Finite Group Databasebeta

← #28344808

Subgroups — #28344808

Order 13696.

644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812085C212yesmaximal
2C2×C42×D107C211yesmaximal, characteristic
3#20292534C212yesmaximal, complemented
4#20292523C214yesmaximal, complemented
5#20292497C211yesmaximal, characteristic, complemented
6#20292484C214yesmaximal
7C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107C2211yescharacteristic
9C22×C4×D107C2212yes—
10#16812085—28no—
11C2×C4⋊Dic107C2212yes—
12C2×C4⋊Dic107C2214yes—
13C2×Dic107⋊C4C2214yes—
14C2×(C2×D107)⋊C4C2214yes—
15C2×C4×Dic107C2212yes—
16C2×C4×Dic107C2211yescharacteristic
17C42×D107—24no—
18C107⋊#95C2218yes—
19C22×Q8×C107C2212yes—
20C4⋊Q8×C107—24no—
21C2×C4⋊C4×C107C2214yes—
22C2×C42×C107C2211yescharacteristic
23C23×D107C2311yescharacteristic
24C22×Dic107C2314yes—
25C22×Dic107C2311yescharacteristic
26C22×Dic107C2312yes—
27C2×C4×D107—22no—
28C2×C4×D107—24no—
29C2×C4×D107—22no—
30C2×C4×D107—24no—
31C2×C4×D107D414yes—
32C2×C4×D107Q818yescomplemented
33C4⋊Dic107—24no—
34C4⋊Dic107—28no—
35Dic107⋊C4—28no—
36(C2×D107)⋊C4—28no—
37C4×Dic107—22no—
38C4×Dic107—24no—
39C2×Q8×C107—24no—
40C2×Q8×C107—24no—
41C2×Q8×C107—24no—
42C22×C4×C107C2314yes—
43C22×C4×C107C2311yescharacteristic
44C22×C4×C107C2312yes—
45C4⋊C4×C107—28no—
46C42×C107D414yes—
47C23×C107C2411yescharacteristic
48C22×D107—21no—
49C22×D107—21no—
50C22×D107—22no—
51C22×D107C2×D412yes—
52C22×D107C2×Q814yes—
53Q8×C107—48no—
54C2×C4×C107—24no—
55C2×C4×C107—24no—
56C2×C4×C107—24no—
57C2×C4×C107C2×D412yes—
58C2×C4×C107C2×D412yes—
59C2×C4×C107C2×D414yes—
60C2×C4×C107C2×Q814yes—
61C2×C4×C107#4112yes—
62C2×C4×C107#4114yes—
63C2×Dic107—21no—
64C2×Dic107—22no—
65C2×Dic107—21no—
66C2×Dic107—22no—
67C2×Dic107—24no—
68C2×Dic107—28no—
69C2×Dic107C2×D412yes—
70C2×Dic107C2×Q814yes—
71C4×D107—28no—
72C4×D107—216no—
73C22×C107C22×D411yescharacteristic
74C22×C107C22×D411yescharacteristic
75C22×C107C22×Q812yes—
76C22×C107C2×#4111yescharacteristic
77C22×C107C2×#4112yes—
78C2×D107—22no—
79C2×D107—24no—
80C2×D107—22no—
81C2×D107—24no—
82C2×D107C4⋊Q814yescomplemented
83C4×C107—44no—
84C4×C107#12114yes—
85C4×C107C22⋊Q818yescomplemented
86Dic107—22no—
87Dic107—24no—
88Dic107—44no—
89C2×C107C2×#12111yescharacteristic
90C2×C107C2×C22⋊Q812yes—
91C2×C107C2×C4⋊Q811yescharacteristic
92C2×C107#53211yescharacteristic
93C2×C107#53312yes—
94D107—24no—
95#2857—1071nomaximal, Sylow
96C107#285711yescharacteristic, complemented, Sylow
97C2×C4⋊Q8D10711yescharacteristic, complemented
98C2×C22⋊Q8—1072no—
99C22×C42—1071no—
100#391—1072no—
101#389—1074no—
102#384—1071no—
103#380—1074no—
104C22×Q8C2×D10712yes—
105C23×C4—1071no—
106C23×C4—1072no—
107C4⋊Q8—24no—
108C22⋊Q8—2148no—
109C2×C4⋊C4—1074no—
110C2×C4⋊C4—1072no—
111C2×C4⋊C4—1074no—
112C2×C4⋊C4C2×D10714yes—
113C2×C22⋊C4—1074no—
114C2×C42—1072no—
115C2×C42—1071no—
116C2×C42—2144no—
117C2×C42C2×D10711yescharacteristic
118#95—1078no—
119C24—1071no—
120C2×Q8—24no—
121C2×Q8—24no—
122C2×Q8—24no—
123C22×C4—1072no—
124C22×C4—1071no—
125C22×C4—1074no—
126C22×C4—1078no—
127C22×C4—1074no—
128C22×C4—2142no—
129C22×C4—2144no—
130C22×C4—2142no—
131C22×C4—2144no—
132C22×C4C22×D10714yes—
133C22×C4C22×D10711yescharacteristic
134C22×C4C22×D10712yes—
135C4⋊C4—28no—
136C4⋊C4—2148no—
137C4⋊C4—2144no—
138C4⋊C4—2148no—
139C22⋊C4—2148no—
140C42—2142no—
141C42—2144no—
142C42C22⋊D10714yes—
143C23—1072no—
144C23—1074no—
145C23—2141no—
146C23—2141no—
147C23—2142no—
148C23C23×D10711yescharacteristic
149Q8—48no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—1072no—
154C2×C4—1074no—
155C2×C4—2144no—
156C2×C4—2148no—
157C2×C4—2142no—
158C2×C4—2141no—
159C2×C4—2141no—
160C2×C4—2142no—
161C2×C4—2148no—
162C2×C4—21416no—
163C2×C4D4×D10712yes—
164C2×C4C2×C22⋊D10712yes—
165C2×C4C2×C22⋊D10714yes—
166C2×C4Q8×D10714yes—
167C2×C4#1313381712yes—
168C2×C4#1313381914yes—
169C22—1074no—
170C22—2144no—
171C22—2142no—
172C22—2144no—
173C22—2142no—
174C22C2×D4×D10711yescharacteristic
175C22C22×C22⋊D10711yescharacteristic
176C22C2×Q8×D10712yes—
177C22C2×#1313381711yescharacteristic
178C22C2×#1313381912yes—
179C4—44no—
180C4—4284no—
181C4—2144no—
182C4—2142no—
183C4#1681208014yes—
184C4#1681208518yescomplemented
185C2—2144no—
186C2C2×#1681208011yescharacteristic
187C2C2×#1681208512yes—
188C2C4⋊Q8×D10711yescharacteristic
189C2#2029317811yescharacteristic
190C2#2029318212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4⋊Q8×C107C2
prime radicalC23×D107C23
hypercenterC2×C4⋊Q8D107
hypocenterC107#2857
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20292534C2semidirect
4. #20292523C2semidirect
5. #20292497C2semidirect
7. C2×C4⋊Q8×C107C2semidirect
32. C2×C4×D107Q8semidirect
82. C2×D107C4⋊Q8semidirect
85. C4×C107C22⋊Q8semidirect
96. C107#2857semidirect
97. C2×C4⋊Q8D107semidirect
184. C4#16812085semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C2×C42×D1072
#202925342
#202925232
#202924972
#202924842
C2×C4⋊Q8×C1072
#2857107