Subgroups — #34182038

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763559	C₂	1	2	yes	maximal
2	#23763557	C₂	1	1	yes	maximal, characteristic
3	#23763527	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579094	—	2	4	no	—
7	C₂×#14579093	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#19098433	C₂²	1	2	yes	—
11	#19098325	—	2	4	no	—
12	C₈×C₇₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×#106×C₇₉	—	2	2	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#384×C₇₉	C₂²	1	2	yes	—
16	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
17	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₇₉	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	#14579094	—	4	8	no	—
22	#14579093	—	4	4	no	—
23	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
26	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	#106×C₇₉	—	4	4	no	—
33	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₉	Q₈	1	2	yes	—
35	C₂²×Dic₇₉	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	2	4	no	—
37	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₇₉⋊C₈	SD₁₆	1	8	yes	—
40	C₄⋊Dic₇₉	—	4	4	no	—
41	C₄⋊Dic₇₉	—	4	4	no	—
42	C₄⋊Dic₇₉	—	4	4	no	—
43	C₂²×C₄×C₇₉	—	4	2	no	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₇₉	D₈	1	4	yes	—
49	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
50	C₄⋊C₄×C₇₉	—	4	4	no	—
51	C₄⋊C₄×C₇₉	—	4	4	no	—
52	C₄⋊C₄×C₇₉	—	4	4	no	—
53	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₇₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₉	—	8	2	no	—
59	C₂×C₄×C₇₉	—	8	2	no	—
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₇₉	#135	1	4	yes	—
71	C₂×Dic₇₉	—	8	2	no	—
72	C₂×Dic₇₉	—	8	2	no	—
73	C₂×Dic₇₉	—	8	2	no	—
74	C₈×C₇₉	#107	1	8	yes	complemented
75	C₇₉⋊C₈	—	2	4	no	—
76	C₇₉⋊C₈	#106	1	8	yes	complemented
77	C₂²×C₇₉	#384	1	1	yes	characteristic
78	C₂²×C₇₉	#492	1	2	yes	—
79	C₂²×C₇₉	#493	1	1	yes	characteristic
80	C₂²×C₇₉	#494	1	1	yes	characteristic
81	C₂²×C₇₉	#495	1	2	yes	—
82	C₄×C₇₉	—	16	2	no	—
83	C₄×C₇₉	C₂×#106	1	2	yes	—
84	C₄×C₇₉	C₂×#107	1	2	yes	—
85	C₄×C₇₉	#427	1	2	yes	—
86	C₄×C₇₉	#498	1	2	yes	—
87	C₄×C₇₉	#499	1	2	yes	—
88	C₄×C₇₉	#500	1	2	yes	—
89	Dic₇₉	—	16	2	no	—
90	#8814	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	#1706	1	2	yes	—
92	C₂×C₇₉	#1707	1	2	yes	—
93	C₂×C₇₉	#1844	1	1	yes	characteristic
94	C₂×C₇₉	#1845	1	1	yes	characteristic
95	C₂×C₇₉	#1846	1	1	yes	characteristic
96	#1845	—	79	2	no	—
97	#1844	—	79	1	no	—
98	#1844	D₇₉	1	1	yes	characteristic
99	#1834	—	79	2	no	—
100	C₂×C₈²	—	79	1	no	—
101	C₇₉	#8814	1	1	yes	characteristic, complemented, Sylow
102	C₂×#107	—	158	4	no	—
103	C₂×#106	—	2	2	no	—
104	C₂×#106	—	158	2	no	—
105	C₂×C₄×C₈	—	79	1	no	—
106	C₂×C₄×C₈	—	79	1	no	—
107	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
108	#384	—	79	2	no	—
109	#384	C₂×D₇₉	1	2	yes	—
110	#340	—	158	4	no	—
111	C₈²	—	79	4	no	—
112	C₂²×C₈	—	79	2	no	—
113	C₂²×C₈	—	79	2	no	—
114	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	158	2	no	—
119	C₂×C₄⋊C₄	—	158	2	no	—
120	C₂×C₄⋊C₄	—	158	2	no	—
121	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	#107	—	316	8	no	—
123	#106	—	4	4	no	—
124	#106	—	316	4	no	—
125	C₄×C₈	—	79	2	no	—
126	C₄×C₈	—	79	2	no	—
127	C₄×C₈	—	79	2	no	—
128	C₄×C₈	—	79	2	no	—
129	C₄×C₈	Dic₁₅₈	1	2	yes	—
130	C₄×C₈	C₄×D₇₉	1	2	yes	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	316	2	no	—
133	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
136	C₂×C₈	—	79	4	no	—
137	C₂×C₈	—	79	8	no	—
138	C₂×C₈	—	79	4	no	—
139	C₂×C₈	—	158	4	no	—
140	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
141	C₂×C₈	#11929609	1	4	yes	—
142	C₂×C₈	#11929614	1	4	yes	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	316	4	no	—
147	C₄⋊C₄	—	316	4	no	—
148	C₄⋊C₄	—	316	4	no	—
149	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
150	C₄²	Q₈×D₇₉	1	1	yes	characteristic
151	C₄²	Q₈×D₇₉	1	1	yes	characteristic
152	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
153	C₂³	#14579157	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	632	2	no	—
158	C₂×C₄	—	632	2	no	—
159	C₂×C₄	—	632	2	no	—
160	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#14579159	1	1	yes	characteristic
164	C₂×C₄	#14579160	1	1	yes	characteristic
165	C₂×C₄	#14579160	1	1	yes	characteristic
166	C₂×C₄	C₂×#11929609	1	2	yes	—
167	C₂×C₄	SD₁₆×D₇₉	1	4	yes	—
168	C₂×C₄	C₂×#11929614	1	2	yes	—
169	C₂×C₄	#14579182	1	4	yes	—
170	C₈	—	79	8	no	—
171	C₈	—	158	4	no	—
172	C₈	#14579094	1	8	yes	complemented
173	C₂²	#19098432	1	1	yes	characteristic
174	C₂²	#19098934	1	2	yes	—
175	C₂²	#19098939	1	1	yes	characteristic
176	C₂²	#19098942	1	1	yes	characteristic
177	C₂²	#19098944	1	2	yes	—
178	C₄	—	16	2	no	—
179	C₄	—	1264	2	no	—
180	C₄	#106×D₇₉	1	2	yes	—
181	C₄	C₂×#14579094	1	2	yes	—
182	C₄	#19098598	1	2	yes	—
183	C₄	#19098960	1	2	yes	—
184	C₄	#19098963	1	2	yes	—
185	C₄	#19098966	1	2	yes	—
186	C₂	#23762990	1	2	yes	—
187	C₂	#23762995	1	2	yes	—
188	C₂	#23763555	1	1	yes	characteristic
189	C₂	#23763559	1	1	yes	characteristic
190	C₂	#23763561	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1844	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1844	D₇₉
hypocenter	C₇₉	#8814
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
12. C₈×C₇₉⋊C₈	C₄	semidirect
74. C₈×C₇₉	#107	semidirect
76. C₇₉⋊C₈	#106	semidirect
101. C₇₉	#8814	semidirect
172. C₈	#14579094	semidirect

Maximal subgroups

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

subgroup	index
#23763559	2
#23763557	2
#23763527	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1844	2
#8814	79