Subgroups — #31901670

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986379	C₂	1	1	yes	maximal, characteristic
2	#22589292	C₂	1	2	yes	maximal
3	C₂×#17985746	C₂	1	2	yes	maximal
4	#22586175	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈₄	—	2	1	no	—
7	C₂²×Dic₂₈₄	—	2	1	no	—
8	C₂×#14556128	C₂²	1	2	yes	—
9	#17986379	—	2	4	no	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#17985761	—	2	2	no	—
12	#17985761	—	2	2	no	—
13	#17985746	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	—	2	1	no	—
19	C₂²×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₂₈₄	—	2	4	no	—
21	C₂×Dic₂₈₄	—	2	4	no	—
22	C₂×Dic₂₈₄	—	4	2	no	—
23	C₂×Dic₂₈₄	—	4	2	no	—
24	C₂×Dic₂₈₄	—	4	2	no	—
25	C₂×Dic₂₈₄	—	4	2	no	—
26	C₂²×C₇₁⋊C₈	—	2	2	no	—
27	#14556128	—	2	2	no	—
28	#14556128	—	2	2	no	—
29	#14556128	—	2	2	no	—
30	#14556128	D₄	1	4	yes	—
31	C₂×C₄⋊Dic₇₁	—	2	2	no	—
32	C₂×C₄⋊Dic₇₁	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
35	C₂²×C₈×C₇₁	—	2	1	no	—
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	—	2	1	no	—
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	—	2	2	no	—
43	C₄⋊C₈×C₇₁	—	4	2	no	—
44	C₄×C₈×C₇₁	—	2	2	no	—
45	#95×C₇₁	—	2	1	no	—
46	C₂×Dic₁₄₂	—	2	4	no	—
47	C₂×Dic₁₄₂	—	4	2	no	—
48	C₂×Dic₁₄₂	—	4	2	no	—
49	C₂×Dic₁₄₂	—	4	2	no	—
50	C₂×Dic₁₄₂	—	4	2	no	—
51	C₂×Dic₁₄₂	—	4	2	no	—
52	C₂²×Dic₇₁	—	4	2	no	—
53	Dic₂₈₄	—	4	8	no	—
54	Dic₂₈₄	—	4	8	no	—
55	C₂×C₇₁⋊C₈	—	4	2	no	—
56	C₂×C₇₁⋊C₈	—	4	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₄⋊Dic₇₁	—	2	4	no	—
59	C₄⋊Dic₇₁	—	2	4	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₄⋊Dic₇₁	—	4	2	no	—
62	C₂²×C₄×C₇₁	—	2	1	no	—
63	C₂²×C₄×C₇₁	—	2	1	no	—
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	2	1	no	—
70	C₂×C₈×C₇₁	—	2	1	no	—
71	C₂×C₈×C₇₁	—	4	1	no	—
72	C₂×C₈×C₇₁	—	4	1	no	—
73	C₂×C₈×C₇₁	—	4	1	no	—
74	C₂×C₈×C₇₁	D₈	1	2	yes	—
75	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
76	C₄⋊C₄×C₇₁	—	2	1	no	—
77	C₄⋊C₄×C₇₁	—	2	2	no	—
78	C₄⋊C₄×C₇₁	Q₁₆	1	2	yes	complemented
79	C₄²×C₇₁	—	4	1	no	—
80	C₄²×C₇₁	—	4	1	no	—
81	C₄²×C₇₁	C₂×D₄	1	2	yes	—
82	C₄²×C₇₁	#41	1	2	yes	—
83	C₂³×C₇₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₁	—	2	1	no	—
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	4	1	no	—
88	C₂×C₄×C₇₁	—	4	1	no	—
89	C₂×C₄×C₇₁	—	4	1	no	—
90	C₂×C₄×C₇₁	—	4	1	no	—
91	C₂×C₄×C₇₁	—	4	1	no	—
92	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#121	1	2	yes	—
94	C₂×C₄×C₇₁	#123	1	2	yes	—
95	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
100	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#137	1	2	yes	—
103	C₂×Dic₇₁	—	4	4	no	—
104	C₂×Dic₇₁	—	8	2	no	—
105	C₂×Dic₇₁	—	8	2	no	—
106	Dic₁₄₂	—	4	4	no	—
107	Dic₁₄₂	—	4	4	no	—
108	Dic₁₄₂	—	8	2	no	—
109	Dic₁₄₂	—	8	2	no	—
110	C₈×C₇₁	—	2	2	no	—
111	C₈×C₇₁	—	2	2	no	—
112	C₈×C₇₁	—	8	1	no	—
113	C₇₁⋊C₈	—	8	2	no	—
114	C₂²×C₇₁	#394	1	1	yes	characteristic
115	C₂²×C₇₁	#459	1	1	yes	characteristic
116	C₂²×C₇₁	#460	1	1	yes	characteristic
117	C₂²×C₇₁	#467	1	1	yes	characteristic
118	C₂²×C₇₁	#470	1	1	yes	characteristic
119	C₂²×C₇₁	#473	1	1	yes	characteristic
120	C₂²×C₇₁	#473	1	1	yes	characteristic
121	C₄×C₇₁	—	2	2	no	—
122	C₄×C₇₁	—	4	2	no	—
123	C₄×C₇₁	—	8	1	no	—
124	C₄×C₇₁	#447	1	1	yes	characteristic
125	C₄×C₇₁	#450	1	1	yes	characteristic
126	C₄×C₇₁	#462	1	2	yes	—
127	C₄×C₇₁	#464	1	2	yes	—
128	C₄×C₇₁	#486	1	1	yes	characteristic
129	C₄×C₇₁	#490	1	1	yes	characteristic
130	Dic₇₁	—	8	4	no	—
131	#10538	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	#1669	1	1	yes	characteristic
133	C₂×C₇₁	#1670	1	1	yes	characteristic
134	C₂×C₇₁	#1689	1	1	yes	characteristic
135	C₂×C₇₁	#1690	1	1	yes	characteristic
136	C₂×C₇₁	#2041	1	1	yes	characteristic
137	C₂×C₇₁	#2047	1	1	yes	characteristic
138	C₂×C₇₁	#2056	1	1	yes	characteristic
139	C₂×#494	—	71	1	no	—
140	#1988	—	71	2	no	—
141	#1939	D₇₁	1	1	yes	characteristic
142	C₂×#332	—	71	2	no	—
143	#1277	—	71	1	no	—
144	C₇₁	#10538	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₁₆	—	142	1	no	—
146	C₂²×Q₁₆	—	142	1	no	—
147	C₂×C₄⋊Q₈	—	71	2	no	—
148	#494	—	142	4	no	—
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	71	2	no	—
151	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
154	#340	—	142	2	no	—
155	#340	—	142	2	no	—
156	#336	—	2	1	no	—
157	#332	—	142	4	no	—
158	C₂²×Q₈	—	142	2	no	—
159	C₂×Q₁₆	—	284	2	no	—
160	C₂×Q₁₆	—	284	2	no	—
161	C₂×Q₁₆	—	284	2	no	—
162	C₂×Q₁₆	—	284	2	no	—
163	C₂×Q₁₆	—	142	4	no	—
164	C₂×Q₁₆	—	142	4	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	142	2	no	—
167	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	71	4	no	—
170	C₄⋊Q₈	—	142	2	no	—
171	C₄⋊Q₈	—	142	2	no	—
172	C₄⋊Q₈	—	142	2	no	—
173	C₂×C₄⋊C₄	—	142	2	no	—
174	C₂×C₄⋊C₄	—	142	2	no	—
175	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
176	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	4	2	no	—
181	C₄⋊C₈	—	142	2	no	—
182	C₄⋊C₈	—	142	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	2	1	no	—
185	C₂×Q₈	—	284	2	no	—
186	C₂×Q₈	—	284	2	no	—
187	C₂×Q₈	—	284	2	no	—
188	C₂×Q₈	—	284	2	no	—
189	C₂×Q₈	—	284	2	no	—
190	C₂×Q₈	—	142	4	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	284	2	no	—
194	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	C₂²×C₄	#11765865	1	1	yes	characteristic
197	Q₁₆	—	284	8	no	—
198	Q₁₆	—	284	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	284	2	no	—
207	C₂×C₈	—	284	2	no	—
208	C₂×C₈	—	284	2	no	—
209	C₂×C₈	#11765850	1	2	yes	—
210	C₂×C₈	#11765852	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	284	2	no	—
214	C₄⋊C₄	—	284	2	no	—
215	C₄⋊C₄	—	142	4	no	—
216	C₄⋊C₄	—	142	4	no	—
217	C₄⋊C₄	Dic₂₈₄	1	2	yes	complemented
218	C₄²	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	D₄×D₇₁	1	2	yes	—
221	C₄²	#11765866	1	2	yes	—
222	C₂³	#14556102	1	1	yes	characteristic
223	Q₈	—	284	4	no	—
224	Q₈	—	284	4	no	—
225	Q₈	—	568	2	no	—
226	Q₈	—	568	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	284	4	no	—
236	C₂×C₄	—	568	2	no	—
237	C₂×C₄	—	568	2	no	—
238	C₂×C₄	#14556096	1	1	yes	characteristic
239	C₂×C₄	#14556103	1	2	yes	—
240	C₂×C₄	#14556113	1	2	yes	—
241	C₂×C₄	#14556117	1	1	yes	characteristic
242	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
243	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
244	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
245	C₂×C₄	#14556150	1	1	yes	characteristic
246	C₂×C₄	#14556153	1	1	yes	characteristic
247	C₂×C₄	#14556153	1	1	yes	characteristic
248	C₂×C₄	#14556161	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	568	2	no	—
253	C₂²	#17985916	1	1	yes	characteristic
254	C₂²	#17986194	1	1	yes	characteristic
255	C₂²	#17986199	1	1	yes	characteristic
256	C₂²	#17986240	1	1	yes	characteristic
257	C₂²	#17986257	1	1	yes	characteristic
258	C₂²	#17986274	1	1	yes	characteristic
259	C₂²	#17986274	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	1	no	—
263	C₄	—	568	4	no	—
264	C₄	#17986132	1	1	yes	characteristic
265	C₄	#17986149	1	1	yes	characteristic
266	C₄	#17986210	1	2	yes	—
267	C₄	#17986222	1	2	yes	—
268	C₄	#17986347	1	1	yes	characteristic
269	C₄	#17986365	1	1	yes	characteristic
270	C₂	#22587705	1	1	yes	characteristic
271	C₂	#22587710	1	1	yes	characteristic
272	C₂	#22587829	1	1	yes	characteristic
273	C₂	#22587834	1	1	yes	characteristic
274	C₂	#22589574	1	1	yes	characteristic
275	C₂	#22589608	1	1	yes	characteristic
276	C₂	#22589663	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1939	C₂
socle, prime radical	C₂³×C₇₁	#121
hypercenter	#1939	D₇₁
hypocenter	C₇₁	#10538
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
78. C₄⋊C₄×C₇₁	Q₁₆	semidirect
144. C₇₁	#10538	semidirect
217. C₄⋊C₄	Dic₂₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986379	2
#22589292	2
C₂×#17985746	2
#22586175	2
C₇₁×#1939	2
#10538	71