Finite Group Databasebeta

← #1356×D79 #34199441

Subgroups — #1356×D79 #34199441

Order 20224.

314 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 71 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#491×D79C211yesmaximal, characteristic
2#347×D79C212yesmaximal
3#23761435C211yesmaximal, characteristic, complemented
4#23761434C212yesmaximal, complemented
5C79×#1356C211yesmaximal, characteristic, complemented
6C4⋊Q8×D79C411yescharacteristic
7#125×D79C411yescharacteristic
8M32×D79—22no—
9#19098933C411yescharacteristic
10#19098932C2211yescharacteristic
11#19098931C411yescharacteristic
12#19098930—22no—
13#103×D79—22no—
14#97×D79C2211yescharacteristic
15#19098341C2212yes—
16#19098340—22no—
17#491×C79C2211yescharacteristic
18#347×C79C2212yes—
19C2×Q8×D79—21no—
20C2×C8×D79D412yes—
21#14579161C2×C411yescharacteristic
22#14579160—41no—
23#14579159—21no—
24#14579153C2×C411yescharacteristic
25#14579152—21no—
26#14579151—41no—
27C4⋊C4×D79—21no—
28C4⋊C4×D79—41no—
29C4⋊C4×D79—41no—
30C42×D79C2×C411yescharacteristic
31#14579101—22no—
32#14579100—24no—
33C16×D79—24no—
34#14579086—22no—
35#14579085—22no—
36#14579084—22no—
37#14579072C2×C411yescharacteristic
38#14579071D412yes—
39C4⋊Q8×C79C2×C411yescharacteristic
40#125×C79C2×C411yescharacteristic
41M32×C79—22no—
42#103×C79—22no—
43#97×C79C2311yescharacteristic
44C2×Dic158—21no—
45C2×Dic158—41no—
46Q8×D79—42no—
47Q8×D79—42no—
48C2×C4×D79—21no—
49C2×C4×D79—41no—
50C2×C4×D79—41no—
51C2×C4×D79C22⋊C411yescharacteristic
52C2×C79⋊C8C22⋊C412yes—
53C8×D79—22no—
54C8×D79—22no—
55C4⋊Dic79—21no—
56C4⋊Dic79—41no—
57C4⋊Dic79—41no—
58Dic79⋊C4—41no—
59Dic79⋊C4—41no—
60Dic79⋊C4—42no—
61Dic79⋊C4—41no—
62C4×Dic79—21no—
63C4×Dic79C22⋊C411yescharacteristic
64C79⋊C16—24no—
65C2×Q8×C79—21no—
66C2×C8×C79C2×D412yes—
67C4⋊C4×C79—21no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—41no—
70C42×C79C22×C411yescharacteristic
71C16×C79—24no—
72C22×D79#10011yescharacteristic
73Q8×C79—42no—
74C2×C4×C79—21no—
75C2×C4×C79—41no—
76C2×C4×C79—41no—
77C2×C4×C79C2×C22⋊C411yescharacteristic
78C2×Dic79—21no—
79C2×Dic79—41no—
80C2×Dic79—41no—
81C2×Dic79#10011yescharacteristic
82C4×D79—42no—
83C4×D79—82no—
84C4×D79—81no—
85C4×D79—81no—
86C4×D79#10412yes—
87C4×D79#10412yes—
88Dic158—42no—
89Dic158—82no—
90C8×C79—22no—
91C79⋊C8—22no—
92C22×C79C2×#10011yescharacteristic
93C2×D79—22no—
94C2×D79#32911yescharacteristic
95C2×D79#32911yescharacteristic
96C4×C79—41no—
97C4×C79—81no—
98C4×C79—81no—
99C4×C79C2×#10412yes—
100Dic79—41no—
101Dic79—81no—
102Dic79—81no—
103Dic79#32912yes—
104C2×#1356—791nomaximal, Sylow
105C2×C79—21no—
106C2×C79C2×#32911yescharacteristic
107D79—21no—
108D79#135612yesdirect product factor
109C2×#491—791no—
110C2×#347—792no—
111#1356—791no—
112#1356—792no—
113#1356D7911yescharacteristic, direct product factor
114C79C2×#135611yescharacteristic, complemented, Sylow
115C2×C4⋊Q8—791no—
116C2×#125—791no—
117C2×M32—1582no—
118#491—791no—
119#491—791no—
120#491—791no—
121#491—1582no—
122#491C2×D7911yescharacteristic
123C2×#103—1582no—
124C2×#97—791no—
125#347—792no—
126#347—1582no—
127#347C2×D7912yes—
128C22×Q8—1581no—
129C22×C8—792no—
130C4⋊Q8—3161no—
131C4⋊Q8—791no—
132C4⋊Q8—1581no—
133C4⋊Q8C4×D7911yescharacteristic
134#125—3161no—
135#125—791no—
136#125—1581no—
137#125C4×D7911yescharacteristic
138C2×C4⋊C4—3161no—
139C2×C4⋊C4—3161no—
140C2×C4⋊C4—1581no—
141C2×C42—791no—
142M32—22no—
143M32—1584no—
144M32—1582no—
145C2×C16—1584no—
146#103—22no—
147#103—1582no—
148#103—1582no—
149#103—1582no—
150#97—792no—
151#97—791no—
152#97C22×D7911yescharacteristic
153C2×Q8—21no—
154C2×Q8—3161no—
155C2×Q8—3162no—
156C2×Q8—3162no—
157C2×Q8—1581no—
158C22×C4—3161no—
159C22×C4—3161no—
160C22×C4—791no—
161C22×C4—1581no—
162C2×C8—792no—
163C2×C8—1582no—
164C2×C8—1582no—
165C2×C8D4×D7912yes—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—3161no—
170C4⋊C4—3161no—
171C4⋊C4—3161no—
172C4⋊C4—3161no—
173C4⋊C4—3162no—
174C4⋊C4—3161no—
175C4⋊C4—1581no—
176C42—791no—
177C42—1581no—
178C42C2×C4×D7911yescharacteristic
179C16—24no—
180C16—1584no—
181C23—791no—
182Q8—42no—
183Q8—3162no—
184Q8—6322no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—3162no—
189C2×C4—3161no—
190C2×C4—3161no—
191C2×C4—791no—
192C2×C4—792no—
193C2×C4—792no—
194C2×C4—6321no—
195C2×C4—6321no—
196C2×C4—6322no—
197C2×C4—1581no—
198C2×C4C22⋊C4×D7911yescharacteristic
199C8—22no—
200C8—1582no—
201C22—791no—
202C22—791no—
203C22—1582no—
204C22#100×D7911yescharacteristic
205C4—41no—
206C4—81no—
207C4—81no—
208C4—3161no—
209C4—792no—
210C4—6321no—
211C4—6321no—
212C4#104×D7912yes—
213C2—21no—
214C2—792no—
215C2—1581no—
216C2#329×D7911yescharacteristic
subgroupG/H
centerC2#329×D79
commutatorC42×C79C22×C4
Frattini#97C22×D79
FittingC79×#1356C2
socleC2×C79C2×#329
prime radicalC22×D79#100
cosocle#97×C79C23
hypercenter#1356D79
hypocenterC79C2×#1356
LCSG ▷ C42×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
3. #23761435C2semidirect
4. #23761434C2semidirect
5. C79×#1356C2semidirect
108. D79#1356direct (N × H)
113. #1356D79direct (N × H)
114. C79C2×#1356semidirect

Maximal subgroups

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

subgroupindex
#491×D792
#347×D792
#237614352
#237614342
C79×#13562
C2×#135679