Subgroups — #31915542

Order 18176.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985825	C₂	1	2	yes	maximal
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
4	#22586432	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₂×#14556062	C₂²	1	1	yes	characteristic
9	C₂×#14556062	C₂²	1	1	yes	characteristic
10	#17985869	—	2	1	no	—
11	C₄×C₄⋊Dic₇₁	—	2	1	no	—
12	#17985825	—	2	4	no	—
13	#17985762	—	2	1	no	—
14	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₇₁	C₄	1	4	yes	complemented
17	C₂×C₄⋊Dic₇₁	—	2	1	no	—
18	C₂×C₄⋊Dic₇₁	—	2	1	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	4	1	no	—
21	C₂×C₄⋊Dic₇₁	—	4	1	no	—
22	C₂×C₄×Dic₇₁	—	4	1	no	—
23	#14556062	—	2	2	no	—
24	#14556062	—	2	1	no	—
25	#14556062	—	2	1	no	—
26	#14556062	—	2	2	no	—
27	#14556062	—	4	2	no	—
28	#14556062	D₄	1	2	yes	—
29	#14556062	Q₈	1	2	yes	—
30	C₇₁⋊#95	—	4	1	no	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	—	2	2	no	—
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	C₂²×Dic₇₁	—	4	1	no	—
40	C₂²×Dic₇₁	—	4	1	no	—
41	C₂²×Dic₇₁	—	4	2	no	—
42	C₄⋊Dic₇₁	—	2	2	no	—
43	C₄⋊Dic₇₁	—	2	2	no	—
44	C₄⋊Dic₇₁	—	4	1	no	—
45	C₄⋊Dic₇₁	—	4	1	no	—
46	C₄⋊Dic₇₁	—	4	2	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₄⋊Dic₇₁	—	8	2	no	—
49	C₄⋊Dic₇₁	—	8	2	no	—
50	C₄×Dic₇₁	—	8	2	no	—
51	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
63	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₄²×C₇₁	Q₁₆	1	2	yes	—
65	C₁₆×C₇₁	—	2	4	no	—
66	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	—	2	1	no	—
68	C₂×C₄×C₇₁	—	2	1	no	—
69	C₂×C₄×C₇₁	#107	1	2	yes	—
70	C₂×C₄×C₇₁	#107	1	2	yes	—
71	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
72	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
79	C₂×Dic₇₁	—	4	2	no	—
80	C₂×Dic₇₁	—	4	2	no	—
81	C₂×Dic₇₁	—	8	1	no	—
82	C₂×Dic₇₁	—	8	1	no	—
83	C₂×Dic₇₁	—	8	1	no	—
84	C₂×Dic₇₁	—	8	1	no	—
85	C₂×Dic₇₁	—	8	2	no	—
86	C₂×Dic₇₁	—	8	2	no	—
87	C₂×Dic₇₁	—	8	2	no	—
88	C₈×C₇₁	—	2	2	no	—
89	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
90	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
91	C₈×C₇₁	#125	1	1	yes	characteristic
92	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
94	C₂²×C₇₁	#459	1	1	yes	characteristic
95	C₂²×C₇₁	#474	1	1	yes	characteristic
96	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
97	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
98	C₂²×C₇₁	#509	1	1	yes	characteristic
99	C₂²×C₇₁	#510	1	1	yes	characteristic
100	C₄×C₇₁	—	2	2	no	—
101	C₄×C₇₁	#366	1	4	yes	complemented
102	C₄×C₇₁	#389	1	1	yes	characteristic
103	C₄×C₇₁	#428	1	1	yes	characteristic
104	C₄×C₇₁	#462	1	1	yes	characteristic
105	C₄×C₇₁	#477	1	1	yes	characteristic
106	Dic₇₁	—	8	2	no	—
107	Dic₇₁	—	8	2	no	—
108	Dic₇₁	—	16	2	no	—
109	#13176	—	71	1	no	maximal, Sylow
110	C₂×C₇₁	#1947	1	1	yes	characteristic
111	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
112	C₂×C₇₁	#2159	1	1	yes	characteristic
113	C₂×C₇₁	#2206	1	1	yes	characteristic
114	C₂×C₇₁	#2209	1	1	yes	characteristic
115	C₂×C₇₁	#2224	1	1	yes	characteristic
116	C₂×C₇₁	#2227	1	1	yes	characteristic
117	C₂×#366	—	71	2	no	—
118	C₂×C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
119	#1845	—	71	1	no	—
120	C₄×#107	—	71	1	no	—
121	#1380	—	71	2	no	—
122	C₇₁	#13176	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
124	C₂×#107	—	71	1	no	—
125	C₂×#107	—	71	1	no	—
126	C₂×#107	—	71	2	no	—
127	C₂×#107	—	142	1	no	—
128	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
129	#384	—	142	1	no	—
130	C₄×C₄⋊C₄	—	142	1	no	—
131	#366	—	142	4	no	—
132	C₄⋊C₁₆	Dic₇₁	1	4	yes	complemented
133	#340	—	142	1	no	—
134	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	284	1	no	—
137	C₂×C₄⋊C₄	—	284	1	no	—
138	C₂×C₄⋊C₄	—	142	2	no	—
139	C₂×C₄⋊C₄	—	142	1	no	—
140	C₂×C₄⋊C₄	—	142	1	no	—
141	C₂×C₄²	—	284	1	no	—
142	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
146	#107	—	284	2	no	—
147	#107	—	71	2	no	—
148	#107	—	71	2	no	—
149	#107	—	142	2	no	—
150	#107	—	142	2	no	—
151	#107	—	142	1	no	—
152	#107	—	142	1	no	—
153	C₄×C₈	Dic₁₄₂	1	2	yes	—
154	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
155	#95	—	284	1	no	—
156	C₂²×C₄	—	284	1	no	—
157	C₂²×C₄	—	284	1	no	—
158	C₂²×C₄	—	284	2	no	—
159	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
160	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
165	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
166	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
167	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
168	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
169	C₂×C₈	#11765864	1	1	yes	characteristic
170	C₂×C₈	#11765866	1	1	yes	characteristic
171	C₄⋊C₄	—	284	1	no	—
172	C₄⋊C₄	—	284	1	no	—
173	C₄⋊C₄	—	284	2	no	—
174	C₄⋊C₄	—	284	2	no	—
175	C₄⋊C₄	—	568	2	no	—
176	C₄⋊C₄	—	568	2	no	—
177	C₄⋊C₄	—	142	2	no	—
178	C₄⋊C₄	—	142	2	no	—
179	C₄²	—	568	2	no	—
180	C₄²	C₄⋊Dic₇₁	1	2	yes	—
181	C₄²	Dic₂₈₄	1	2	yes	—
182	C₁₆	—	2	4	no	—
183	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	284	2	no	—
187	C₂×C₄	—	284	2	no	—
188	C₂×C₄	—	568	1	no	—
189	C₂×C₄	—	568	1	no	—
190	C₂×C₄	—	568	1	no	—
191	C₂×C₄	—	568	1	no	—
192	C₂×C₄	—	568	2	no	—
193	C₂×C₄	—	568	2	no	—
194	C₂×C₄	—	568	2	no	—
195	C₂×C₄	#14556062	1	2	yes	—
196	C₂×C₄	#14556062	1	2	yes	—
197	C₂×C₄	D₁₁₃₆	1	2	yes	—
198	C₂×C₄	Dic₅₆₈	1	2	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
200	C₂×C₄	#14556103	1	1	yes	characteristic
201	C₂×C₄	#14556108	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
203	C₂×C₄	#14556156	1	1	yes	characteristic
204	C₂×C₄	#14556161	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	D₄×Dic₇₁	1	1	yes	characteristic
207	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
208	C₈	#14556119	1	1	yes	characteristic
209	C₈	#14556127	1	1	yes	characteristic
210	C₂²	C₂×#14556062	1	1	yes	characteristic
211	C₂²	#17986195	1	1	yes	characteristic
212	C₂²	#17986280	1	1	yes	characteristic
213	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
214	C₂²	C₂×Dic₅₆₈	1	1	yes	characteristic
215	C₂²	#17986437	1	1	yes	characteristic
216	C₂²	#17986442	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	568	2	no	—
219	C₄	—	568	2	no	—
220	C₄	—	1136	2	no	—
221	C₄	#17985825	1	4	yes	complemented
222	C₄	#17985898	1	1	yes	characteristic
223	C₄	#17986041	1	1	yes	characteristic
224	C₄	#17986210	1	1	yes	characteristic
225	C₄	#17986295	1	1	yes	characteristic
226	C₂	#22589099	1	1	yes	characteristic
227	C₂	C₂×#17985825	1	1	yes	characteristic
228	C₂	#22590128	1	1	yes	characteristic
229	C₂	#22590376	1	1	yes	characteristic
230	C₂	#22590391	1	1	yes	characteristic
231	C₂	#22590478	1	1	yes	characteristic
232	C₂	#22590493	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄⋊C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#13176
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
16. C₄⋊C₁₆×C₇₁	C₄	semidirect
101. C₄×C₇₁	#366	semidirect
122. C₇₁	#13176	semidirect
132. C₄⋊C₁₆	Dic₇₁	semidirect
221. C₄	#17985825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985825	2
#22588500	2
C₄×#14556062	2
#22586432	2
C₂×C₄⋊C₁₆×C₇₁	2
#13176	71