Finite Group Databasebeta

← #34197704

Subgroups — #34197704

Order 20224.

284 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 69 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#23772037C211yesmaximal, characteristic
2#23765209C212yesmaximal, complemented
3#23761351C211yesmaximal, characteristic, complemented
4#23761335C212yesmaximal
5C79×#1327C211yesmaximal, characteristic, complemented
6C2×#14579118C411yescharacteristic
7#19098623—22no—
8#19098623—22no—
9#19098587C411yescharacteristic
10#19098526C411yescharacteristic
11#19098526C2211yescharacteristic
12M16×Dic79C411yescharacteristic
13M16×Dic79C2211yescharacteristic
14#19098383—22no—
15#19098345C2212yes—
16#19098340—22no—
17#423×C79C2211yescharacteristic
18#349×C79C2212yes—
19C22×C4×D79C2×C411yescharacteristic
20C2×#11929608C2×C411yescharacteristic
21C2×#11929608C2×C411yescharacteristic
22#14579118—22no—
23#14579118—21no—
24#14579118—21no—
25C2×Dic79⋊C4—21no—
26C2×(C2×D79)⋊C4—21no—
27C2×C4×Dic79C2×C411yescharacteristic
28#14579101—22no—
29C4⋊(C79⋊C8)—22no—
30(C2×D79)⋊C8—22no—
31(C2×D79)⋊C8—22no—
32#14579071—21no—
33#14579071D412yes—
34C8×Dic79—21no—
35C8×Dic79D412yes—
36C2×M16×C79C2×C411yescharacteristic
37C2×M16×C79C2311yescharacteristic
38C2×C42×C79C2×C411yescharacteristic
39M32×C79—22no—
40C4⋊C8×C79—22no—
41C23×D79M1611yescharacteristic
42C22×Dic79—21no—
43C22×Dic79M1611yescharacteristic
44C2×C4×D79—21no—
45C2×C4×D79—21no—
46C2×C4×D79—41no—
47C2×C4×D79—41no—
48#11929608—41no—
49#11929608—41no—
50C2×C79⋊C8—21no—
51C2×C79⋊C8C22⋊C412yes—
52Dic79⋊C4—22no—
53Dic79⋊C4—22no—
54(C2×D79)⋊C4—22no—
55(C2×D79)⋊C4—22no—
56C4×Dic79—21no—
57C4×Dic79C22⋊C411yescharacteristic
58C4×Dic79C22⋊C411yescharacteristic
59C79⋊C16—42no—
60C22×C4×C79—21no—
61C22×C4×C79C22×C411yescharacteristic
62M16×C79—41no—
63M16×C79—41no—
64C2×C8×C79—21no—
65C2×C8×C79C2×D412yes—
66C42×C79—21no—
67C42×C79M1612yes—
68C16×C79—42no—
69C23×C79C2×M1611yescharacteristic
70C22×D79—21no—
71C22×D79—41no—
72C22×D79—81no—
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—22no—
76C2×C4×C79—81no—
77C2×C4×C79C2×C22⋊C411yescharacteristic
78C2×C4×C79C2×M1611yescharacteristic
79C2×Dic79—21no—
80C2×Dic79—21no—
81C2×Dic79—21no—
82C2×Dic79—41no—
83C2×Dic79—81no—
84C4×D79—42no—
85C4×D79—41no—
86C4×D79—41no—
87C8×C79—22no—
88C8×C79—41no—
89C79⋊C8—22no—
90C79⋊C8—41no—
91C22×C79—21no—
92C22×C79—41no—
93C22×C79#40711yescharacteristic
94C2×D79—42no—
95C2×D79—41no—
96C2×D79—82no—
97C4×C79—21no—
98C4×C79—42no—
99C4×C79#41011yescharacteristic
100C4×C79#41311yescharacteristic
101Dic79—41no—
102Dic79—41no—
103Dic79—41no—
104#11522—791nomaximal, Sylow
105C2×C79—21no—
106C2×C79—41no—
107C2×C79#146411yescharacteristic
108D79—81no—
109#2921—791no—
110#2165—792no—
111#1327—791no—
112#1327D7911yescharacteristic, complemented
113#1320—792no—
114C79#1152211yescharacteristic, complemented, Sylow
115C2×#117—791no—
116#433—1582no—
117#433—1582no—
118#423—791no—
119#423C2×D7911yescharacteristic
120#407—791no—
121#407—791no—
122C4×M16—791no—
123C4×M16—791no—
124#364—1582no—
125#349—792no—
126#349C2×D7912yes—
127#347—1582no—
128C23×C4—791no—
129C2×M16—791no—
130C2×M16—791no—
131C2×M16C4×D7911yescharacteristic
132C2×M16C22×D7911yescharacteristic
133#117—1581no—
134#117—1581no—
135#117—1582no—
136C2×C4⋊C4—1581no—
137C2×C22⋊C4—1581no—
138C2×C42—791no—
139C2×C42C4×D7911yescharacteristic
140M32—22no—
141M32—1582no—
142C4⋊C8—22no—
143C4⋊C8—1582no—
144C22⋊C8—1582no—
145C22⋊C8—1582no—
146#97—792no—
147#97—1581no—
148C4×C8—792no—
149C4×C8—1581no—
150C24—791no—
151C22×C4—21no—
152C22×C4—3161no—
153C22×C4—3161no—
154C22×C4—791no—
155C22×C4—1581no—
156C22×C4—1581no—
157C22×C4—1581no—
158C22×C4C2×C4×D7911yescharacteristic
159M16—41no—
160M16—41no—
161M16—3161no—
162M16—3161no—
163C2×C8—21no—
164C2×C8—792no—
165C2×C8—1581no—
166C2×C8D4×D7912yes—
167C4⋊C4—1582no—
168C4⋊C4—1582no—
169C22⋊C4—1582no—
170C22⋊C4—1582no—
171C42—21no—
172C42—791no—
173C42—791no—
174C42—1581no—
175C42#1192960712yes—
176C16—42no—
177C16—3162no—
178C23—3161no—
179C23—6321no—
180C23—1581no—
181C23M16×D7911yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—81no—
186C2×C4—3162no—
187C2×C4—3161no—
188C2×C4—3161no—
189C2×C4—3161no—
190C2×C4—6321no—
191C2×C4—1581no—
192C2×C4—1581no—
193C2×C4—1581no—
194C2×C4C22⋊C4×D7911yescharacteristic
195C2×C4C2×#1192960711yescharacteristic
196C8—22no—
197C8—41no—
198C8—3161no—
199C8—1582no—
200C22—21no—
201C22—41no—
202C22—3162no—
203C22—3161no—
204C22—6322no—
205C22#1909852511yescharacteristic
206C4—21no—
207C4—42no—
208C4—3161no—
209C4—3161no—
210C4—3161no—
211C4#1909853511yescharacteristic
212C4#1909854611yescharacteristic
213C2—21no—
214C2—41no—
215C2—6321no—
216C2#2376173311yescharacteristic
subgroupG/H
centerC4#19098535
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#1327C2
socleC2×C79#1464
prime radicalC23×D79M16
cosocleC2×M16×C79C23
hypercenter#1327D79
hypocenterC79#11522
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #23765209C2semidirect
3. #23761351C2semidirect
5. C79×#1327C2semidirect
112. #1327D79semidirect
114. C79#11522semidirect

Maximal subgroups

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

subgroupindex
#237720372
#237652092
#237613512
#237613352
C79×#13272
#1152279