Finite Group Databasebeta

← #28344488

Subgroups — #28344488

Order 13696.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812083C211yesmaximal, characteristic
2#20292562C212yesmaximal, complemented
3#20292557C212yesmaximal, complemented
4#20292544C212yesmaximal, complemented
5#20292542C214yesmaximal, complemented
6#20292538C212yesmaximal, complemented
7#20292490C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107C2212yes—
11#16812083—24no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C2214yes—
14C2×C22⋊Dic107C2212yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2214yes—
17C2×(C2×D107)⋊C4C2214yes—
18C107⋊#95C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2214yes—
21C22×Q8×C107C2211yescharacteristic
22C23×C4×C107C2211yescharacteristic
23C22⋊Q8×C107—24no—
24C2×C4⋊C4×C107C2212yes—
25C2×C4⋊C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2212yes—
27C23×D107C2312yes—
28C2×D428—22no—
29C2×D428—42no—
30C2×C22⋊D107—24no—
31C2×C22⋊D107—44no—
32C22×Dic107C2314yes—
33C22×Dic107C2312yes—
34C4⋊Dic107—22no—
35Dic107⋊C4—44no—
36C22⋊Dic107—42no—
37(C2×D107)⋊C4—24no—
38(C2×D107)⋊C4—44no—
39(C2×D107)⋊C4—44no—
40C24×C107C2311yescharacteristic
41C2×Q8×C107—22no—
42C2×Q8×C107—42no—
43C22×C4×C107—42no—
44C22×C4×C107D414yes—
45C22×C4×C107C2312yes—
46C22×C4×C107C2312yes—
47C22×C4×C107C2311yescharacteristic
48C22×C4×C107C2311yescharacteristic
49C4⋊C4×C107—22no—
50C4⋊C4×C107—24no—
51C22⋊C4×C107—24no—
52C23×C107—22no—
53C23×C107—22no—
54C23×C107—22no—
55C23×C107C2×D412yes—
56C23×C107C2411yescharacteristic
57C22×D107—22no—
58C22×D107—24no—
59C22×D107—24no—
60C22×D107—24no—
61Q8×C107—44no—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—24no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—44no—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D412yes—
72C22⋊D107—48no—
73C2×Dic107—22no—
74C2×Dic107—24no—
75C2×Dic107—24no—
76C2×Dic107—24no—
77C2×Dic107—24no—
78D428—44no—
79C22×C107—22no—
80C22×C107—22no—
81C22×C107—22no—
82C22×C107—44no—
83C22×C107C22×D411yescharacteristic
84C22×C107H(5,𝔽2)12yes—
85C22×C107H(5,𝔽2)12yes—
86C22×C1072−1+412yes—
87C2×D107—42no—
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—48no—
91C4×C107—22no—
92C4×C107—22no—
93C4×C107—42no—
94C4×C107—42no—
95Dic107—42no—
96Dic107—44no—
97C2×C107—42no—
98C2×C107#53411yescharacteristic
99C2×C107#53511yescharacteristic
100C2×C107#53611yescharacteristic
101C2×C107#56214yes—
102D107—82no—
103#2816—1071nomaximal, Sylow
104C107#281611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8D10711yescharacteristic, complemented
106C2×#121—1071no—
107#399—1072no—
108#398—1072no—
109#394—1072no—
110#394—1074no—
111#393—1072no—
112#381—1071no—
113C22×Q8C2×D10711yescharacteristic
114C22×D4—1072no—
115C22×D4—1071no—
116C23×C4C2×D10711yescharacteristic
117C22⋊Q8—24no—
118#121—2144no—
119C2×C4⋊C4—1074no—
120C2×C4⋊C4—1071no—
121C2×C4⋊C4C2×D10712yes—
122C2×C4⋊C4C2×D10711yescharacteristic
123C2×C22⋊C4—1072no—
124C2×C22⋊C4—1074no—
125C2×C22⋊C4—1074no—
126C2×C22⋊C4—1072no—
127C2×C22⋊C4C2×D10712yes—
128#95—1072no—
129#95—1072no—
130#95—1074no—
131C24—1072no—
132C24C22×D10711yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—4282no—
136C2×D4—4284no—
137C2×D4—2144no—
138C2×D4—2142no—
139C22×C4—42no—
140C22×C4—1074no—
141C22×C4—1072no—
142C22×C4C22⋊D10714yes—
143C22×C4C22×D10712yes—
144C22×C4C22×D10712yes—
145C22×C4C22×D10711yescharacteristic
146C22×C4C22×D10711yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—4284no—
150C4⋊C4—2142no—
151C22⋊C4—24no—
152C22⋊C4—4284no—
153C22⋊C4—4284no—
154C22⋊C4—4282no—
155C22⋊C4—2144no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—2142no—
160C23—2144no—
161C23—2144no—
162C23—2144no—
163C23C2×C22⋊D10712yes—
164C23C23×D10711yescharacteristic
165Q8—44no—
166D4—4284no—
167D4—4288no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—2142no—
177C2×C4—2144no—
178C2×C4—2144no—
179C2×C4—2144no—
180C2×C4—2144no—
181C2×C4C2×C22⋊D10712yes—
182C2×C4C2×C22⋊D10712yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—4282no—
188C22—4282no—
189C22—4282no—
190C22—4288no—
191C22C22×C22⋊D10711yescharacteristic
192C22#1681215412yes—
193C22#1681215512yes—
194C22#1681215712yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—4282no—
200C4—4284no—
201C2—42no—
202C2—8562no—
203C2#2029318611yescharacteristic
204C2#2029319111yescharacteristic
205C2#2029320111yescharacteristic
206C2#2029337314yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC2×#16812083C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#2816
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20292562C2semidirect
3. #20292557C2semidirect
4. #20292544C2semidirect
5. #20292542C2semidirect
6. #20292538C2semidirect
7. #20292490C2semidirect
8. C2×C22⋊Q8×C107C2semidirect
104. C107#2816semidirect
105. C2×C22⋊Q8D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
#202925622
#202925572
#202925442
#202925422
#202925382
#202924902
C2×C22⋊Q8×C1072
#2816107