Finite Group Databasebeta

← #28338148

Subgroups — #28338148

Order 13696.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#20293201C211yesmaximal, characteristic
2C2×#16812028C211yesmaximal, characteristic, complemented
3#20292425C212yesmaximal, complemented
4#20292423C212yesmaximal, complemented
5C2×#101×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C411yescharacteristic
7C2×#13133802—21no—
8#16812107C411yescharacteristic
9#16812107C2211yescharacteristic
10#16812098C411yescharacteristic
11#16812098C2211yescharacteristic
12#16812092—21no—
13#16812092—21no—
14#16812085—21no—
15#16812085—21no—
16#16812068C411yescharacteristic
17#16812028—21no—
18#16812028C2212yes—
19C22×Q8×C107C2211yescharacteristic
20C2×M16×C107—21no—
21#101×C107—21no—
22#101×C107C2212yes—
23#13133818—41no—
24#13133818—41no—
25C2×Dic214C2×C411yescharacteristic
26C2×D428C2×C411yescharacteristic
27C2×C22⋊D107—21no—
28C2×C4×D107—21no—
29#13133802—22no—
30#13133802—41no—
31C2×C107⋊C8—41no—
32C4⋊Dic107—21no—
33Dic107⋊C4—21no—
34Dic107⋊C4—21no—
35Dic107⋊C4—21no—
36Dic107⋊C4—21no—
37C22⋊Dic107—21no—
38(C2×D107)⋊C4—21no—
39(C2×D107)⋊C4—21no—
40(C2×D107)⋊C4—21no—
41(C2×D107)⋊C4—21no—
42C4×Dic107C2×C411yescharacteristic
43C4×Dic107C2×C411yescharacteristic
44C2×Q8×C107—41no—
45C2×Q8×C107—41no—
46C2×Q8×C107C2×C411yescharacteristic
47C2×Q8×C107D411yescharacteristic
48C2×Q8×C107D411yescharacteristic
49C2×Q8×C107C2311yescharacteristic
50C22×C4×C107C2×C411yescharacteristic
51C22×C4×C107D411yescharacteristic
52C22×C4×C107D411yescharacteristic
53M16×C107—22no—
54M16×C107—41no—
55C2×C8×C107—41no—
56C23×C107C22⋊C411yescharacteristic
57C22×D107—21no—
58Q8×C107—41no—
59Q8×C107—41no—
60Q8×C107—41no—
61Q8×C107—41no—
62C2×C4×C107—21no—
63C2×C4×C107—21no—
64C2×C4×C107—21no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107C22⋊C411yescharacteristic
69C2×C4×C107C22⋊C411yescharacteristic
70C2×C4×C107C22⋊C411yescharacteristic
71C2×C4×C107C22×C411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C22⋊D107—41no—
75C22⋊D107—41no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—21no—
79D428—41no—
80C4×D107—42no—
81Dic214—41no—
82C8×C107—42no—
83C107⋊C8—42no—
84C22×C107—41no—
85C22×C107#9911yescharacteristic
86C22×C107#9911yescharacteristic
87C22×C107C2×C22⋊C411yescharacteristic
88C2×D107—42no—
89C2×D107—41no—
90C4×C107—21no—
91C4×C107—21no—
92C4×C107—21no—
93C4×C107—21no—
94C4×C107—21no—
95C4×C107—21no—
96Dic107—41no—
97Dic107—41no—
98Dic107—41no—
99C2×C107—21no—
100C2×C107—21no—
101C2×C107—21no—
102C2×C107C2×#9911yescharacteristic
103D107—81no—
104#2133—1071nomaximal, Sylow
105C107#213311yescharacteristic, complemented, Sylow
106#536—1071no—
107C2×#101—1071no—
108C2×#101D10711yescharacteristic, complemented
109#356—1072no—
110#355—1072no—
111C2×#41—1071no—
112C22×Q8C2×D10711yescharacteristic
113C2×M16—21no—
114C2×M16—2141no—
115C4⋊Q8—1071no—
116C4⋊Q8—1071no—
117#124—1071no—
118#124—1071no—
119#123—2141no—
120#123—2141no—
121C22⋊Q8—2141no—
122C22⋊Q8—2141no—
123#117—1071no—
124#101—21no—
125#101—1072no—
126#101—2141no—
127#101C2×D10712yes—
128#41—4281no—
129#41—4281no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—1071no—
133C2×Q8C4×D10711yescharacteristic
134C2×Q8D42811yescharacteristic
135C2×Q8C22⋊D10711yescharacteristic
136C2×Q8C22×D10711yescharacteristic
137C2×D4—1071no—
138C2×D4—2141no—
139C22×C4—2141no—
140C22×C4C4×D10711yescharacteristic
141C22×C4D42811yescharacteristic
142C22×C4C22⋊D10711yescharacteristic
143M16—22no—
144M16—41no—
145M16—4281no—
146M16—2142no—
147C2×C8—41no—
148C2×C8—4281no—
149C4⋊C4—2141no—
150C4⋊C4—2141no—
151C4⋊C4—2141no—
152C4⋊C4—2141no—
153C4⋊C4—2141no—
154C22⋊C4—2141no—
155C22⋊C4—2141no—
156C22⋊C4—2141no—
157C22⋊C4—2141no—
158C22⋊C4—2141no—
159C42—1071no—
160C42—1071no—
161C23—2141no—
162C23(C2×D107)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—4281no—
168D4—4281no—
169D4—4281no—
170D4—4281no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—4282no—
178C2×C4—2141no—
179C2×C4—2141no—
180C2×C4—2141no—
181C2×C4(C2×D107)⋊C411yescharacteristic
182C2×C4(C2×D107)⋊C411yescharacteristic
183C2×C4(C2×D107)⋊C411yescharacteristic
184C2×C4C2×C4×D10711yescharacteristic
185C2×C4C2×C22⋊D10711yescharacteristic
186C2×C4C2×D42811yescharacteristic
187C8—42no—
188C8—4282no—
189C22—41no—
190C22—4282no—
191C22—4281no—
192C22#1681202311yescharacteristic
193C22#1681202311yescharacteristic
194C22C2×(C2×D107)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—4281no—
202C4—4281no—
203C4—4281no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—8561no—
208C2C2×#1681202311yescharacteristic
subgroupG/H
centerC2C2×#16812023
commutatorC2×C4×C107C22×C4
FrattiniC2×Q8C22×D107
FittingC2×#101×C107C2
socleC2×C107C2×#99
prime radicalC2×#13133818C4
cosocleC2×Q8×C107C23
hypercenterC2×#101D107
hypocenterC107#2133
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16812028C2semidirect
3. #20292425C2semidirect
4. #20292423C2semidirect
5. C2×#101×C107C2semidirect
105. C107#2133semidirect
108. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
#202932012
C2×#168120282
#202924252
#202924232
C2×#101×C1072
#2133107