22597258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#1451	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#1451	D₇₁
hypocenter	C₇₁	#18072
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #22587250	C₂	semidirect
4. #22587213	C₂	semidirect
5. #22587160	C₂	semidirect
149. C₇₁	#18072	semidirect

Maximal subgroups

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

subgroup	index
#22598926	2
#22597287	2
#22587250	2
#22587213	2
#22587160	2
C₂×#17985750	2
#22586744	2
C₇₁×#1451	2
#18072	71