Finite Group Databasebeta

← #28336997

Subgroups — #28336997

Order 13696.

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

normal subgroups only

This group has 314 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#20293203C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic, complemented
3#20292645C211yesmaximal, characteristic
4#20292643C211yesmaximal, characteristic
5C2×#16812027C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10C2×#13133802C2211yescharacteristic
11#16812109C412yescomplemented
12#16812097C412yescomplemented
13#16812093—22no—
14#16812089—22no—
15C4×Dic214—22no—
16#16812074—22no—
17#16812068C2211yescharacteristic
18#16812065—22no—
19#16812037—22no—
20#16812035—22no—
21#16812027—22no—
22C2×M16×C107C2211yescharacteristic
23#117×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#13133818—21no—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214C2×C411yescharacteristic
34C2×Dic214C2×C412yes—
35C2×Dic214D412yes—
36C2×D428C2×C411yescharacteristic
37C2×C22⋊D107—21no—
38C22×Dic107D412yes—
39C2×C4×D107—21no—
40#13133802—21no—
41#13133802—21no—
42C2×C107⋊C8—21no—
43C4⋊Dic107—21no—
44C4⋊Dic107—24no—
45Dic107⋊C4—24no—
46C22⋊Dic107—21no—
47(C2×D107)⋊C4—24no—
48C4×Dic107—24no—
49C4×Dic107D412yes—
50C22×C4×C107C2311yescharacteristic
51M16×C107—21no—
52M16×C107—21no—
53C2×C8×C107—21no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C22⋊C4×C107—21no—
57C22⋊C4×C107—21no—
58C42×C107C2×C412yes—
59C42×C107D412yes—
60C23×C107C2×D411yescharacteristic
61C22×D107—21no—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107C22×C411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107#4111yescharacteristic
69C2×C4×C107#4111yescharacteristic
70C22⋊D107—41no—
71C22⋊D107—41no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—21no—
75C2×Dic107—22no—
76C2×Dic107C22⋊C414yes—
77D428—21no—
78D428—21no—
79C4×D107—41no—
80C4×D107—41no—
81Dic214—22no—
82Dic214—22no—
83Dic214—22no—
84Dic214—22no—
85Dic214—21no—
86Dic214—21no—
87C8×C107—41no—
88C107⋊C8—41no—
89C22×C107—41no—
90C22×C107C2×C22⋊C411yescharacteristic
91C22×C107#12011yescharacteristic
92C22×C107#12311yescharacteristic
93C2×D107—42no—
94C2×D107—41no—
95C4×C107—42no—
96C4×C107—42no—
97C4×C107C4×D411yescharacteristic
98C4×C107C4×D411yescharacteristic
99C4×C107#12111yescharacteristic
100C4×C107#12111yescharacteristic
101Dic107—24no—
102Dic107—42no—
103Dic107—41no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107#38611yescharacteristic
108D107—81no—
109#1912—1071nomaximal, Sylow
110C107#191211yescharacteristic, complemented, Sylow
111#536—1071no—
112#519—1071no—
113#421—1071no—
114#421—1071no—
115C2×#101—1071no—
116H(3,ℤ/4)—1071no—
117H(3,ℤ/4)D10711yescharacteristic, complemented
118C2×#41—1071no—
119C22×Q8—1071no—
120C2×M16—1071no—
121C2×M16C2×D10711yescharacteristic
122C4⋊Q8—1072no—
123#124—1072no—
124#123—2142no—
125C22⋊Q8—2142no—
126C4×Q8—2142no—
127C4×Q8—2142no—
128#117—1071no—
129#117—2142no—
130#117C2×D10711yescharacteristic
131#117C2×D10711yescharacteristic
132#104—2142no—
133#104—2142no—
134#101—2142no—
135#41—2141no—
136#41—2141no—
137#41—2141no—
138#41—2141no—
139C2×Q8—1072no—
140C2×Q8—1072no—
141C2×Q8—1071no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8—2141no—
146C2×D4—1071no—
147C2×D4—2141no—
148C22×C4—1072no—
149C22×C4—2141no—
150C22×C4C22×D10711yescharacteristic
151M16—21no—
152M16—21no—
153M16—2141no—
154M16—2141no—
155C2×C8—21no—
156C2×C8—2141no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—2144no—
160C4⋊C4—2144no—
161C4⋊C4—2141no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—2144no—
165C22⋊C4—2141no—
166C42—1072no—
167C42—2144no—
168C42C4×D10712yes—
169C42D42812yes—
170C23—2141no—
171C23D4×D10711yescharacteristic
172Q8—2141no—
173Q8—2141no—
174Q8—2142no—
175Q8—2142no—
176Q8—2142no—
177Q8—2142no—
178D4—4281no—
179D4—4281no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1074no—
185C2×C4—4281no—
186C2×C4—4281no—
187C2×C4—2141no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4C2×D42811yescharacteristic
195C2×C4#1313381811yescharacteristic
196C2×C4#1313381911yescharacteristic
197C8—41no—
198C8—4281no—
199C22—41no—
200C22—4282no—
201C22—4281no—
202C22C22⋊C4×D10711yescharacteristic
203C22#1681207911yescharacteristic
204C22#1681209411yescharacteristic
205C4—42no—
206C4—42no—
207C4—4282no—
208C4—4281no—
209C4—2144no—
210C4#1681207111yescharacteristic
211C4C4×D42811yescharacteristic
212C4#1681208211yescharacteristic
213C4#1681208411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—8561no—
218C2#2029250511yescharacteristic
subgroupG/H
centerC2#20292505
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingH(3,ℤ/4)×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenterH(3,ℤ/4)D107
hypocenterC107#1912
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #20293104C2semidirect
5. C2×#16812027C2semidirect
6. #20292383C2semidirect
7. H(3,ℤ/4)×C107C2semidirect
11. #16812109C4semidirect
12. #16812097C4semidirect
110. C107#1912semidirect
117. H(3,ℤ/4)D107semidirect

Maximal subgroups

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

subgroupindex
#202932032
#202931042
#202926452
#202926432
C2×#168120272
#202923832
H(3,ℤ/4)×C1072
#1912107