Finite Group Databasebeta

← #34201611

Subgroups — #34201611

Order 20224.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765612C212yesmaximal, complemented
2#23764953C211yesmaximal, characteristic, complemented
3#23761385C212yesmaximal, complemented
4#23761338C211yesmaximal, characteristic, complemented
5C79×#3199C211yesmaximal, characteristic
6#19098612C2211yescharacteristic, complemented
7#19098589—21no—
8#19098580—21no—
9#19098385—22no—
10#19098376—22no—
11#19098345C2212yescomplemented
12#19098320C2211yescharacteristic, complemented
13#535×C79—21no—
14#482×C79—22no—
15#469×C79C2212yes—
16#466×C79—21no—
17#447×C79—22no—
18#408×C79C2211yescharacteristic
19C2×#11929608—21no—
20#14579120—21no—
21#14579101—22no—
22#14579097—22no—
23Dic79⋊C8—42no—
24#14579088—42no—
25C2×#41×C79D411yescharacteristic
26C22×D4×C79—21no—
27C2×D8×C79—21no—
28C2×D8×C79D412yes—
29C2×M16×C79C2311yescharacteristic
30C22×C8×C79D411yescharacteristic
31#127×C79—41no—
32#124×C79—41no—
33#121×C79—22no—
34#121×C79—22no—
35#120×C79—41no—
36#120×C79—41no—
37#117×C79—21no—
38#107×C79—21no—
39#106×C79D412yes—
40#102×C79—22no—
41#102×C79—22no—
42C22⋊C8×C79—22no—
43#11929608—41no—
44#11929608—41no—
45C2×C79⋊C8—41no—
46C4⋊Dic79—41no—
47C22⋊Dic79—41no—
48C4×Dic79—22no—
49C79⋊C16—42no—
50C24×C79—41no—
51#41×C79—41no—
52#41×C79—41no—
53C2×Q8×C79SD1611yescharacteristic
54C2×D4×C79—21no—
55C2×D4×C79—22no—
56C2×D4×C79—41no—
57C2×D4×C79—42no—
58C2×D4×C79—42no—
59C2×D4×C79—81no—
60C2×D4×C79—81no—
61C2×D4×C79SD1611yescharacteristic
62C22×C4×C79—21no—
63C22×C4×C79C2×D411yescharacteristic
64D8×C79—42no—
65D8×C79—81no—
66M16×C79—41no—
67C2×C8×C79—21no—
68C2×C8×C79—41no—
69C2×C8×C79C2×D412yes—
70C4⋊C4×C79—22no—
71C22⋊C4×C79—42no—
72C22⋊C4×C79—41no—
73C22⋊C4×C79—42no—
74C42×C79—41no—
75C23×C79—21no—
76C23×C79—42no—
77C23×C79—82no—
78C23×C79—162no—
79C23×C79AGL(1,ℤ/8)11yescharacteristic
80Q8×C79—41no—
81D4×C79—41no—
82D4×C79—42no—
83D4×C79—42no—
84D4×C79—81no—
85D4×C79—82no—
86D4×C79—162no—
87D4×C79—162no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—42no—
94C2×C4×C79#12011yescharacteristic
95C2×C4×C79C2×SD1611yescharacteristic
96C2×Dic79—42no—
97C8×C79—22no—
98C8×C79—41no—
99C79⋊C8—81no—
100C22×C79—21no—
101C22×C79—41no—
102C22×C79—42no—
103C22×C79—41no—
104C22×C79—82no—
105C22×C79—82no—
106C22×C79—82no—
107C22×C79—164no—
108C22×C79#45011yescharacteristic
109C4×C79—21no—
110C4×C79—41no—
111C4×C79—82no—
112C4×C79#45311yescharacteristic
113C4×C79#45811yescharacteristic
114Dic79—82no—
115#12301—791nomaximal, Sylow
116C2×C79—21no—
117C2×C79—41no—
118C2×C79—81no—
119C2×C79—162no—
120C2×C79#161311yescharacteristic
121#3199D7911yescharacteristic
122#2238—792no—
123#2098—791no—
124#1339—792no—
125#1321—791no—
126C79#1230111yescharacteristic, complemented, Sylow
127#535—21no—
128#482—22no—
129#469C2×D7912yes—
130#466—21no—
131#447—22no—
132#430—791no—
133#424—1581no—
134#421—1581no—
135#408C2×D7911yescharacteristic
136#365—1582no—
137#361—1582no—
138#349—792no—
139H(3,ℤ/4)—791no—
140C2×#41C22⋊D7911yescharacteristic
141C22×D4—21no—
142C2×D8—21no—
143C2×D8C22⋊D7912yes—
144C2×M16—1581no—
145C2×M16C22×D7911yescharacteristic
146C22×C8C22⋊D7911yescharacteristic
147#127—41no—
148#124—41no—
149#121—22no—
150#121—22no—
151#120—41no—
152#120—41no—
153#117—21no—
154#117—1581no—
155M32—1582no—
156#108—1582no—
157#107—21no—
158#106C22⋊D7912yes—
159C4⋊C8—3162no—
160#104—3162no—
161#102—22no—
162#102—22no—
163C22⋊C8—22no—
164C24—41no—
165#41—41no—
166#41—41no—
167C2×Q8#1192961211yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4#1192961211yescharacteristic
176C22×C4—21no—
177C22×C4C2×C22⋊D7911yescharacteristic
178D8—42no—
179D8—81no—
180M16—41no—
181M16—3161no—
182M16—3161no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—3161no—
186C2×C8C2×C22⋊D7912yes—
187C4⋊C4—22no—
188C4⋊C4—3161no—
189C22⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—3161no—
193C42—41no—
194C42—1582no—
195C16—3162no—
196C23—21no—
197C23—42no—
198C23—82no—
199C23—162no—
200C23#1457918711yescharacteristic
201Q8—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—82no—
207D4—162no—
208D4—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—3162no—
216C2×C4#1457913011yescharacteristic
217C2×C4C2×#1192961211yescharacteristic
218C8—22no—
219C8—41no—
220C8—6321no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—164no—
229C22#1909871411yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—6322no—
234C4#1909873111yescharacteristic
235C4#1909875311yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—162no—
240C2#2376242011yescharacteristic
subgroupG/H
centerC2#23762420
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
Fitting, prime radicalC79×#3199C2
socleC2×C79#1613
hypercenter#3199D79
hypocenterC79#12301
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3199

Complements

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

NH ≅ G/Ntype
1. #23765612C2semidirect
2. #23764953C2semidirect
3. #23761385C2semidirect
4. #23761338C2semidirect
6. #19098612C22semidirect
11. #19098345C22semidirect
12. #19098320C22semidirect
126. C79#12301semidirect

Maximal subgroups

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

subgroupindex
#237656122
#237649532
#237613852
#237613382
C79×#31992
#1230179