Subgroups — #31888707

Order 18176.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556059	C₂	1	1	yes	maximal, characteristic
2	#22588207	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985751	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	#22586181	C₂	1	1	yes	maximal, characteristic
6	#22586179	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556059	C₄	1	2	yes	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14556038	—	2	1	no	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
14	#17985761	C₄	1	2	yes	—
15	#17985753	—	2	1	no	—
16	#17985751	C₄	1	4	yes	complemented
17	#17985721	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	#336×C₇₁	—	2	1	no	—
23	C₂²×C₇₁⋊C₈	—	2	1	no	—
24	C₂²×C₇₁⋊C₈	—	2	1	no	—
25	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₇₁	—	2	1	no	—
29	#14556059	—	2	4	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
31	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
32	#14556038	—	4	2	no	—
33	C₄×C₇₁⋊C₈	—	2	1	no	—
34	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
37	C₇₁⋊#95	—	2	1	no	—
38	C₂²×C₈×C₇₁	—	2	1	no	—
39	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₁	—	2	1	no	—
43	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₁	—	2	1	no	—
45	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
46	#97×C₇₁	—	4	2	no	—
47	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₁	Q₈	1	2	yes	—
49	#95×C₇₁	—	2	1	no	—
50	C₂²×Dic₇₁	—	2	2	no	—
51	C₂×C₇₁⋊C₈	—	2	2	no	—
52	C₂×C₇₁⋊C₈	—	2	2	no	—
53	C₂×C₇₁⋊C₈	—	2	2	no	—
54	C₂×C₇₁⋊C₈	—	4	2	no	—
55	C₂×C₇₁⋊C₈	—	4	1	no	—
56	C₂×C₇₁⋊C₈	—	4	1	no	—
57	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
58	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
59	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
60	C₄⋊Dic₇₁	—	2	2	no	—
61	C₄⋊Dic₇₁	—	2	2	no	—
62	C₄×Dic₇₁	—	4	1	no	—
63	C₄×Dic₇₁	—	4	1	no	—
64	C₂²×C₄×C₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
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	—	2	1	no	—
72	C₂×C₈×C₇₁	—	2	2	no	—
73	C₂×C₈×C₇₁	—	4	1	no	—
74	C₂×C₈×C₇₁	—	4	1	no	—
75	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
77	C₄⋊C₄×C₇₁	—	2	2	no	—
78	C₄⋊C₄×C₇₁	—	2	2	no	—
79	C₄²×C₇₁	—	4	1	no	—
80	C₄²×C₇₁	—	4	1	no	—
81	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₇₁	#41	1	1	yes	characteristic
85	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	4	2	no	—
88	C₂×C₄×C₇₁	—	4	2	no	—
89	C₂×C₄×C₇₁	—	4	2	no	—
90	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇₁	#131	1	2	yes	—
98	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
101	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
102	C₂×Dic₇₁	—	4	2	no	—
103	C₂×Dic₇₁	—	4	2	no	—
104	C₂×Dic₇₁	—	4	2	no	—
105	C₈×C₇₁	—	2	2	no	—
106	C₈×C₇₁	—	2	2	no	—
107	C₈×C₇₁	—	4	2	no	—
108	C₇₁⋊C₈	—	2	2	no	—
109	C₇₁⋊C₈	—	4	2	no	—
110	C₇₁⋊C₈	—	4	2	no	—
111	C₇₁⋊C₈	#97	1	4	yes	complemented
112	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₇₁	#436	1	2	yes	—
114	C₂²×C₇₁	#440	1	1	yes	characteristic
115	C₂²×C₇₁	#440	1	1	yes	characteristic
116	C₂²×C₇₁	#444	1	2	yes	—
117	C₄×C₇₁	—	2	1	no	—
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	8	2	no	—
120	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
121	C₄×C₇₁	#405	1	1	yes	characteristic
122	C₄×C₇₁	#428	1	1	yes	characteristic
123	C₄×C₇₁	#428	1	1	yes	characteristic
124	C₄×C₇₁	#441	1	1	yes	characteristic
125	C₄×C₇₁	#442	1	1	yes	characteristic
126	C₄×C₇₁	#446	1	2	yes	—
127	Dic₇₁	—	8	2	no	—
128	#8286	—	71	1	no	maximal, Sylow
129	C₂×C₇₁	#1592	1	2	yes	—
130	C₂×C₇₁	#1592	1	2	yes	—
131	C₂×C₇₁	#1770	1	1	yes	characteristic
132	C₂×C₇₁	#1773	1	1	yes	characteristic
133	C₂×C₇₁	#1776	1	1	yes	characteristic
134	C₄×#106	—	71	1	no	—
135	#1770	—	71	1	no	—
136	#1770	D₇₁	1	1	yes	characteristic
137	C₂×#335	—	71	1	no	—
138	C₂×#322	—	71	1	no	—
139	#1279	—	71	1	no	—
140	#1279	—	71	1	no	—
141	C₇₁	#8286	1	1	yes	characteristic, complemented, Sylow
142	C₂×#106	—	71	2	no	—
143	C₂×C₄⋊C₈	—	71	1	no	—
144	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
145	C₂×#97	—	2	1	no	—
146	C₂×#97	—	142	1	no	—
147	C₂×C₄×C₈	—	71	1	no	—
148	C₂×C₄×C₈	—	71	1	no	—
149	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
150	C₄×C₄⋊C₄	—	71	1	no	—
151	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
152	#340	—	71	2	no	—
153	#336	—	2	1	no	—
154	#336	—	142	1	no	—
155	#335	—	71	4	no	—
156	#322	—	71	4	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	71	1	no	—
159	C₂²×C₈	—	71	1	no	—
160	C₂²×C₈	—	142	1	no	—
161	C₂²×C₈	—	142	1	no	—
162	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	71	2	no	—
165	C₂×C₄⋊C₄	C₄×D₇₁	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	142	1	no	—
168	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
169	#106	—	142	4	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	71	2	no	—
172	C₄⋊C₈	—	142	1	no	—
173	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
174	#97	—	4	2	no	—
175	#97	—	284	2	no	—
176	C₄×C₈	—	71	2	no	—
177	C₄×C₈	—	71	2	no	—
178	C₄×C₈	—	71	2	no	—
179	C₄×C₈	—	142	1	no	—
180	C₄×C₈	Dic₁₄₂	1	2	yes	—
181	C₄×C₈	C₄×D₇₁	1	2	yes	—
182	#95	—	2	1	no	—
183	#95	—	142	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	142	2	no	—
186	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
188	C₂²×C₄	#11765865	1	1	yes	characteristic
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	284	2	no	—
197	C₂×C₈	—	284	1	no	—
198	C₂×C₈	—	284	1	no	—
199	C₂×C₈	—	71	2	no	—
200	C₂×C₈	—	71	4	no	—
201	C₂×C₈	—	71	2	no	—
202	C₂×C₈	—	142	2	no	—
203	C₂×C₈	—	142	2	no	—
204	C₂×C₈	—	142	2	no	—
205	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
206	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	142	2	no	—
210	C₄⋊C₄	—	142	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	284	1	no	—
214	C₄²	—	284	1	no	—
215	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
217	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
218	C₄²	#11765865	1	1	yes	characteristic
219	C₂³	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	284	2	no	—
225	C₂×C₄	—	284	2	no	—
226	C₂×C₄	—	284	2	no	—
227	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#14556085	1	1	yes	characteristic
230	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
231	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
232	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
233	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
234	C₂×C₄	#14556136	1	2	yes	—
235	C₂×C₄	#14556153	1	1	yes	characteristic
236	C₂×C₄	#14556154	1	1	yes	characteristic
237	C₂×C₄	#14556158	1	1	yes	characteristic
238	C₂×C₄	#14556159	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	284	2	no	—
243	C₈	—	284	2	no	—
244	C₈	—	71	4	no	—
245	C₈	—	142	2	no	—
246	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
247	C₂²	#17986075	1	2	yes	—
248	C₂²	#17986097	1	1	yes	characteristic
249	C₂²	#17986099	1	1	yes	characteristic
250	C₂²	#17986119	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	—	568	2	no	—
255	C₄	#97×D₇₁	1	1	yes	characteristic
256	C₄	#17985956	1	1	yes	characteristic
257	C₄	#17986039	1	1	yes	characteristic
258	C₄	#17986042	1	1	yes	characteristic
259	C₄	#17986104	1	1	yes	characteristic
260	C₄	#17986109	1	1	yes	characteristic
261	C₄	#17986127	1	2	yes	—
262	C₂	#22587240	1	2	yes	—
263	C₂	#22587242	1	2	yes	—
264	C₂	#22588208	1	1	yes	characteristic
265	C₂	#22588225	1	1	yes	characteristic
266	C₂	#22588238	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×#1770	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1770	D₇₁
hypocenter	C₇₁	#8286
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
16. #17985751	C₄	semidirect
17. #17985721	C₄	semidirect
111. C₇₁⋊C₈	#97	semidirect
141. C₇₁	#8286	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556059	2
#22588207	2
C₂×#17985751	2
C₂×#17985721	2
#22586181	2
#22586179	2
C₇₁×#1770	2
#8286	71