Finite Group Databasebeta

← #34197481

Subgroups — #34197481

Order 20224.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#23772040C211yesmaximal, characteristic
2#23765200C212yesmaximal, complemented
3#23761341C212yesmaximal, complemented
4#23761326C211yesmaximal, characteristic
5C79×#1315C211yesmaximal, characteristic
6#41×Dic79C411yescharacteristic
7#19098621—22no—
8#19098613—22no—
9#19098591C411yescharacteristic
10#19098589C2211yescharacteristic
11#19098530C411yescharacteristic
12M16×Dic79C411yescharacteristic
13C2×C8×Dic79C2211yescharacteristic
14Dic79⋊C16—22no—
15C22⋊(C79⋊C16)C2212yes—
16#19098338—22no—
17#408×C79C2211yescharacteristic
18C22⋊C16×C79C2212yes—
19C2×#11929608C2×C411yescharacteristic
20C22×C79⋊C8C2×C411yescharacteristic
21Q8×Dic79C812yes—
22D4×Dic79—22no—
23D4×Dic79C812yes—
24#14579120—21no—
25#14579120C2×C411yescharacteristic
26C2×C4×Dic79—21no—
27C2×C4×Dic79C2×C411yescharacteristic
28C2×C79⋊C16—22no—
29Dic79⋊C8—22no—
30Dic79⋊C8—22no—
31C22⋊(C79⋊C8)—22no—
32#14579071—21no—
33C8×Dic79—21no—
34C8×Dic79D412yes—
35C8×Dic79D412yes—
36C2×#41×C79C2×C411yescharacteristic
37C2×M16×C79C2×C411yescharacteristic
38C22×C8×C79C2311yescharacteristic
39C2×C16×C79—22no—
40C22⋊C8×C79—22no—
41C22×Dic79—22no—
42C22×Dic79M1612yes—
43#11929608—41no—
44C2×C79⋊C8—21no—
45C2×C79⋊C8—41no—
46C2×C79⋊C8C22⋊C412yes—
47C4⋊Dic79—22no—
48C4⋊Dic79C2×C812yes—
49C22⋊Dic79—22no—
50C22⋊Dic79—21no—
51C4×Dic79—22no—
52C4×Dic79—21no—
53C4×Dic79C22⋊C411yescharacteristic
54C4×Dic79C22⋊C411yescharacteristic
55C4×Dic79C2×C812yes—
56C79⋊C16—42no—
57#41×C79—41no—
58#41×C79—41no—
59C2×Q8×C79C2×C811yescharacteristic
60C2×D4×C79—21no—
61C2×D4×C79C2×C811yescharacteristic
62C22×C4×C79—21no—
63C22×C4×C79C22×C411yescharacteristic
64M16×C79—41no—
65C2×C8×C79—21no—
66C2×C8×C79—41no—
67C2×C8×C79C2×D412yes—
68C16×C79—42no—
69C23×C79—21no—
70C23×C79C2×M1611yescharacteristic
71Q8×C79—41no—
72D4×C79—41no—
73D4×C79—42no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—21no—
77C2×C4×C79—41no—
78C2×C4×C79—41no—
79C2×C4×C79C2×C22⋊C411yescharacteristic
80C2×C4×C79C22×C811yescharacteristic
81C2×Dic79—22no—
82C2×Dic79—42no—
83C2×Dic79—44no—
84C2×Dic79C22⋊C812yes—
85C2×Dic79C22⋊C812yes—
86C8×C79—22no—
87C8×C79—41no—
88C79⋊C8—22no—
89C79⋊C8—41no—
90C22×C79—22no—
91C22×C79—21no—
92C22×C79—42no—
93C22×C79—41no—
94C22×C79C2×C22⋊C811yescharacteristic
95C4×C79—21no—
96C4×C79—41no—
97C4×C79#41011yescharacteristic
98C4×C79#41311yescharacteristic
99Dic79—22no—
100Dic79—22no—
101Dic79—42no—
102#11463—791nomaximal, Sylow
103C2×C79—41no—
104C2×C79—81no—
105C2×C79#145811yescharacteristic
106C2×C79#211712yes—
107#2922—791no—
108#2163—792no—
109#1323—792no—
110#1315—791no—
111#1315D7911yescharacteristic
112C79#1146311yescharacteristic, complemented, Sylow
113C4×#41—791no—
114#433—1582no—
115#431—1582no—
116#424—791no—
117#424—791no—
118#408—791no—
119#408C2×D7911yescharacteristic
120C4×M16—791no—
121C2×C4×C8—791no—
122C4⋊C16—1582no—
123C22⋊C16—792no—
124C22⋊C16C2×D7912yes—
125#346—1582no—
126C2×#41C4×D7911yescharacteristic
127C2×M16—791no—
128C2×M16C4×D7911yescharacteristic
129C22×C8—791no—
130C22×C8C22×D7911yescharacteristic
131C4×Q8—792no—
132C4×D4—792no—
133C4×D4—1582no—
134#117—791no—
135#117—1581no—
136C2×C42—791no—
137C2×C42—1581no—
138C2×C16—22no—
139C2×C16—1582no—
140C4⋊C8—1582no—
141C4⋊C8—1582no—
142C22⋊C8—22no—
143C22⋊C8—1582no—
144#97—1581no—
145C4×C8—792no—
146C4×C8—792no—
147C4×C8—1581no—
148#41—41no—
149#41—41no—
150C2×Q8C8×D7911yescharacteristic
151C2×D4—21no—
152C2×D4C8×D7911yescharacteristic
153C22×C4—21no—
154C22×C4—792no—
155C22×C4—1582no—
156C22×C4C2×C4×D7911yescharacteristic
157M16—41no—
158M16—3161no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—3161no—
162C2×C8—792no—
163C2×C8—1581no—
164C2×C8D4×D7912yes—
165C4⋊C4—792no—
166C4⋊C4—1582no—
167C22⋊C4—1582no—
168C22⋊C4—1581no—
169C42—791no—
170C42—791no—
171C42—792no—
172C42—1582no—
173C42—1581no—
174C16—42no—
175C16—3162no—
176C23—21no—
177C23M16×D7911yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3164no—
187C2×C4—3162no—
188C2×C4—792no—
189C2×C4—792no—
190C2×C4—1582no—
191C2×C4C22⋊C4×D7911yescharacteristic
192C2×C4C2×C8×D7911yescharacteristic
193C8—22no—
194C8—41no—
195C8—3161no—
196C8—1582no—
197C22—22no—
198C22—21no—
199C22—42no—
200C22—41no—
201C22C22⋊C8×D7911yescharacteristic
202C4—21no—
203C4—41no—
204C4—3162no—
205C4—1582no—
206C4—1582no—
207C4#1909853511yescharacteristic
208C4#1909854411yescharacteristic
209C2—41no—
210C2—81no—
211C2#2376171111yescharacteristic
212C2#2376502612yes—
subgroupG/H
centerC2×C4C22⋊C4×D79
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC79×#1315C2
socleC22×C79C2×C22⋊C8
prime radicalC2×#41×C79C2×C4
cosocleC22×C8×C79C23
hypercenter#1315D79
hypocenterC79#11463
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #23765200C2semidirect
3. #23761341C2semidirect
112. C79#11463semidirect

Maximal subgroups

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

subgroupindex
#237720402
#237652002
#237613412
#237613262
C79×#13152
#1146379