Subgroups — #34187643

Order 20224.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	1	yes	maximal, characteristic
2	#23763564	C₂	1	1	yes	maximal, characteristic
3	#23761965	C₂	1	4	yes	maximal
4	C₇₉×#1450	C₂	1	1	yes	maximal, characteristic
5	#19099120	C₂²	1	2	yes	—
6	#19098968	C₄	1	2	yes	complemented
7	#19098965	—	2	1	no	—
8	#19098965	—	2	1	no	—
9	#19098961	C₄	1	2	yes	complemented
10	#19098601	—	2	1	no	—
11	#19098601	—	2	1	no	—
12	C₂×#14579095	—	2	1	no	—
13	C₂×#14579093	—	2	1	no	—
14	#19098429	C₂²	1	2	yes	—
15	#19098301	—	2	8	no	—
16	#431×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#322×C₇₉	—	2	1	no	—
19	C₈²×C₇₉	—	2	1	no	—
20	#14579161	C₂×C₄	1	4	yes	—
21	#14579153	—	2	2	no	—
22	#14579140	—	2	4	no	—
23	C₄×Dic₁₅₈	—	2	4	no	—
24	#14579120	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	#14579095	—	2	2	no	—
27	#14579095	—	2	2	no	—
28	#14579095	—	4	2	no	—
29	#14579093	—	2	2	no	—
30	#14579093	—	2	2	no	—
31	#14579093	—	4	2	no	—
32	C₇₉⋊#95	—	2	4	no	—
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₂²⋊C₈×C₇₉	D₄	1	4	yes	—
38	C₄×C₈×C₇₉	—	2	2	no	—
39	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂×Dic₁₅₈	—	2	4	no	—
44	C₂²×Dic₇₉	—	4	2	no	—
45	C₄⋊Dic₇₉	—	2	4	no	—
46	C₄⋊Dic₇₉	—	2	4	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	Dic₇₉⋊C₄	—	4	4	no	—
50	C₂²⋊Dic₇₉	—	4	2	no	—
51	C₄×Dic₇₉	—	4	2	no	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	—	2	4	no	—
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
60	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
61	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
62	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
63	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₉	#41	1	2	yes	—
66	C₂³×C₇₉	#127	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₇₉	#121	1	2	yes	—
71	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₇₉	—	4	4	no	—
77	C₂×Dic₇₉	—	8	4	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	Dic₁₅₈	—	8	4	no	—
80	C₈×C₇₉	—	4	4	no	—
81	C₈×C₇₉	#103	1	2	yes	—
82	C₈×C₇₉	#103	1	2	yes	—
83	C₈×C₇₉	#103	1	2	yes	—
84	C₈×C₇₉	#103	1	2	yes	—
85	C₂²×C₇₉	—	2	2	no	—
86	C₂²×C₇₉	#390	1	1	yes	characteristic
87	C₂²×C₇₉	#492	1	1	yes	characteristic
88	C₂²×C₇₉	#494	1	1	yes	characteristic
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	2	2	no	—
91	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
92	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
93	C₄×C₇₉	#487	1	1	yes	characteristic
94	C₄×C₇₉	#487	1	1	yes	characteristic
95	Dic₇₉	—	8	4	no	—
96	Dic₇₉	—	16	2	no	—
97	#9755	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	1	no	—
99	C₂×C₇₉	—	2	1	no	—
100	C₂×C₇₉	#1956	1	1	yes	characteristic
101	C₂×C₇₉	#1976	1	2	yes	—
102	#3160	—	79	1	no	—
103	#1846	—	79	1	no	—
104	#1518	—	79	4	no	—
105	#1450	D₇₉	1	1	yes	characteristic
106	C₇₉	#9755	1	1	yes	characteristic, complemented, Sylow
107	#533	—	79	2	no	—
108	#500	—	79	2	no	—
109	#499	—	158	1	no	—
110	#499	—	158	1	no	—
111	#498	—	79	2	no	—
112	#431	C₂×D₇₉	1	2	yes	—
113	#427	—	158	1	no	—
114	#427	—	158	1	no	—
115	C₂×#107	—	158	1	no	—
116	C₂×#106	—	158	1	no	—
117	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	#382	—	79	2	no	—
119	#328	—	158	8	no	—
120	#322	—	2	1	no	—
121	C₈²	—	2	1	no	—
122	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
123	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
124	C₄⋊Q₈	—	79	4	no	—
125	#125	—	158	2	no	—
126	C₂²⋊Q₈	—	158	4	no	—
127	C₄×Q₈	—	158	4	no	—
128	#117	—	158	2	no	—
129	C₂×C₄⋊C₄	—	158	2	no	—
130	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
131	#107	—	316	2	no	—
132	#107	—	158	2	no	—
133	#107	—	158	2	no	—
134	#106	—	316	2	no	—
135	#106	—	158	2	no	—
136	#106	—	158	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	D₃₁₆	1	4	yes	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
141	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
144	#95	—	158	4	no	—
145	C₂×Q₈	—	158	4	no	—
146	C₂²×C₄	—	316	2	no	—
147	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
148	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
152	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
153	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
154	C₂×C₈	#11929613	1	2	yes	—
155	C₂×C₈	#11929613	1	2	yes	—
156	C₂×C₈	Dic₃₁₆	1	2	yes	—
157	C₂×C₈	Dic₃₁₆	1	2	yes	—
158	C₄⋊C₄	—	316	4	no	—
159	C₄⋊C₄	—	316	2	no	—
160	C₄⋊C₄	—	316	2	no	—
161	C₄⋊C₄	—	158	4	no	—
162	C₄⋊C₄	—	158	4	no	—
163	C₂²⋊C₄	—	316	2	no	—
164	C₄²	—	316	2	no	—
165	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
167	C₄²	#11929624	1	2	yes	—
168	C₂³	#14579158	1	1	yes	characteristic
169	Q₈	—	632	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	316	4	no	—
172	C₂×C₄	—	632	4	no	—
173	C₂×C₄	—	632	2	no	—
174	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
176	C₂×C₄	#14579135	1	2	yes	—
177	C₂×C₄	#14579149	1	1	yes	characteristic
178	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
179	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
181	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
182	C₈	—	4	4	no	—
183	C₈	#14579086	1	2	yes	—
184	C₈	#14579086	1	2	yes	—
185	C₈	#14579086	1	2	yes	—
186	C₈	#14579086	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#19098465	1	1	yes	characteristic
189	C₂²	#19098936	1	1	yes	characteristic
190	C₂²	#19098943	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	632	4	no	—
194	C₄	—	1264	2	no	—
195	C₄	C₂×#14579086	1	1	yes	characteristic
196	C₄	C₂×#14579086	1	1	yes	characteristic
197	C₄	#19098914	1	1	yes	characteristic
198	C₄	#19098914	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#23764202	1	1	yes	characteristic
202	C₂	#23764298	1	2	yes	—

	subgroup	G/H
center	C₂²	#19098465
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1450	C₂
socle	C₂²×C₇₉	#390
prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1450	D₇₉
hypocenter	C₇₉	#9755
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
6. #19098968	C₄	semidirect
9. #19098961	C₄	semidirect
106. C₇₉	#9755	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
#23763564	2
#23761965	2
C₇₉×#1450	2
#9755	79