Subgroups — #31913659

Order 18176.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985829	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985827	C₂	1	1	yes	maximal, characteristic, complemented
3	#22588404	C₂	1	2	yes	maximal
4	#22586449	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	C₂²	1	2	yes	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	#17986000	—	2	2	no	—
9	#17985829	—	2	2	no	—
10	#17985827	—	2	1	no	—
11	#17985827	—	2	1	no	—
12	#17985767	—	2	2	no	—
13	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
14	#504×C₇₁	—	2	2	no	—
15	#504×C₇₁	C₄	1	4	yes	complemented
16	C₂×M₃₂×C₇₁	—	2	1	no	—
17	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₁	—	2	1	no	—
19	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	#14556064	—	2	2	no	—
23	#14556064	—	2	2	no	—
24	#14556062	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556052	—	4	2	no	—
27	#14556049	—	4	2	no	—
28	C₂×#41×C₇₁	D₄	1	1	yes	characteristic
29	#131×C₇₁	—	2	1	no	—
30	#131×C₇₁	—	2	1	no	—
31	#131×C₇₁	C₂×C₄	1	2	yes	—
32	#131×C₇₁	Q₈	1	2	yes	—
33	C₂×M₁₆×C₇₁	—	2	1	no	—
34	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	—	2	1	no	—
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	M₃₂×C₇₁	—	2	1	no	—
38	M₃₂×C₇₁	—	2	4	no	—
39	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
40	C₂×C₁₆×C₇₁	—	2	1	no	—
41	C₂×C₁₆×C₇₁	—	2	4	no	—
42	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
46	C₂²×Dic₇₁	—	4	2	no	—
47	#11765849	—	4	2	no	—
48	#11765849	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₄⋊Dic₇₁	—	4	2	no	—
51	C₄⋊Dic₇₁	—	4	2	no	—
52	#41×C₇₁	—	2	1	no	—
53	#41×C₇₁	—	2	1	no	—
54	#41×C₇₁	C₄⋊C₄	1	2	yes	—
55	#41×C₇₁	Q₁₆	1	2	yes	—
56	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
57	C₂×D₄×C₇₁	—	2	1	no	—
58	C₂×D₄×C₇₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	—	2	1	no	—
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	M₁₆×C₇₁	—	2	1	no	—
62	M₁₆×C₇₁	—	2	2	no	—
63	M₁₆×C₇₁	—	2	2	no	—
64	M₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	2	no	—
67	C₂×C₈×C₇₁	—	2	2	no	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
73	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
74	C₁₆×C₇₁	—	2	2	no	—
75	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
77	C₂³×C₇₁	—	2	1	no	—
78	C₂³×C₇₁	#135	1	1	yes	characteristic
79	Q₈×C₇₁	—	2	1	no	—
80	Q₈×C₇₁	#107	1	2	yes	—
81	D₄×C₇₁	—	2	2	no	—
82	D₄×C₇₁	—	2	2	no	—
83	D₄×C₇₁	—	2	1	no	—
84	D₄×C₇₁	#107	1	2	yes	—
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
94	C₂×Dic₇₁	—	8	2	no	—
95	C₂×Dic₇₁	—	16	2	no	—
96	C₈×C₇₁	—	2	2	no	—
97	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₈×C₇₁	#117	1	1	yes	characteristic
99	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
101	C₇₁⋊C₈	—	8	2	no	—
102	C₂²×C₇₁	—	2	2	no	—
103	C₂²×C₇₁	—	2	2	no	—
104	C₂²×C₇₁	—	4	2	no	—
105	C₂²×C₇₁	#427	1	1	yes	characteristic
106	C₂²×C₇₁	#466	1	1	yes	characteristic
107	C₂²×C₇₁	#467	1	1	yes	characteristic
108	C₄×C₇₁	—	2	2	no	—
109	C₄×C₇₁	#380	1	1	yes	characteristic
110	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
111	C₄×C₇₁	#466	1	1	yes	characteristic
112	C₄×C₇₁	#467	1	1	yes	characteristic
113	Dic₇₁	—	16	2	no	—
114	#12843	—	71	1	no	maximal, Sylow
115	C₂×C₇₁	—	2	1	no	—
116	C₂×C₇₁	—	2	1	no	—
117	C₂×C₇₁	—	4	2	no	—
118	C₂×C₇₁	#1808	1	1	yes	characteristic
119	C₂×C₇₁	#2223	1	2	yes	—
120	C₂×#504	D₇₁	1	1	yes	characteristic
121	C₂×#368	—	71	1	no	—
122	C₂×#367	—	71	1	no	—
123	#1816	—	71	2	no	—
124	#1387	—	71	2	no	—
125	C₇₁	#12843	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
127	#504	—	2	2	no	—
128	#504	Dic₇₁	1	4	yes	complemented
129	C₂×M₃₂	—	2	1	no	—
130	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
131	C₂²×C₁₆	—	2	1	no	—
132	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
133	C₂×#108	—	71	2	no	—
134	C₂×#107	—	71	2	no	—
135	#417	—	142	2	no	—
136	#368	—	142	2	no	—
137	#367	—	142	1	no	—
138	#367	—	142	1	no	—
139	#343	—	142	2	no	—
140	C₂×#41	D₂₈₄	1	1	yes	characteristic
141	#131	—	2	1	no	—
142	#131	—	2	1	no	—
143	#131	Dic₁₄₂	1	2	yes	—
144	#131	C₂×Dic₇₁	1	2	yes	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	—	142	2	no	—
147	C₂×M₁₆	D₂₈₄	1	1	yes	characteristic
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	142	2	no	—
151	M₃₂	—	2	1	no	—
152	M₃₂	—	2	4	no	—
153	M₃₂	C₂×Dic₇₁	1	2	yes	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	—	2	4	no	—
156	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
157	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
158	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
159	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
160	#108	—	142	2	no	—
161	#108	—	142	2	no	—
162	#107	—	142	2	no	—
163	#107	—	142	2	no	—
164	#103	—	284	2	no	—
165	#102	—	284	2	no	—
166	#41	—	2	1	no	—
167	#41	—	2	1	no	—
168	#41	C₄⋊Dic₇₁	1	2	yes	—
169	#41	Dic₂₈₄	1	2	yes	—
170	C₂×Q₈	D₅₆₈	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	D₅₆₈	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	284	2	no	—
175	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
176	M₁₆	—	2	1	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	284	2	no	—
180	M₁₆	—	284	2	no	—
181	M₁₆	C₄⋊Dic₇₁	1	2	yes	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	284	2	no	—
186	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
189	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
190	C₂×C₈	#11765865	1	1	yes	characteristic
191	C₂×C₈	#11765865	1	1	yes	characteristic
192	C₄⋊C₄	—	284	2	no	—
193	C₄⋊C₄	—	284	2	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂²⋊Dic₇₁	1	2	yes	—
196	C₁₆	C₂²⋊Dic₇₁	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	#14556150	1	1	yes	characteristic
199	Q₈	—	2	1	no	—
200	Q₈	#14556062	1	2	yes	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	#14556062	1	2	yes	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	568	2	no	—
209	C₂×C₄	—	1136	2	no	—
210	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
211	C₂×C₄	#14556102	1	1	yes	characteristic
212	C₂×C₄	#14556102	1	1	yes	characteristic
213	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
214	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
215	C₂×C₄	#14556150	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	568	2	no	—
218	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
219	C₈	#14556087	1	1	yes	characteristic
220	C₈	#14556107	1	1	yes	characteristic
221	C₈	#14556107	1	1	yes	characteristic
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	#17986037	1	1	yes	characteristic
226	C₂²	#17986235	1	1	yes	characteristic
227	C₂²	#17986240	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	1136	2	no	—
230	C₄	#17985859	1	1	yes	characteristic
231	C₄	C₂×#14556062	1	1	yes	characteristic
232	C₄	#17986235	1	1	yes	characteristic
233	C₄	#17986240	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	#22588369	1	1	yes	characteristic
238	C₂	#22590470	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986235
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#504×C₇₁	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂×#41×C₇₁	D₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#504	D₇₁
hypocenter	C₇₁	#12843
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#17985829	C₂	semidirect
2. C₂×#17985827	C₂	semidirect
4. #22586449	C₂	semidirect
15. #504×C₇₁	C₄	semidirect
125. C₇₁	#12843	semidirect
128. #504	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985829	2
C₂×#17985827	2
#22588404	2
#22586449	2
C₂×#504×C₇₁	2
#12843	71