Finite Group Databasebeta

← #28336726

Subgroups — #28336726

Order 13696.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16812074C211yesmaximal, characteristic
2C2×#16812034C211yesmaximal, characteristic
3C2×#16812032C211yesmaximal, characteristic
4#20292523C211yesmaximal, characteristic
5#20292389C211yesmaximal, characteristic
6#20292381C211yesmaximal, characteristic
7C2×#106×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C22×C107⋊C8C2211yescharacteristic
10#16812074—22no—
11#16812074—22no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4—21no—
14C2×Dic107⋊C4—21no—
15C2×C4×Dic107—21no—
16C2×C4×Dic107C2211yescharacteristic
17#16812034C414yescomplemented
18#16812032—22no—
19C107⋊#95—21no—
20C22×C8×C107C2211yescharacteristic
21C2×C4⋊C4×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23#106×C107—22no—
24C2×Dic214—21no—
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214C2×C412yes—
28C2×Dic214D412yes—
29C2×Dic214Q812yes—
30C22×Dic107—21no—
31C22×Dic107—21no—
32C22×Dic107D411yescharacteristic
33C22×Dic107D411yescharacteristic
34C2×C107⋊C8—21no—
35C2×C107⋊C8—21no—
36C2×C107⋊C8—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107C2×C412yes—
39Dic107⋊C4—24no—
40Dic107⋊C4—42no—
41C4×Dic107—24no—
42C4×Dic107—21no—
43C4×Dic107—21no—
44C22×C4×C107—21no—
45C22×C4×C107—21no—
46C22×C4×C107C2311yescharacteristic
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C2×C8×C107C2×C412yes—
50C4⋊C4×C107—21no—
51C4⋊C4×C107—21no—
52C4⋊C4×C107—21no—
53C4⋊C4×C107D412yes—
54C23×C107C2×D411yescharacteristic
55C2×C4×C107—22no—
56C2×C4×C107—41no—
57C2×C4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107—41no—
60C2×C4×C107—41no—
61C2×C4×C107C22×C411yescharacteristic
62C2×C4×C107C2×D411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×Q811yescharacteristic
65C2×C4×C107#4111yescharacteristic
66C2×C4×C107#4111yescharacteristic
67C2×Dic107—22no—
68C2×Dic107—21no—
69C2×Dic107—22no—
70C2×Dic107—22no—
71C2×Dic107—21no—
72C2×Dic107—22no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78C2×Dic107SD1612yes—
79Dic214—22no—
80Dic214—22no—
81Dic214—22no—
82Dic214C4⋊C414yescomplemented
83C8×C107—22no—
84C107⋊C8—41no—
85C22×C107C4×D411yescharacteristic
86C22×C107#12011yescharacteristic
87C22×C107C22⋊Q811yescharacteristic
88C22×C107C2×SD1611yescharacteristic
89C22×C107#13511yescharacteristic
90C22×C107#13711yescharacteristic
91C22×C107#13711yescharacteristic
92C4×C107—42no—
93C4×C107—81no—
94C4×C107C2×C4⋊C411yescharacteristic
95C4×C107C4×D411yescharacteristic
96C4×C107C22⋊Q811yescharacteristic
97C4×C107#12311yescharacteristic
98Dic107—24no—
99Dic107—22no—
100Dic107—42no—
101Dic107—41no—
102C2×C107#38511yescharacteristic
103C2×C107C4×SD1611yescharacteristic
104C2×C107#44111yescharacteristic
105C2×C107#44811yescharacteristic
106C2×C107#45211yescharacteristic
107C2×C107#47511yescharacteristic
108C2×C107#47911yescharacteristic
109#1865—1071nomaximal, Sylow
110C107#186511yescharacteristic, complemented, Sylow
111C2×C4×Q8—1071no—
112C2×#106D10711yescharacteristic
113C2×#103—1071no—
114C2×#103—1071no—
115#389—1071no—
116#340—1071no—
117#336—1071no—
118C22×Q8—1071no—
119C22×C8—1071no—
120C22×C8C2×D10711yescharacteristic
121C4×Q8—2142no—
122C4×Q8—2142no—
123C2×C4⋊C4—1071no—
124C2×C4⋊C4—2141no—
125C2×C4⋊C4—2141no—
126C2×C4⋊C4C2×D10711yescharacteristic
127C2×C4⋊C4C2×D10711yescharacteristic
128C2×C42—1071no—
129C2×C42—2141no—
130#106—22no—
131#103—1074no—
132#103—2142no—
133#95—2141no—
134C2×Q8—1072no—
135C2×Q8—1072no—
136C2×Q8—1072no—
137C2×Q8—2141no—
138C2×Q8—2141no—
139C2×Q8—2141no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—1071no—
143C22×C4—1071no—
144C22×C4—2141no—
145C22×C4—2141no—
146C22×C4C22×D10711yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—2141no—
150C2×C8—2141no—
151C2×C8—2141no—
152C2×C8C4×D10712yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1072no—
157C4⋊C4—4282no—
158C4⋊C4—2144no—
159C4⋊C4—2141no—
160C4⋊C4C22⋊D10712yes—
161C42—2144no—
162C42—2141no—
163C42—2141no—
164C23D4×D10711yescharacteristic
165Q8—1074no—
166Q8—2142no—
167Q8—2142no—
168Q8—2142no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—1072no—
176C2×C4—4281no—
177C2×C4—4281no—
178C2×C4—2142no—
179C2×C4—2141no—
180C2×C4—2141no—
181C2×C4—2141no—
182C2×C4—2142no—
183C2×C4—2141no—
184C2×C4—2142no—
185C2×C4—2142no—
186C2×C4—2141no—
187C2×C4C2×C4×D10711yescharacteristic
188C2×C4D4×D10711yescharacteristic
189C2×C4C2×C22⋊D10711yescharacteristic
190C2×C4Q8×D10711yescharacteristic
191C2×C4#1313381811yescharacteristic
192C2×C4#1313381911yescharacteristic
193C8—22no—
194C8—4281no—
195C22#1681207111yescharacteristic
196C22#1681207711yescharacteristic
197C22#1681208711yescharacteristic
198C22SD16×D10711yescharacteristic
199C22#1681213011yescharacteristic
200C22#1681213811yescharacteristic
201C22#1681213811yescharacteristic
202C4—42no—
203C4—81no—
204C4—4282no—
205C4—4281no—
206C4—2144no—
207C4—2142no—
208C4C4⋊C4×D10711yescharacteristic
209C4C4×C22⋊D10711yescharacteristic
210C4#1681208511yescharacteristic
211C4#1681209411yescharacteristic
212C2#2029249811yescharacteristic
213C2#2029271411yescharacteristic
214C2#2029273311yescharacteristic
215C2#2029276411yescharacteristic
216C2#2029278611yescharacteristic
217C2#2029291111yescharacteristic
218C2#2029293711yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#106×C107C2
socle, prime radicalC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenterC2×#106D107
hypocenterC107#1865
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
17. #16812034C4semidirect
82. Dic214C4⋊C4semidirect
110. C107#1865semidirect

Maximal subgroups

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

subgroupindex
C2×#168120742
C2×#168120342
C2×#168120322
#202925232
#202923892
#202923812
C2×#106×C1072
#1865107