Finite Group Databasebeta

← #28336958

Subgroups — #28336958

Order 13696.

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

normal subgroups only

This group has 310 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
1C2×#16812139C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic, complemented
3#20292644C211yesmaximal, characteristic
4#20292635C211yesmaximal, characteristic, complemented
5#20292397C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7#410×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9#16812139—22no—
10#16812139—22no—
11C2×#13133808C412yescomplemented
12C2×#13133806C412yescomplemented
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C4×Dic214—22no—
16#16812074—22no—
17#16812068C2211yescharacteristic
18#16812065—22no—
19#16812036—22no—
20#16812032—22no—
21C2×#41×C107C2211yescharacteristic
22#117×C107C2211yescharacteristic
23#117×C107C2211yescharacteristic
24#99×C107—22no—
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214C2×C412yes—
29C2×Dic214D412yes—
30C2×Dic214D412yes—
31C22×Dic107—21no—
32#13133808—24no—
33#13133806—24no—
34#13133802—21no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38C2×C107⋊C8—21no—
39C2×C107⋊C8C2×C412yes—
40C4⋊Dic107—21no—
41Dic107⋊C4—24no—
42C22⋊Dic107—21no—
43C4×Dic107—24no—
44C4×Dic107D412yes—
45#41×C107—21no—
46#41×C107—21no—
47#41×C107—21no—
48#41×C107—21no—
49C2×Q8×C107C2×C411yescharacteristic
50C2×D4×C107—21no—
51C2×D4×C107C2×C411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C22⋊C4×C107—21no—
57C22⋊C4×C107D412yes—
58C42×C107—21no—
59C42×C107—21no—
60C23×C107—21no—
61C23×C107C2×D411yescharacteristic
62Q8×C107—21no—
63Q8×C107—21no—
64D4×C107—21no—
65D4×C107—21no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107#4111yescharacteristic
78C2×C4×C107#4111yescharacteristic
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83Dic214—22no—
84Dic214—22no—
85Dic214—22no—
86Dic214C22⋊C414yes—
87C107⋊C8—22no—
88C107⋊C8—41no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107—81no—
92C22×C107C4×D411yescharacteristic
93C22×C107#12011yescharacteristic
94C22×C107#12111yescharacteristic
95C4×C107—42no—
96C4×C107—41no—
97C4×C107—81no—
98C4×C107C2×C22⋊C411yescharacteristic
99C4×C107C4×D411yescharacteristic
100C4×C107#12111yescharacteristic
101C4×C107#12311yescharacteristic
102Dic107—24no—
103Dic107—42no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107—81no—
108C2×C107#38611yescharacteristic
109#1905—1071nomaximal, Sylow
110C107#190511yescharacteristic, complemented, Sylow
111C2×#137—1071no—
112#519—1071no—
113#421—1071no—
114#419—1071no—
115#410D10711yescharacteristic
116#344—1071no—
117H(3,ℤ/4)—1071no—
118C2×#41C2×D10711yescharacteristic
119C22×Q8—1071no—
120#137—2142no—
121#137—2142no—
122C2×Q16—1072no—
123C2×SD16—1072no—
124C2×M16—1071no—
125C2×M16—1071no—
126C4×Q8—2142no—
127C4×Q8—2142no—
128#117—1071no—
129#117—2142no—
130#117C2×D10711yescharacteristic
131#117C2×D10711yescharacteristic
132#104—2142no—
133#103—2142no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—1072no—
140C2×Q8—1072no—
141C2×Q8—1072no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8C4×D10711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D10711yescharacteristic
148C22×C4—21no—
149C22×C4—2141no—
150C22×C4C22×D10711yescharacteristic
151Q16—2144no—
152SD16—2144no—
153M16—2141no—
154M16—2141no—
155M16—2141no—
156M16—2141no—
157C2×C8—1072no—
158C2×C8—2141no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2144no—
162C4⋊C4—2141no—
163C22⋊C4—21no—
164C22⋊C4—2141no—
165C22⋊C4C22⋊D10712yes—
166C42—21no—
167C42—21no—
168C42—1072no—
169C42—2144no—
170C23—21no—
171C23C2×C22⋊D10711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1074no—
175Q8—2142no—
176Q8—2142no—
177Q8—2142no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4D4×D10711yescharacteristic
195C2×C4#1313381711yescharacteristic
196C2×C4#1313381811yescharacteristic
197C8—4281no—
198C8—2142no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D10711yescharacteristic
203C22#1681207711yescharacteristic
204C22#1681208011yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—4282no—
209C4—2144no—
210C4C22⋊C4×D10711yescharacteristic
211C4(C22⋊D107)⋊C411yescharacteristic
212C4#1681208211yescharacteristic
213C4#1681209111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2029250311yescharacteristic
subgroupG/H
centerC2#20292503
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#386
prime radicalC2×#41×C107C22
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1905
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20293104C2semidirect
4. #20292635C2semidirect
5. #20292397C2semidirect
6. #20292383C2semidirect
11. C2×#13133808C4semidirect
12. C2×#13133806C4semidirect
110. C107#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
#202931042
#202926442
#202926352
#202923972
#202923832
#410×C1072
#1905107