Finite Group Databasebeta

← #34198813

Subgroups — #34198813

Order 20224.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774050C211yesmaximal, characteristic
2#23763155C211yesmaximal, characteristic, complemented
3#23761392C212yesmaximal, complemented
4#23761386C212yesmaximal, complemented
5C79×#2166C211yesmaximal, characteristic, complemented
6#19099168—21no—
7#19098965C411yescharacteristic, complemented
8#19098940C411yescharacteristic, complemented
9#19098928—21no—
10#19098816C2212yes—
11#19098799—21no—
12#19098676C412yescomplemented
13C4×D632—21no—
14#19098518—21no—
15C79⋊#338C2211yescharacteristic
16#19098320C2211yescharacteristic
17#19098319—21no—
18#405×C79C2211yescharacteristic
19#364×C79—21no—
20#349×C79C2212yes—
21#347×C79—21no—
22C2×D632—21no—
23C2×D632C2×C412yes—
24C2×#11929608—21no—
25#14579158—21no—
26#14579153—21no—
27#14579146—42no—
28#14579135—22no—
29#14579134—42no—
30C4×D316—22no—
31#14579120—21no—
32#14579095—21no—
33#14579093C2×C412yes—
34#14579083—22no—
35#14579083—22no—
36#14579071—22no—
37C8×Dic79—22no—
38C2×M16×C79C2311yescharacteristic
39C22×C8×C79D411yescharacteristic
40#117×C79D411yescharacteristic
41M32×C79—22no—
42#97×C79C2×C411yescharacteristic
43#97×C79D411yescharacteristic
44C4×C8×C79C2×C411yescharacteristic
45C4×C8×C79D411yescharacteristic
46C2×D316—22no—
47C2×D316—42no—
48C2×C22⋊D79—42no—
49C2×C4×D79—42no—
50D632—42no—
51D632—41no—
52D632—41no—
53#11929608—41no—
54#11929608—41no—
55C2×C79⋊C8—22no—
56C4⋊Dic79—22no—
57C4⋊Dic79—41no—
58Dic79⋊C4—42no—
59C22⋊Dic79—41no—
60(C2×D79)⋊C4—42no—
61(C2×D79)⋊C4—42no—
62C4×Dic79—22no—
63C22×C4×C79C2×D411yescharacteristic
64M16×C79—41no—
65C2×C8×C79—21no—
66C2×C8×C79—21no—
67C2×C8×C79C22⋊C411yescharacteristic
68C2×C8×C79C22⋊C411yescharacteristic
69C2×C8×C79C22×C411yescharacteristic
70C2×C8×C79C2×D411yescharacteristic
71C4⋊C4×C79—21no—
72C22⋊C4×C79—21no—
73C42×C79C22⋊C411yescharacteristic
74C42×C79C22⋊C411yescharacteristic
75C16×C79—42no—
76C23×C79AGL(1,ℤ/8)11yescharacteristic
77C22×D79—42no—
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79C2×C22⋊C411yescharacteristic
83C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
84C22⋊D79—162no—
85C2×Dic79—42no—
86C2×Dic79—42no—
87D316—42no—
88D316—42no—
89D316—162no—
90C4×D79—162no—
91C8×C79—21no—
92C8×C79—21no—
93C8×C79—21no—
94C8×C79#10411yescharacteristic
95C8×C79#10411yescharacteristic
96C79⋊C8—42no—
97C22×C79—21no—
98C22×C79—41no—
99C22×C79#41811yescharacteristic
100C2×D79—82no—
101C2×D79—82no—
102C2×D79—82no—
103C4×C79—21no—
104C4×C79—41no—
105C4×C79—41no—
106C4×C79#41011yescharacteristic
107C4×C79C2×#10411yescharacteristic
108Dic79—82no—
109Dic79—82no—
110#11770—791nomaximal, Sylow
111C2×C79—21no—
112C2×C79—41no—
113C2×C79#150511yescharacteristic
114D79—162no—
115#3176—791no—
116#2166D7911yescharacteristic, complemented
117#1747—791no—
118#1341—792no—
119#1339—792no—
120C79#1177011yescharacteristic, complemented, Sylow
121#540—1581no—
122#499—791no—
123#493—791no—
124#490—1581no—
125#469—792no—
126#466—1581no—
127#442—792no—
128C4×D8—1581no—
129#405—1581no—
130#405C2×D7911yescharacteristic
131#364—21no—
132#349C2×D7912yes—
133#347—21no—
134#338—791no—
135H(3,ℤ/4)—791no—
136H(3,ℤ/4)—1581no—
137C2×D8—792no—
138C2×D8—1581no—
139C2×M16—1581no—
140C2×M16C22×D7911yescharacteristic
141C22×C8D31611yescharacteristic
142#127—1581no—
143#125—1581no—
144#123—3162no—
145#121—3162no—
146#121—1582no—
147C4×D4—1582no—
148#117—1581no—
149#117D31611yescharacteristic
150M32—22no—
151#107—1581no—
152#106—792no—
153#102—1582no—
154#102—1582no—
155#97—1582no—
156#97C4×D7911yescharacteristic
157#97C22⋊D7911yescharacteristic
158C4×C8—1582no—
159C4×C8C4×D7911yescharacteristic
160C4×C8C22⋊D7911yescharacteristic
161C2×D4—3162no—
162C2×D4—3162no—
163C2×D4—1582no—
164C22×C4—3162no—
165C22×C4C2×D31611yescharacteristic
166D8—3162no—
167D8—3161no—
168D8—3161no—
169M16—41no—
170M16—3161no—
171M16—3161no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1582no—
175C2×C8(C2×D79)⋊C411yescharacteristic
176C2×C8(C2×D79)⋊C411yescharacteristic
177C2×C8C2×C4×D7911yescharacteristic
178C2×C8C2×C22⋊D7911yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—3162no—
181C4⋊C4—3161no—
182C4⋊C4—1582no—
183C22⋊C4—21no—
184C22⋊C4—3162no—
185C22⋊C4—3161no—
186C22⋊C4—3162no—
187C42—1582no—
188C42(C2×D79)⋊C411yescharacteristic
189C42(C2×D79)⋊C411yescharacteristic
190C16—42no—
191C23—3162no—
192C23#1457918911yescharacteristic
193D4—3162no—
194D4—3162no—
195D4—12642no—
196D4—12642no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—3162no—
202C2×C4—3162no—
203C2×C4—12642no—
204C2×C4C2×(C2×D79)⋊C411yescharacteristic
205C2×C4#1457918911yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—3162no—
210C8#1457908711yescharacteristic
211C8#1457908711yescharacteristic
212C22—21no—
213C22—41no—
214C22—6322no—
215C22—6322no—
216C22—6322no—
217C22#1909856911yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—6322no—
222C4—6322no—
223C4#1909853611yescharacteristic
224C4C2×#1457908711yescharacteristic
225C2—21no—
226C2—41no—
227C2—12642no—
228C2#2376190911yescharacteristic
subgroupG/H
centerC2#23761909
commutatorC2×C8×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#2166C2
socleC2×C79#1505
prime radical#23774050C2
cosocleC2×M16×C79C23
hypercenter#2166D79
hypocenterC79#11770
LCSG ▷ C2×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #23763155C2semidirect
3. #23761392C2semidirect
4. #23761386C2semidirect
5. C79×#2166C2semidirect
7. #19098965C4semidirect
8. #19098940C4semidirect
12. #19098676C4semidirect
116. #2166D79semidirect
120. C79#11770semidirect

Maximal subgroups

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

subgroupindex
#237740502
#237631552
#237613922
#237613862
C79×#21662
#1177079