Subgroups — #34206881

Order 20224.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#23761497	C₂	1	4	yes	maximal
4	C₇₉×#1848	C₂	1	1	yes	maximal, characteristic
5	#19099120	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
7	#19098968	C₄	1	2	yes	—
8	#19098965	—	2	1	no	—
9	#19098965	C₄	1	2	yes	—
10	#19098961	—	2	1	no	—
11	#19098601	—	2	1	no	—
12	#19098601	C₂²	1	2	yes	—
13	C₄×C₇₉⋊C₁₆	—	2	1	no	—
14	C₄×C₇₉⋊C₁₆	—	2	1	no	—
15	C₂×#108×C₇₉	—	2	1	no	—
16	C₂×#108×C₇₉	C₂²	1	2	yes	—
17	#423×C₇₉	—	2	1	no	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
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	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	complemented
27	#14579095	—	2	2	no	—
28	#14579095	C₂×C₄	1	4	yes	—
29	#14579093	—	2	2	no	—
30	#14579093	—	2	2	no	—
31	C₂×M₁₆×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
35	#108×C₇₉	—	2	4	no	—
36	#108×C₇₉	—	4	2	no	—
37	C₄⋊C₈×C₇₉	—	4	2	no	—
38	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₇₉	—	4	2	no	—
52	M₁₆×C₇₉	—	4	2	no	—
53	C₂×C₈×C₇₉	—	4	2	no	—
54	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
57	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
58	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₇₉	#41	1	2	yes	—
62	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₇₉	D₈	1	2	yes	—
65	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₉	—	2	4	no	—
67	C₂×C₄×C₇₉	#102	1	2	yes	—
68	C₂×C₄×C₇₉	#102	1	2	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₈	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×Dic₇₉	—	4	4	no	—
76	Dic₁₅₈	—	8	4	no	—
77	C₈×C₇₉	—	8	2	no	—
78	C₈×C₇₉	#103	1	2	yes	—
79	C₈×C₇₉	#103	1	2	yes	—
80	C₈×C₇₉	C₄×Q₈	1	2	yes	—
81	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
82	C₂²×C₇₉	—	2	2	no	—
83	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
84	C₂²×C₇₉	#487	1	1	yes	characteristic
85	C₂²×C₇₉	#498	1	1	yes	characteristic
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	#359	1	4	yes	—
88	C₄×C₇₉	#391	1	1	yes	characteristic
89	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
90	C₄×C₇₉	#486	1	1	yes	characteristic
91	C₄×C₇₉	#500	1	1	yes	characteristic
92	Dic₇₉	—	8	4	no	—
93	#13285	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	—	2	1	no	—
95	C₂×C₇₉	—	2	1	no	—
96	C₂×C₇₉	#1990	1	1	yes	characteristic
97	C₂×C₇₉	C₂×#359	1	2	yes	—
98	#3160	—	79	1	no	—
99	C₂×C₄×C₁₆	—	79	1	no	—
100	#1848	D₇₉	1	1	yes	characteristic
101	#1382	—	79	4	no	—
102	C₇₉	#13285	1	1	yes	characteristic, complemented, Sylow
103	#533	—	158	1	no	—
104	C₂²×C₁₆	—	79	2	no	—
105	#500	—	79	2	no	—
106	#499	—	79	2	no	—
107	#499	—	158	1	no	—
108	#498	—	158	1	no	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂×D₇₉	1	2	yes	—
111	#427	—	79	2	no	—
112	#427	—	158	1	no	—
113	#423	—	2	1	no	—
114	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	C₄×C₁₆	—	158	1	no	—
116	C₄×C₁₆	—	158	1	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	158	2	no	—
121	#125	—	158	2	no	—
122	C₂²⋊Q₈	—	316	2	no	—
123	C₄×Q₈	—	316	2	no	—
124	#117	—	158	2	no	—
125	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
126	C₂×C₁₆	—	79	4	no	—
127	C₂×C₁₆	—	158	2	no	—
128	C₂×C₁₆	—	158	2	no	—
129	#108	—	2	4	no	—
130	#108	—	4	2	no	—
131	#107	—	79	4	no	—
132	#107	—	158	2	no	—
133	#106	—	158	2	no	—
134	#106	—	158	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
138	C₄×C₈	D₃₁₆	1	1	yes	characteristic
139	C₄×C₈	D₃₁₆	1	1	yes	characteristic
140	C₂×Q₈	—	316	2	no	—
141	C₂²×C₄	#11929611	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
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	—	316	4	no	—
155	C₄⋊C₄	—	316	2	no	—
156	C₄⋊C₄	—	158	4	no	—
157	C₂²⋊C₄	—	316	2	no	—
158	C₄²	—	316	2	no	—
159	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
160	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
161	C₄²	#11929609	1	2	yes	—
162	C₁₆	—	158	4	no	—
163	C₂³	C₂×#11929611	1	1	yes	characteristic
164	Q₈	—	632	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	316	4	no	—
167	C₂×C₄	#14579081	1	2	yes	—
168	C₂×C₄	#14579081	1	2	yes	—
169	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#14579148	1	1	yes	characteristic
171	C₂×C₄	#14579159	1	1	yes	characteristic
172	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
173	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
174	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
175	C₈	—	8	2	no	—
176	C₈	#14579084	1	2	yes	—
177	C₈	#14579084	1	2	yes	—
178	C₈	#14579126	1	2	yes	—
179	C₈	#14579141	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14579081	1	1	yes	characteristic
182	C₂²	#19098913	1	1	yes	characteristic
183	C₂²	#19098959	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	632	4	no	—
186	C₄	#19098369	1	4	yes	—
187	C₄	#19098468	1	1	yes	characteristic
188	C₄	C₂×#14579084	1	1	yes	characteristic
189	C₄	#19098909	1	1	yes	characteristic
190	C₄	#19098966	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#23764360	1	1	yes	characteristic
194	C₂	C₂×#19098369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1848	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1848	D₇₉
hypocenter	C₇₉	#13285
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
26. C₂×C₇₉⋊C₁₆	Q₈	semidirect
102. C₇₉	#13285	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
C₂×C₄×C₇₉⋊C₁₆	2
#23761497	2
C₇₉×#1848	2
#13285	79