Finite Group Databasebeta

← #28336980

Subgroups — #28336980

Order 13696.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293393C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic
3#20292636C211yesmaximal, characteristic, complemented
4#20292600C211yesmaximal, characteristic
5#20292384C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7#419×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812117—22no—
10#16812115—22no—
11C2×#13133802C2211yescharacteristic
12M16×D107—22no—
13#16812107C412yescomplemented
14#16812098C412yescomplemented
15#16812092—22no—
16#16812085—22no—
17#16812068C2211yescharacteristic
18#16812068C2211yescharacteristic
19#16812033—22no—
20#16812023—22no—
21C22×Q8×C107C2211yescharacteristic
22C2×M16×C107C2211yescharacteristic
23#117×C107C2211yescharacteristic
24#103×C107—22no—
25#13133818—21no—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29C2×Dic214C2×C411yescharacteristic
30C2×D428C2×C411yescharacteristic
31C2×C22⋊D107D412yes—
32C2×C4×D107D412yes—
33#13133802—21no—
34#13133802—21no—
35#13133801—24no—
36C2×C107⋊C8D412yes—
37C8×D107—24no—
38C4⋊Dic107—21no—
39C4⋊Dic107—21no—
40Dic107⋊C4—24no—
41C22⋊Dic107—21no—
42C22⋊Dic107—21no—
43(C2×D107)⋊C4—24no—
44C4×Dic107—21no—
45C4×Dic107C2×C412yes—
46C2×Q8×C107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107C2×C412yes—
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54M16×C107—21no—
55M16×C107—21no—
56C2×C8×C107D412yes—
57C4⋊C4×C107—21no—
58C22⋊C4×C107—21no—
59C42×C107—21no—
60C23×C107C2×D411yescharacteristic
61C22×D107C22⋊C412yes—
62Q8×C107—22no—
63Q8×C107—22no—
64Q8×C107—41no—
65Q8×C107—41no—
66C2×C4×C107—22no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107C22×C411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107#4111yescharacteristic
75C2×C4×C107#4111yescharacteristic
76C22⋊D107—22no—
77C22⋊D107—22no—
78C2×Dic107—22no—
79C2×Dic107—41no—
80C2×Dic107C22⋊C412yes—
81D428—21no—
82D428—21no—
83C4×D107—22no—
84C4×D107—22no—
85Dic214—21no—
86Dic214—21no—
87C8×C107—22no—
88C107⋊C8—22no—
89C22×C107—41no—
90C22×C107C2×C22⋊C411yescharacteristic
91C22×C107#12111yescharacteristic
92C22×C107#12111yescharacteristic
93C2×D107—22no—
94C2×D107—42no—
95C4×C107—42no—
96C4×C107—81no—
97C4×C107C4×D411yescharacteristic
98C4×C107C4×D411yescharacteristic
99C4×C107#12011yescharacteristic
100C4×C107#12311yescharacteristic
101Dic107—22no—
102Dic107—42no—
103Dic107—81no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107#38611yescharacteristic
108D107—42no—
109#1909—1071nomaximal, Sylow
110C107#190911yescharacteristic, complemented, Sylow
111#568—1071no—
112#536—1071no—
113#419—1071no—
114#419D10711yescharacteristic, complemented
115#410—1071no—
116H(3,ℤ/4)—1071no—
117H(3,ℤ/4)—1071no—
118C2×#41—1071no—
119C22×Q8C2×D10711yescharacteristic
120#131—2142no—
121#131—2142no—
122C2×M16—1071no—
123C2×M16—2142no—
124C2×M16C2×D10711yescharacteristic
125C4⋊Q8—1072no—
126#124—1072no—
127#123—2142no—
128C22⋊Q8—2142no—
129#117—1071no—
130#117—1071no—
131#117C2×D10711yescharacteristic
132#103—22no—
133#103—2142no—
134#99—2142no—
135#41—2141no—
136#41—2141no—
137#41—2141no—
138#41—2141no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—1071no—
145C2×Q8C4×D10712yes—
146C2×D4—1072no—
147C2×D4—1071no—
148C22×C4—21no—
149C22×C4—1072no—
150C22×C4C22×D10711yescharacteristic
151M16—21no—
152M16—21no—
153M16—2144no—
154M16—2141no—
155M16—2141no—
156C2×C8—1072no—
157C2×C8—2144no—
158C2×C8C22⋊D10712yes—
159C4⋊C4—21no—
160C4⋊C4—2144no—
161C4⋊C4—2141no—
162C4⋊C4—2141no—
163C22⋊C4—21no—
164C22⋊C4—2144no—
165C22⋊C4—2141no—
166C22⋊C4—2141no—
167C42—21no—
168C42—1072no—
169C42—2141no—
170C23—1072no—
171C23D4×D10711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—2141no—
177Q8—2141no—
178D4—2142no—
179D4—2142no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1072no—
187C2×C4—4281no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4C2×C22⋊D10711yescharacteristic
195C2×C4#1313381711yescharacteristic
196C2×C4#1313381811yescharacteristic
197C8—22no—
198C8—2142no—
199C22—41no—
200C22—4282no—
201C22—2142no—
202C22C22⋊C4×D10711yescharacteristic
203C22#1681208011yescharacteristic
204C22#1681208211yescharacteristic
205C4—42no—
206C4—81no—
207C4—8561no—
208C4—4282no—
209C4—2142no—
210C4(C22⋊D107)⋊C411yescharacteristic
211C4C4×C22⋊D10711yescharacteristic
212C4#1681207711yescharacteristic
213C4#1681209111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—4282no—
218C2#2029250311yescharacteristic
subgroupG/H
centerC2#20292503
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#419×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenter#419D107
hypocenterC107#1909
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20292636C2semidirect
5. #20292384C2semidirect
6. #20292383C2semidirect
7. #419×C107C2semidirect
13. #16812107C4semidirect
14. #16812098C4semidirect
110. C107#1909semidirect
114. #419D107semidirect

Maximal subgroups

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

subgroupindex
#202933932
#202932012
#202926362
#202926002
#202923842
#202923832
#419×C1072
#1909107