Subgroups — #32152940

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	#22590620	C₂	1	2	yes	maximal, complemented
3	#22590616	C₂	1	4	yes	maximal, complemented
4	#22590613	C₂	1	2	yes	maximal, complemented
5	#22590122	C₂	1	4	yes	maximal
6	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
8	#17986556	—	2	1	no	—
9	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	#17986404	C₂²	1	2	yes	—
11	#17986401	—	2	1	no	—
12	#17986401	C₂²	1	2	yes	—
13	#17986397	—	2	1	no	—
14	#17986037	—	2	1	no	—
15	#17986037	C₂²	1	2	yes	—
16	#17985826	C₂²	1	8	yes	—
17	#17985824	C₂²	1	8	yes	—
18	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
20	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
21	#533×C₇₁	—	2	1	no	—
22	#500×C₇₁	C₂²	1	2	yes	—
23	#499×C₇₁	—	2	1	no	—
24	#499×C₇₁	C₂²	1	2	yes	—
25	#498×C₇₁	—	2	1	no	—
26	#427×C₇₁	—	2	1	no	—
27	#427×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	#14556128	—	2	2	no	—
30	#14556120	—	2	2	no	—
31	#14556107	—	4	2	no	—
32	C₄×Dic₁₄₂	—	4	2	no	—
33	#14556087	—	2	2	no	—
34	C₂×C₇₁⋊C₁₆	Q₈	1	8	yes	complemented
35	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
36	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
37	#14556062	—	2	2	no	—
38	#14556062	C₂³	1	4	yes	—
39	#14556060	—	2	2	no	—
40	#14556060	—	2	2	no	—
41	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₇₁	—	2	2	no	—
44	#125×C₇₁	—	2	2	no	—
45	C₂²⋊Q₈×C₇₁	—	4	2	no	—
46	C₄×Q₈×C₇₁	—	4	2	no	—
47	#117×C₇₁	—	2	2	no	—
48	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
49	#107×C₇₁	—	2	2	no	—
50	#107×C₇₁	C₂³	1	4	yes	—
51	#106×C₇₁	—	2	2	no	—
52	#106×C₇₁	—	2	2	no	—
53	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₄₂	—	4	2	no	—
58	C₄⋊Dic₇₁	—	2	4	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	Dic₇₁⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₇₁	—	4	2	no	—
62	C₄×Dic₇₁	—	4	2	no	—
63	C₇₁⋊C₁₆	C₂×Q₈	1	4	yes	—
64	C₇₁⋊C₁₆	C₂×Q₈	1	4	yes	—
65	C₂×Q₈×C₇₁	—	4	2	no	—
66	C₂²×C₄×C₇₁	D₈	1	2	yes	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
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×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₇₁	—	2	4	no	—
75	C₄⋊C₄×C₇₁	—	4	4	no	—
76	C₄⋊C₄×C₇₁	—	4	2	no	—
77	C₂²⋊C₄×C₇₁	—	4	2	no	—
78	C₄²×C₇₁	—	4	2	no	—
79	C₄²×C₇₁	D₈	1	2	yes	—
80	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₇₁	—	8	4	no	—
84	C₂×C₄×C₇₁	—	2	4	no	—
85	C₂×C₄×C₇₁	—	4	4	no	—
86	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₇₁	—	4	4	no	—
93	Dic₁₄₂	—	8	4	no	—
94	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
95	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
96	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
97	C₈×C₇₁	C₂²×Q₈	1	2	yes	—
98	C₂²×C₇₁	—	2	2	no	—
99	C₂²×C₇₁	#500	1	2	yes	—
100	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₇₁	—	2	2	no	—
102	C₄×C₇₁	—	8	4	no	—
103	C₄×C₇₁	#500	1	2	yes	—
104	C₄×C₇₁	#508	1	4	yes	—
105	C₄×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
107	Dic₇₁	—	8	4	no	—
108	#33726	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	—	2	2	no	—
110	C₂×C₇₁	C₂×#500	1	1	yes	characteristic
111	C₂×C₇₁	C₂×#508	1	2	yes	—
112	#3160	—	71	1	no	—
113	#3160	D₇₁	1	1	yes	characteristic
114	#2254	—	71	2	no	—
115	#2253	—	71	4	no	—
116	#2252	—	71	2	no	—
117	#2158	—	71	4	no	—
118	C₂×C₄×C₁₆	—	71	1	no	—
119	C₇₁	#33726	1	1	yes	characteristic, complemented, Sylow
120	#533	—	2	1	no	—
121	#533	—	142	1	no	—
122	C₂²×C₁₆	—	71	2	no	—
123	#500	—	71	2	no	—
124	#500	C₂×D₇₁	1	2	yes	—
125	#499	—	2	1	no	—
126	#499	—	71	2	no	—
127	#499	—	142	1	no	—
128	#499	C₂×D₇₁	1	2	yes	—
129	#498	—	2	1	no	—
130	#498	—	142	1	no	—
131	#427	—	2	1	no	—
132	#427	—	71	2	no	—
133	#427	—	142	1	no	—
134	#427	C₂×D₇₁	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
136	#367	—	71	8	no	—
137	#366	—	71	8	no	—
138	C₄×C₁₆	—	71	1	no	—
139	C₄×C₁₆	—	71	2	no	—
140	C₄×C₁₆	—	71	1	no	—
141	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	142	2	no	—
145	#125	—	2	2	no	—
146	#125	—	142	2	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	C₂²⋊Q₈	—	284	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	C₄×Q₈	—	284	2	no	—
151	#117	—	2	2	no	—
152	#117	—	142	2	no	—
153	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
154	C₂×C₁₆	—	71	8	no	—
155	C₂×C₁₆	—	71	2	no	—
156	C₂×C₁₆	—	71	2	no	—
157	#107	—	2	2	no	—
158	#107	—	71	4	no	—
159	#107	—	142	2	no	—
160	#107	C₂²×D₇₁	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	#106	—	142	2	no	—
164	#106	—	142	2	no	—
165	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	284	2	no	—
171	C₂²×C₄	#11765850	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂×C₈	#11765855	1	4	yes	—
174	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₇₁	1	4	yes	—
178	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	284	4	no	—
183	C₄⋊C₄	—	284	2	no	—
184	C₄⋊C₄	—	142	4	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	284	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	284	2	no	—
189	C₄²	#11765850	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
192	C₁₆	—	71	4	no	—
193	C₁₆	—	71	4	no	—
194	C₂³	C₂×#11765850	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	568	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	284	4	no	—
200	C₂×C₄	#14556126	1	2	yes	—
201	C₂×C₄	C₂×#11765850	1	2	yes	—
202	C₂×C₄	C₂×#11765850	1	2	yes	—
203	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
204	C₂×C₄	C₂×#11765855	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
206	C₈	#14556126	1	2	yes	—
207	C₈	C₂×#11765855	1	2	yes	—
208	C₈	C₂×#11765855	1	2	yes	—
209	C₈	C₂×Q₈×D₇₁	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#17986402	1	2	yes	—
212	C₂²	C₂²×#11765850	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	568	4	no	—
216	C₄	#17986402	1	2	yes	—
217	C₄	#17986429	1	4	yes	—
218	C₄	C₂×#14556126	1	1	yes	characteristic
219	C₄	C₂²×#11765855	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#17986402	1	1	yes	characteristic
222	C₂	C₂×#17986429	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#33726
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22590620	C₂	semidirect
3. #22590616	C₂	semidirect
4. #22590613	C₂	semidirect
34. C₂×C₇₁⋊C₁₆	Q₈	semidirect
119. C₇₁	#33726	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
#22590620	2
#22590616	2
#22590613	2
#22590122	2
C₂×C₄×C₇₁⋊C₁₆	2
C₇₁×#3160	2
#33726	71