Subgroups — #34206893

Order 20224.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 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	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
3	#23761507	C₂	1	4	yes	maximal
4	C₇₉×#1846	C₂	1	1	yes	maximal, characteristic
5	#19099120	—	2	1	no	—
6	C₂×#14579101	C₂²	1	2	yes	—
7	#19098968	C₄	1	2	yes	complemented
8	#19098965	—	2	1	no	—
9	#19098965	C₄	1	2	yes	complemented
10	#19098961	—	2	1	no	—
11	#19098601	—	2	1	no	—
12	#19098601	C₂²	1	2	yes	—
13	#19098339	—	2	1	no	—
14	C₄×C₇₉⋊C₁₆	—	2	1	no	—
15	C₂×#107×C₇₉	—	2	1	no	—
16	C₂×#106×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#382×C₇₉	—	2	1	no	—
19	#14579161	—	2	2	no	—
20	#14579153	—	2	2	no	—
21	#14579140	—	4	2	no	—
22	C₄×Dic₁₅₈	—	4	2	no	—
23	#14579120	—	2	2	no	—
24	#14579101	—	2	2	no	—
25	#14579101	Q₈	1	4	yes	complemented
26	C₂×C₇₉⋊C₁₆	—	2	2	no	—
27	#14579095	—	2	2	no	—
28	#14579095	C₂×C₄	1	4	yes	—
29	#14579093	—	2	2	no	—
30	#14579093	—	2	2	no	—
31	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
34	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
35	#107×C₇₉	—	4	2	no	—
36	#106×C₇₉	—	2	4	no	—
37	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
41	#95×C₇₉	—	4	2	no	—
42	C₂×Dic₁₅₈	—	4	2	no	—
43	C₄⋊Dic₇₉	—	2	4	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	Dic₇₉⋊C₄	—	4	4	no	—
46	C₂²⋊Dic₇₉	—	4	2	no	—
47	C₄×Dic₇₉	—	4	2	no	—
48	C₇₉⋊C₁₆	—	2	4	no	—
49	C₂²×C₄×C₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	D₈	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
55	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
56	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₇₉	#41	1	2	yes	—
60	C₄⋊C₄×C₇₉	—	4	2	no	—
61	C₄⋊C₄×C₇₉	—	4	2	no	—
62	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₇₉	SD₁₆	1	2	yes	—
65	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	—	8	4	no	—
68	C₂×C₄×C₇₉	—	8	2	no	—
69	C₂×C₄×C₇₉	#102	1	2	yes	—
70	C₂×C₄×C₇₉	#102	1	2	yes	—
71	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₇₉	—	4	4	no	—
78	Dic₁₅₈	—	8	4	no	—
79	C₈×C₇₉	#103	1	2	yes	—
80	C₈×C₇₉	#103	1	2	yes	—
81	C₈×C₇₉	C₄×Q₈	1	2	yes	—
82	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
83	C₂²×C₇₉	—	2	2	no	—
84	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
85	C₂²×C₇₉	#486	1	1	yes	characteristic
86	C₂²×C₇₉	#500	1	1	yes	characteristic
87	C₄×C₇₉	—	2	2	no	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	—	16	2	no	—
90	C₄×C₇₉	#391	1	1	yes	characteristic
91	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
92	C₄×C₇₉	#487	1	1	yes	characteristic
93	C₄×C₇₉	#498	1	1	yes	characteristic
94	Dic₇₉	—	8	4	no	—
95	#13288	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	2	1	no	—
97	C₂×C₇₉	—	2	1	no	—
98	C₂×C₇₉	#1990	1	1	yes	characteristic
99	C₂×C₇₉	#2145	1	2	yes	—
100	#3160	—	79	1	no	—
101	C₄×M₃₂	—	79	1	no	—
102	#1846	D₇₉	1	1	yes	characteristic
103	#1386	—	79	4	no	—
104	C₇₉	#13288	1	1	yes	characteristic, complemented, Sylow
105	#533	—	158	1	no	—
106	C₂×M₃₂	—	79	2	no	—
107	#500	—	79	2	no	—
108	#499	—	79	2	no	—
109	#499	—	158	1	no	—
110	#498	—	158	1	no	—
111	#427	—	79	2	no	—
112	#427	—	158	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂×D₇₉	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
116	#382	—	2	1	no	—
117	#346	—	158	1	no	—
118	C₄×C₁₆	—	158	1	no	—
119	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
121	C₄⋊Q₈	—	158	2	no	—
122	#125	—	158	2	no	—
123	C₂²⋊Q₈	—	316	2	no	—
124	C₄×Q₈	—	316	2	no	—
125	#117	—	158	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
128	M₃₂	—	79	4	no	—
129	M₃₂	—	158	2	no	—
130	C₂×C₁₆	—	158	2	no	—
131	#107	—	4	2	no	—
132	#107	—	79	4	no	—
133	#107	—	158	2	no	—
134	#106	—	2	4	no	—
135	#106	—	158	2	no	—
136	#106	—	158	2	no	—
137	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
138	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
139	C₄×C₈	D₃₁₆	1	1	yes	characteristic
140	C₄×C₈	D₃₁₆	1	1	yes	characteristic
141	#95	—	4	2	no	—
142	C₂×Q₈	—	316	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	#11929609	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#11929612	1	2	yes	—
149	C₂×C₈	#11929614	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
151	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₇₉	1	2	yes	—
153	C₂×C₈	#11929623	1	2	yes	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	316	4	no	—
157	C₄⋊C₄	—	316	2	no	—
158	C₄⋊C₄	—	158	4	no	—
159	C₂²⋊C₄	—	316	2	no	—
160	C₄²	—	316	2	no	—
161	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
162	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
163	C₄²	#11929611	1	2	yes	—
164	C₁₆	—	158	4	no	—
165	C₂³	C₂×#11929609	1	1	yes	characteristic
166	Q₈	—	632	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	316	4	no	—
171	C₂×C₄	#14579081	1	2	yes	—
172	C₂×C₄	#14579081	1	2	yes	—
173	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14579148	1	1	yes	characteristic
175	C₂×C₄	#14579159	1	1	yes	characteristic
176	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
177	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
178	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
179	C₈	#14579084	1	2	yes	—
180	C₈	#14579084	1	2	yes	—
181	C₈	#14579126	1	2	yes	—
182	C₈	#14579141	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#14579081	1	1	yes	characteristic
185	C₂²	#19098909	1	1	yes	characteristic
186	C₂²	#19098966	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	16	2	no	—
190	C₄	—	632	4	no	—
191	C₄	#19098468	1	1	yes	characteristic
192	C₄	C₂×#14579084	1	1	yes	characteristic
193	C₄	#19098913	1	1	yes	characteristic
194	C₄	#19098959	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#23764360	1	1	yes	characteristic
198	C₂	#23765135	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14579081
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1846	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1846	D₇₉
hypocenter	C₇₉	#13288
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
7. #19098968	C₄	semidirect
9. #19098965	C₄	semidirect
25. #14579101	Q₈	semidirect
104. C₇₉	#13288	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
C₄×#14579101	2
#23761507	2
C₇₉×#1846	2
#13288	79