Subgroups — #31888673

Order 18176.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985751	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985749	C₂	1	1	yes	maximal, characteristic
5	#22586174	C₂	1	1	yes	maximal, characteristic
6	#22586174	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#17985754	—	2	1	no	—
14	#17985754	C₄	1	4	yes	—
15	#17985751	C₄	1	4	yes	complemented
16	#17985749	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₁	—	2	1	no	—
20	C₄³×C₇₁	C₂²	1	1	yes	characteristic
21	#336×C₇₁	—	2	1	no	—
22	C₂²×C₇₁⋊C₈	—	2	1	no	—
23	C₂²×C₇₁⋊C₈	—	2	1	no	—
24	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
28	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
29	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
30	C₄×C₇₁⋊C₈	—	2	1	no	—
31	C₄×C₇₁⋊C₈	—	2	1	no	—
32	C₄×C₇₁⋊C₈	—	4	2	no	—
33	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₇₁	—	2	1	no	—
36	C₂²×C₈×C₇₁	—	2	1	no	—
37	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	—	2	1	no	—
43	C₄⋊C₈×C₇₁	—	2	1	no	—
44	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
46	#97×C₇₁	—	4	2	no	—
47	C₂×C₇₁⋊C₈	—	2	4	no	—
48	C₂×C₇₁⋊C₈	—	2	2	no	—
49	C₂×C₇₁⋊C₈	—	2	2	no	—
50	C₂×C₇₁⋊C₈	—	2	2	no	—
51	C₂×C₇₁⋊C₈	—	2	4	no	—
52	C₂×C₇₁⋊C₈	—	4	1	no	—
53	C₂×C₇₁⋊C₈	—	4	1	no	—
54	C₂×C₇₁⋊C₈	—	4	1	no	—
55	C₂×C₇₁⋊C₈	—	4	1	no	—
56	C₂×C₇₁⋊C₈	C₄²	1	4	yes	—
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	—	4	1	no	—
65	C₂×C₈×C₇₁	—	4	1	no	—
66	C₂×C₈×C₇₁	—	4	1	no	—
67	C₂×C₈×C₇₁	—	4	1	no	—
68	C₄²×C₇₁	—	2	4	no	—
69	C₄²×C₇₁	—	2	4	no	—
70	C₄²×C₇₁	—	2	1	no	—
71	C₄²×C₇₁	—	2	1	no	—
72	C₄²×C₇₁	—	2	1	no	—
73	C₄²×C₇₁	—	2	1	no	—
74	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇₁	#41	1	1	yes	characteristic
77	C₄²×C₇₁	#41	1	1	yes	characteristic
78	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	2	2	no	—
85	C₂×C₄×C₇₁	#108	1	4	yes	—
86	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#131	1	4	yes	—
93	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
97	C₈×C₇₁	—	4	2	no	—
98	C₈×C₇₁	—	4	2	no	—
99	C₇₁⋊C₈	—	2	4	no	—
100	C₇₁⋊C₈	—	4	2	no	—
101	C₇₁⋊C₈	—	4	2	no	—
102	C₇₁⋊C₈	—	4	2	no	—
103	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₇₁	#424	1	2	yes	—
105	C₂²×C₇₁	#428	1	1	yes	characteristic
106	C₂²×C₇₁	#428	1	1	yes	characteristic
107	C₂²×C₇₁	C₂×#108	1	2	yes	—
108	C₄×C₇₁	—	2	4	no	—
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	—	4	2	no	—
112	C₄×C₇₁	#405	1	2	yes	—
113	C₄×C₇₁	#433	1	2	yes	—
114	C₄×C₇₁	#440	1	1	yes	characteristic
115	C₄×C₇₁	#440	1	1	yes	characteristic
116	C₄×C₇₁	#441	1	1	yes	characteristic
117	C₄×C₇₁	#442	1	1	yes	characteristic
118	#8279	—	71	1	no	maximal, Sylow
119	C₂×C₇₁	#1573	1	2	yes	—
120	C₂×C₇₁	#1574	1	2	yes	—
121	C₂×C₇₁	#1768	1	1	yes	characteristic
122	C₂×C₇₁	#1776	1	1	yes	characteristic
123	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
124	#1767	D₇₁	1	1	yes	characteristic
125	C₄×C₄⋊C₈	—	71	1	no	—
126	#1744	—	71	1	no	—
127	C₂×#335	—	71	1	no	—
128	C₂×#334	—	71	1	no	—
129	#1277	—	71	1	no	—
130	#1277	—	71	1	no	—
131	C₇₁	#8279	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊C₈	—	71	1	no	—
133	C₂×C₄⋊C₈	—	71	1	no	—
134	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	—	71	1	no	—
138	C₂×C₄×C₈	—	71	1	no	—
139	C₂×C₄×C₈	—	142	1	no	—
140	C₄³	C₂×D₇₁	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#336	—	71	4	no	—
143	#336	—	142	1	no	—
144	#335	—	71	4	no	—
145	#334	—	71	4	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	71	2	no	—
149	C₂²×C₈	—	71	2	no	—
150	C₂²×C₈	—	142	1	no	—
151	C₂²×C₈	—	142	1	no	—
152	C₂×C₄²	Dic₁₄₂	1	2	yes	—
153	C₂×C₄²	C₄×D₇₁	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	71	2	no	—
160	C₄⋊C₈	—	71	2	no	—
161	C₄⋊C₈	—	142	1	no	—
162	C₄⋊C₈	—	142	1	no	—
163	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
164	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
165	#97	—	4	2	no	—
166	C₄×C₈	—	284	2	no	—
167	C₄×C₈	—	71	2	no	—
168	C₄×C₈	—	71	2	no	—
169	C₄×C₈	—	142	1	no	—
170	C₄×C₈	—	142	1	no	—
171	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
172	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	284	1	no	—
183	C₂×C₈	—	284	1	no	—
184	C₂×C₈	—	284	1	no	—
185	C₂×C₈	—	284	1	no	—
186	C₂×C₈	—	71	4	no	—
187	C₂×C₈	—	142	2	no	—
188	C₂×C₈	—	142	2	no	—
189	C₂×C₈	—	142	2	no	—
190	C₂×C₈	—	142	4	no	—
191	C₂×C₈	—	142	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
199	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
200	C₄²	#11765865	1	1	yes	characteristic
201	C₄²	#11765865	1	1	yes	characteristic
202	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	#14556063	1	4	yes	—
210	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
211	C₂×C₄	#14556085	1	1	yes	characteristic
212	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
213	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
215	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
216	C₂×C₄	#14556136	1	4	yes	—
217	C₂×C₄	#14556153	1	1	yes	characteristic
218	C₂×C₄	#14556154	1	1	yes	characteristic
219	C₂×C₄	#14556158	1	1	yes	characteristic
220	C₂×C₄	#14556159	1	1	yes	characteristic
221	C₈	—	4	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	284	2	no	—
224	C₈	—	284	2	no	—
225	C₈	—	284	2	no	—
226	C₈	—	142	4	no	—
227	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
228	C₂²	#17986027	1	2	yes	—
229	C₂²	#17986039	1	1	yes	characteristic
230	C₂²	#17986042	1	1	yes	characteristic
231	C₂²	C₂×#14556063	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	#17985956	1	2	yes	—
237	C₄	#17986061	1	2	yes	—
238	C₄	#17986097	1	1	yes	characteristic
239	C₄	#17986099	1	1	yes	characteristic
240	C₄	#17986104	1	1	yes	characteristic
241	C₄	#17986109	1	1	yes	characteristic
242	C₂	#22587126	1	2	yes	—
243	C₂	#22587131	1	2	yes	—
244	C₂	#22588200	1	1	yes	characteristic
245	C₂	#22588238	1	1	yes	characteristic
246	C₂	C₄×#14556063	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1767	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1767	D₇₁
hypocenter	C₇₁	#8279
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
15. #17985751	C₄	semidirect
16. #17985749	C₄	semidirect
131. C₇₁	#8279	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₁⋊C₈)	2
#22588086	2
C₂×#17985751	2
C₂×#17985749	2
#22586174	2
C₇₁×#1767	2
#8279	71