Finite Group Databasebeta

← #28337010

Subgroups — #28337010

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#20293201C211yesmaximal, characteristic
3#20292609C211yesmaximal, characteristic
4C2×#16812028C211yesmaximal, characteristic, complemented
5#20292399C211yesmaximal, characteristic, complemented
6#20292398C211yesmaximal, characteristic, complemented
7C2×#101×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812117—22no—
10#16812115—22no—
11C2×#13133802C2211yescharacteristic
12C2×#13133802C2211yescharacteristic
13M16×D107—22no—
14#16812107—21no—
15#16812098—21no—
16#16812092C414yes—
17#16812085—22no—
18#16812068C2211yescharacteristic
19#16812028—22no—
20#16812027—21no—
21#16812025—21no—
22C22×Q8×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25#101×C107—22no—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30C2×Dic214D411yescharacteristic
31C2×D428D411yescharacteristic
32C2×C22⋊D107C2×C412yes—
33C2×C4×D107D412yes—
34#13133802—21no—
35#13133802—21no—
36#13133802—21no—
37#13133802D412yes—
38#13133801—24no—
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C8×D107—24no—
42C4⋊Dic107—21no—
43Dic107⋊C4—24no—
44C22⋊Dic107C2×C412yes—
45(C2×D107)⋊C4—24no—
46C4×Dic107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107—21no—
52C2×Q8×C107—21no—
53C22×C4×C107C2×C412yes—
54C22×C4×C107C2311yescharacteristic
55M16×C107—21no—
56M16×C107—21no—
57M16×C107—21no—
58M16×C107D412yes—
59C2×C8×C107—21no—
60C2×C8×C107—21no—
61C23×C107C22×C411yescharacteristic
62C22×D107C22⋊C412yes—
63Q8×C107—41no—
64Q8×C107—41no—
65Q8×C107—41no—
66Q8×C107—41no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107C2×D411yescharacteristic
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×Dic107C22⋊C412yes—
80D428—21no—
81D428—21no—
82C4×D107—22no—
83C4×D107—22no—
84Dic214—21no—
85Dic214—21no—
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×#101D10711yescharacteristic, complemented
115#344—1071no—
116#344—1071no—
117C2×#41—1071no—
118C22×Q8C2×D10711yescharacteristic
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#117—1071no—
131#101—22no—
132#101—2141no—
133#101—2142no—
134#100—2141no—
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—21no—
145C2×Q8—1071no—
146C2×D4—1072no—
147C2×D4—1071no—
148C22×C4—1072no—
149C22×C4C4×D10712yes—
150C22×C4C22×D10711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—1072no—
155M16—2144no—
156M16—2141no—
157M16—2141no—
158M16—2141no—
159M16C22⋊D10712yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—2144no—
163C2×C8—2141no—
164C2×C8—2141no—
165C4⋊C4—2144no—
166C4⋊C4—2141no—
167C22⋊C4—1072no—
168C22⋊C4—2144no—
169C42—2141no—
170C23—1072no—
171C23C2×C4×D10711yescharacteristic
172Q8—41no—
173Q8—41no—
174Q8—41no—
175Q8—41no—
176Q8—2141no—
177Q8—2141no—
178D4—2142no—
179D4—2142no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—1072no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4D4×D10711yescharacteristic
190C2×C4D4×D10711yescharacteristic
191C2×C4D4×D10711yescharacteristic
192C2×C4C2×C22⋊D10711yescharacteristic
193C2×C4#1313381711yescharacteristic
194C2×C4#1313381811yescharacteristic
195C8—22no—
196C8—41no—
197C8—4281no—
198C8—2142no—
199C22—41no—
200C22—4282no—
201C22—2142no—
202C22C22⋊C4×D10711yescharacteristic
203C22(C22⋊D107)⋊C411yescharacteristic
204C22C4×C22⋊D10711yescharacteristic
205C4—42no—
206C4—4282no—
207C4—2142no—
208C4#1681207711yescharacteristic
209C4#1681208011yescharacteristic
210C4#1681208211yescharacteristic
211C4#1681209111yescharacteristic
212C2—21no—
213C2—21no—
214C2—21no—
215C2—4282no—
216C2#2029250311yescharacteristic
subgroupG/H
centerC2#20292503
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#101×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenterC2×#101D107
hypocenterC107#1915
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
4. C2×#16812028C2semidirect
5. #20292399C2semidirect
6. #20292398C2semidirect
7. C2×#101×C107C2semidirect
109. C107#1915semidirect
114. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
#202933932
#202932012
#202926092
C2×#168120282
#202923992
#202923982
C2×#101×C1072
#1915107