Finite Group Databasebeta

← #28337012

Subgroups — #28337012

Order 13696.

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

normal subgroups only

This group has 308 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#20293203C211yesmaximal, characteristic
3#20292609C211yesmaximal, characteristic
4C2×#16812027C211yesmaximal, characteristic, complemented
5C2×#16812027C211yesmaximal, characteristic, complemented
6#20292397C211yesmaximal, characteristic, complemented
7#344×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10#16812117—22no—
11#16812115—22no—
12C2×#13133802C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14M16×D107—22no—
15#16812109—21no—
16#16812097—21no—
17#16812093C414yes—
18#16812089—22no—
19#16812027—21no—
20#16812027—22no—
21#16812027—22no—
22#16812025—21no—
23C2×M16×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25#117×C107C2211yescharacteristic
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214—21no—
34C2×Dic214—21no—
35C2×Dic214—21no—
36C2×Dic214D411yescharacteristic
37C2×D428D411yescharacteristic
38C2×C22⋊D107C2×C412yes—
39C22×Dic107C2×C412yes—
40C2×C4×D107D412yes—
41#13133802—21no—
42#13133802—21no—
43#13133802—21no—
44#13133802D412yes—
45#13133801—24no—
46C2×C107⋊C8—21no—
47C2×C107⋊C8—21no—
48C8×D107—24no—
49C4⋊Dic107—24no—
50(C2×D107)⋊C4—24no—
51C22×C4×C107C2311yescharacteristic
52M16×C107—21no—
53M16×C107—21no—
54M16×C107—21no—
55M16×C107D412yes—
56C2×C8×C107—21no—
57C2×C8×C107—21no—
58C4⋊C4×C107—21no—
59C22⋊C4×C107C2×C412yes—
60C42×C107—21no—
61C23×C107C22×C411yescharacteristic
62C22×D107C22⋊C412yes—
63C2×C4×C107—22no—
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107#4111yescharacteristic
69C2×C4×C107#4111yescharacteristic
70C22⋊D107—22no—
71C22⋊D107—22no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107C22⋊C412yes—
76D428—21no—
77D428—21no—
78C4×D107—22no—
79C4×D107—22no—
80Dic214—21no—
81Dic214—21no—
82Dic214—41no—
83Dic214—41no—
84Dic214—41no—
85Dic214—41no—
86C8×C107—22no—
87C8×C107—41no—
88C107⋊C8—22no—
89C107⋊C8—41no—
90C22×C107—41no—
91C22×C107C2×C22⋊C411yescharacteristic
92C22×C107C4×D411yescharacteristic
93C22×C107C4×D411yescharacteristic
94C2×D107—22no—
95C2×D107—42no—
96C4×C107—42no—
97C4×C107#12011yescharacteristic
98C4×C107#12111yescharacteristic
99C4×C107#12111yescharacteristic
100C4×C107#12311yescharacteristic
101Dic107—22no—
102Dic107—42no—
103C2×C107—21no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107#38611yescharacteristic
107D107—42no—
108#1915—1071nomaximal, Sylow
109C107#191511yescharacteristic, complemented, Sylow
110#568—1071no—
111#536—1071no—
112#413—1071no—
113C2×#101—1071no—
114C2×#101—1071no—
115#344—1071no—
116#344D10711yescharacteristic, complemented
117C2×#41—1071no—
118C22×Q8—1071no—
119#131—2142no—
120#131—2142no—
121C2×M16—1071no—
122C2×M16—1071no—
123C2×M16—2142no—
124C2×M16C2×D10711yescharacteristic
125C2×M16C2×D10711yescharacteristic
126C4⋊Q8—2141no—
127#124—2141no—
128#123—1074no—
129C22⋊Q8—2142no—
130#117C2×D10711yescharacteristic
131#101—2142no—
132#101—2141no—
133#101—2142no—
134#100—2141no—
135#41—2141no—
136#41—2141no—
137#41—2141no—
138#41—2141no—
139C2×Q8—1071no—
140C2×Q8—2141no—
141C2×Q8—2141no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8—2141no—
146C2×D4—1072no—
147C2×D4—1071no—
148C22×C4—1072no—
149C22×C4—1072no—
150C22×C4C22×D10711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—1072no—
155M16—2144no—
156M16—2141no—
157M16—2141no—
158M16—2141no—
159M16D42812yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—2144no—
163C2×C8—2141no—
164C2×C8—2141no—
165C4⋊C4—21no—
166C4⋊C4—2144no—
167C22⋊C4—2144no—
168C22⋊C4C4×D10712yes—
169C42—21no—
170C23—1072no—
171C23C2×C4×D10711yescharacteristic
172Q8—4281no—
173Q8—4281no—
174Q8—4281no—
175Q8—4281no—
176Q8—2141no—
177Q8—2141no—
178D4—2142no—
179D4—2142no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—1072no—
184C2×C4—2142no—
185C2×C4—2142no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4D4×D10711yescharacteristic
190C2×C4D4×D10711yescharacteristic
191C2×C4D4×D10711yescharacteristic
192C2×C4C2×D42811yescharacteristic
193C2×C4#1313381811yescharacteristic
194C2×C4#1313381911yescharacteristic
195C8—22no—
196C8—41no—
197C8—4281no—
198C8—2142no—
199C22—41no—
200C22—4282no—
201C22—2142no—
202C22C22⋊C4×D10711yescharacteristic
203C22#1681207111yescharacteristic
204C22C4×D42811yescharacteristic
205C4—42no—
206C4—4282no—
207C4—2142no—
208C4#1681207911yescharacteristic
209C4#1681208211yescharacteristic
210C4#1681208411yescharacteristic
211C4#1681209411yescharacteristic
212C2—21no—
213C2—21no—
214C2—21no—
215C2—4282no—
216C2#2029250511yescharacteristic
subgroupG/H
centerC2#20292505
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#344×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenter#344D107
hypocenterC107#1915
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. C2×#16812027C2semidirect
5. C2×#16812027C2semidirect
6. #20292397C2semidirect
7. #344×C107C2semidirect
109. C107#1915semidirect
116. #344D107semidirect

Maximal subgroups

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

subgroupindex
#202933932
#202932032
#202926092
C2×#168120272
#202923972
#344×C1072
#1915107