Subgroups — #31916084

Order 18176.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
3	#22586450	C₂	1	4	yes	maximal
4	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	C₂×#14556068	C₂²	1	2	yes	—
7	#17986404	—	2	1	no	—
8	#17986404	—	2	1	no	—
9	#17986404	C₄	1	2	yes	—
10	#17986404	C₄	1	2	yes	—
11	C₂×#14556062	—	2	1	no	—
12	C₂×#14556062	C₂²	1	2	yes	—
13	#17985775	—	2	1	no	—
14	C₄×C₇₁⋊C₁₆	—	2	1	no	—
15	C₂×#108×C₇₁	—	2	1	no	—
16	C₂×#108×C₇₁	C₂²	1	2	yes	—
17	#423×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₄₂	—	4	1	no	—
20	#14556128	—	2	2	no	—
21	#14556128	—	2	2	no	—
22	#14556128	—	4	1	no	—
23	#14556128	—	4	1	no	—
24	C₂×C₄⋊Dic₇₁	—	2	2	no	—
25	C₂×C₄⋊Dic₇₁	—	4	1	no	—
26	#14556068	—	2	2	no	—
27	#14556068	Q₈	1	4	yes	complemented
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	#14556062	—	2	2	no	—
30	#14556062	—	2	2	no	—
31	#14556062	—	2	2	no	—
32	#14556062	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₇₁	—	2	2	no	—
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	#108×C₇₁	—	2	4	no	—
38	#108×C₇₁	—	4	2	no	—
39	C₄⋊C₈×C₇₁	—	4	1	no	—
40	C₄⋊C₈×C₇₁	—	4	1	no	—
41	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂×Dic₁₄₂	—	4	2	no	—
46	C₂×Dic₁₄₂	—	4	2	no	—
47	C₂×Dic₁₄₂	—	8	4	no	—
48	C₂²×Dic₇₁	—	4	2	no	—
49	C₄⋊Dic₇₁	—	2	4	no	—
50	C₄⋊Dic₇₁	—	4	2	no	—
51	C₄⋊Dic₇₁	—	4	2	no	—
52	C₄⋊Dic₇₁	—	4	2	no	—
53	C₇₁⋊C₁₆	—	2	4	no	—
54	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	M₁₆×C₇₁	—	4	2	no	—
58	M₁₆×C₇₁	—	4	2	no	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
63	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
64	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₇₁	#41	1	2	yes	—
68	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₁	D₈	1	1	yes	characteristic
71	C₄²×C₇₁	D₈	1	1	yes	characteristic
72	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×Dic₇₁	—	4	4	no	—
86	C₂×Dic₇₁	—	8	4	no	—
87	Dic₁₄₂	—	8	4	no	—
88	Dic₁₄₂	—	8	4	no	—
89	C₈×C₇₁	—	8	2	no	—
90	C₈×C₇₁	#103	1	2	yes	—
91	C₈×C₇₁	#103	1	2	yes	—
92	C₈×C₇₁	C₄×Q₈	1	2	yes	—
93	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	—	2	1	no	—
96	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
97	C₂²×C₇₁	#487	1	1	yes	characteristic
98	C₂²×C₇₁	#498	1	1	yes	characteristic
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	—	2	1	no	—
101	C₄×C₇₁	#361	1	2	yes	—
102	C₄×C₇₁	#362	1	2	yes	—
103	C₄×C₇₁	#391	1	1	yes	characteristic
104	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
105	C₄×C₇₁	#486	1	1	yes	characteristic
106	C₄×C₇₁	#500	1	1	yes	characteristic
107	Dic₇₁	—	8	4	no	—
108	#13290	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	—	2	1	no	—
111	C₂×C₇₁	#1990	1	1	yes	characteristic
112	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
114	C₂×#500	—	71	1	no	—
115	C₄×M₃₂	—	71	1	no	—
116	#1848	D₇₁	1	1	yes	characteristic
117	#1387	—	71	4	no	—
118	C₇₁	#13290	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄⋊Q₈	—	142	1	no	—
120	C₂×M₃₂	—	71	2	no	—
121	#500	—	71	2	no	—
122	#500	—	71	2	no	—
123	#500	—	142	1	no	—
124	#500	—	142	1	no	—
125	C₂×#108	—	2	1	no	—
126	C₂×#108	C₂×D₇₁	1	2	yes	—
127	C₂×#107	—	71	2	no	—
128	C₂×#107	—	142	1	no	—
129	#423	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
131	#346	—	142	1	no	—
132	C₄×C₁₆	—	142	1	no	—
133	C₂²×Q₈	—	284	1	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
137	C₄⋊Q₈	—	284	1	no	—
138	C₄⋊Q₈	—	284	1	no	—
139	C₄⋊Q₈	—	142	2	no	—
140	C₄⋊Q₈	—	142	2	no	—
141	C₂×C₄⋊C₄	—	284	1	no	—
142	C₂×C₄⋊C₄	—	142	2	no	—
143	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
144	M₃₂	—	71	4	no	—
145	M₃₂	—	142	2	no	—
146	C₂×C₁₆	—	142	2	no	—
147	#108	—	2	4	no	—
148	#108	—	4	2	no	—
149	#107	—	71	4	no	—
150	#107	—	142	2	no	—
151	#107	—	142	2	no	—
152	#107	—	142	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄⋊C₈	—	4	1	no	—
155	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
156	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
157	C₄×C₈	D₂₈₄	1	1	yes	characteristic
158	C₄×C₈	D₂₈₄	1	1	yes	characteristic
159	C₂×Q₈	—	284	2	no	—
160	C₂×Q₈	—	284	2	no	—
161	C₂×Q₈	—	568	4	no	—
162	C₂²×C₄	—	284	2	no	—
163	C₂²×C₄	#11765852	1	1	yes	characteristic
164	C₂²×C₄	#11765852	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
166	M₁₆	—	4	2	no	—
167	M₁₆	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
171	C₂×C₈	#11765853	1	2	yes	—
172	C₂×C₈	#11765855	1	2	yes	—
173	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
175	C₂×C₈	Q₈×D₇₁	1	2	yes	—
176	C₂×C₈	#11765864	1	2	yes	—
177	C₄⋊C₄	—	284	2	no	—
178	C₄⋊C₄	—	284	2	no	—
179	C₄⋊C₄	—	284	2	no	—
180	C₄⋊C₄	—	142	4	no	—
181	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
182	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
183	C₄²	#11765850	1	1	yes	characteristic
184	C₄²	#11765850	1	1	yes	characteristic
185	C₁₆	—	142	4	no	—
186	C₂³	C₂×#11765852	1	1	yes	characteristic
187	Q₈	—	568	4	no	—
188	Q₈	—	568	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	284	4	no	—
192	C₂×C₄	—	568	4	no	—
193	C₂×C₄	#14556048	1	1	yes	characteristic
194	C₂×C₄	#14556048	1	1	yes	characteristic
195	C₂×C₄	#14556048	1	1	yes	characteristic
196	C₂×C₄	#14556048	1	1	yes	characteristic
197	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#14556115	1	1	yes	characteristic
199	C₂×C₄	#14556126	1	1	yes	characteristic
200	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
201	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
202	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
203	C₈	—	8	2	no	—
204	C₈	#14556051	1	2	yes	—
205	C₈	#14556051	1	2	yes	—
206	C₈	#14556093	1	2	yes	—
207	C₈	#14556108	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#14556048	1	1	yes	characteristic
211	C₂²	#17986349	1	1	yes	characteristic
212	C₂²	#17986395	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	568	4	no	—
216	C₄	#17985811	1	2	yes	—
217	C₄	#17985814	1	2	yes	—
218	C₄	#17985904	1	1	yes	characteristic
219	C₄	C₂×#14556051	1	1	yes	characteristic
220	C₄	#17986345	1	1	yes	characteristic
221	C₄	#17986402	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#22589300	1	1	yes	characteristic
225	C₂	C₂×#17985811	1	1	yes	characteristic
226	C₂	C₂×#17985814	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556048
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#13290
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
27. #14556068	Q₈	semidirect
118. C₇₁	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
C₄×#14556068	2
#22586450	2
C₇₁×#1848	2
#13290	71