Finite Group Databasebeta

← #28334983

Subgroups — #28334983

Order 13696.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#20293158C211yesmaximal, characteristic
2#20292681C211yesmaximal, characteristic, complemented
3#20292478C211yesmaximal, characteristic, complemented
4#20292365C211yesmaximal, characteristic, complemented
5#20292363C211yesmaximal, characteristic, complemented
6#20292361C211yesmaximal, characteristic
7#327×C107C211yesmaximal, characteristic
8#16812102—21no—
9#16812100C411yescharacteristic, complemented
10#16812096C411yescharacteristic, complemented
11#16812090—21no—
12#16812088C2211yescharacteristic
13#16812074C411yescharacteristic, complemented
14D4×Dic107—21no—
15D4×Dic107C411yescharacteristic, complemented
16#16812065—21no—
17C2×Dic107⋊C4—21no—
18C2×C4×Dic107—21no—
19C2×C4×Dic107C2211yescharacteristic
20Dic107⋊C8—21no—
21C22⋊(C107⋊C8)C2211yescharacteristic
22#16812019—21no—
23C107⋊#95C2211yescharacteristic
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C2×Dic214C2×C411yescharacteristic
28C22×Dic107—21no—
29C22×Dic107—21no—
30C22×Dic107D411yescharacteristic
31C22×Dic107D411yescharacteristic
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107C2×C411yescharacteristic
35Dic107⋊C4—24no—
36Dic107⋊C4—21no—
37Dic107⋊C4—21no—
38Dic107⋊C4—21no—
39C22⋊Dic107—21no—
40C22⋊Dic107—21no—
41C22⋊Dic107—21no—
42C22⋊Dic107—21no—
43C4×Dic107—21no—
44C4×Dic107—42no—
45C4×Dic107C2×C411yescharacteristic
46C4×Dic107C2×C411yescharacteristic
47C4×Dic107D411yescharacteristic
48C4×Dic107D411yescharacteristic
49C2×D4×C107—21no—
50C2×D4×C107C2×C411yescharacteristic
51C22×C4×C107—21no—
52C22×C4×C107C2311yescharacteristic
53C2×C8×C107—21no—
54C4⋊C4×C107C2×C411yescharacteristic
55C22⋊C4×C107—21no—
56C23×C107—21no—
57C23×C107C2×D411yescharacteristic
58D4×C107—41no—
59D4×C107—41no—
60D4×C107—41no—
61C2×C4×C107—22no—
62C2×C4×C107—21no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107C22×C411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×Dic107—21no—
69C2×Dic107—21no—
70C2×Dic107—22no—
71C2×Dic107—22no—
72C2×Dic107—21no—
73C2×Dic107—21no—
74C2×Dic107—41no—
75C2×Dic107—41no—
76C2×Dic107—42no—
77C2×Dic107C22⋊C411yescharacteristic
78C2×Dic107C22⋊C411yescharacteristic
79C2×Dic107C22⋊C411yescharacteristic
80C2×Dic107C22⋊C411yescharacteristic
81Dic214—41no—
82C8×C107—41no—
83C107⋊C8—41no—
84C22×C107—21no—
85C22×C107—21no—
86C22×C107—41no—
87C22×C107—41no—
88C22×C107—41no—
89C22×C107C2×C22⋊C411yescharacteristic
90C22×C107AGL(1,ℤ/8)11yescharacteristic
91C22×C107#13711yescharacteristic
92C4×C107—21no—
93C4×C107—21no—
94C4×C107—42no—
95C4×C107—41no—
96Dic107—21no—
97Dic107—21no—
98Dic107—21no—
99Dic107—42no—
100Dic107—41no—
101Dic107—41no—
102Dic107#10412yes—
103C2×C107—21no—
104C2×C107—21no—
105C2×C107—81no—
106C2×C107#41011yescharacteristic
107C2×C107#41711yescharacteristic
108C2×C107C2×#10411yescharacteristic
109#1509—1071nomaximal, Sylow
110C107#150911yescharacteristic, complemented, Sylow
111#529—1071no—
112#432—1071no—
113C4×C4⋊C4—1071no—
114#328—1071no—
115#328—1071no—
116#327—1071no—
117#327D10711yescharacteristic
118#126—2141no—
119#125—1071no—
120#124—1071no—
121#123—2141no—
122C22⋊Q8—1071no—
123#121C2×D10711yescharacteristic
124C4×Q8—1071no—
125C4×D4—1071no—
126C4×D4—2141no—
127#117—2141no—
128C2×C4⋊C4—2141no—
129C2×C42—1071no—
130C2×C42—2141no—
131C4⋊C8—2141no—
132C22⋊C8—1071no—
133C22⋊C8C2×D10711yescharacteristic
134#97—2141no—
135#95—1071no—
136#95C2×D10711yescharacteristic
137C2×Q8—1071no—
138C2×D4—21no—
139C2×D4C4×D10711yescharacteristic
140C22×C4—21no—
141C22×C4—1071no—
142C22×C4—1071no—
143C22×C4—2141no—
144C22×C4—2141no—
145C22×C4C22×D10711yescharacteristic
146C2×C8—21no—
147C2×C8—2141no—
148C4⋊C4—1071no—
149C4⋊C4—2141no—
150C4⋊C4—2141no—
151C4⋊C4—2141no—
152C4⋊C4—2144no—
153C4⋊C4—2141no—
154C4⋊C4C4×D10711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2141no—
157C22⋊C4—2141no—
158C22⋊C4—2141no—
159C22⋊C4—2141no—
160C42—1071no—
161C42—1071no—
162C42—1071no—
163C42—1071no—
164C42—4282no—
165C42—2141no—
166C23—21no—
167C23D4×D10711yescharacteristic
168Q8—4281no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—1071no—
178C2×C4—1071no—
179C2×C4—1071no—
180C2×C4—1071no—
181C2×C4—4282no—
182C2×C4—4281no—
183C2×C4—4281no—
184C2×C4—2142no—
185C2×C4—2142no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4—2141no—
189C2×C4—2141no—
190C2×C4C2×C4×D10711yescharacteristic
191C2×C4D4×D10711yescharacteristic
192C8—41no—
193C8—4281no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D10711yescharacteristic
200C22#1681213311yescharacteristic
201C22#1681213811yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—1072no—
207C4—4282no—
208C4—4281no—
209C4—4281no—
210C4—2141no—
211C4—2141no—
212C4—2141no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2029259911yescharacteristic
217C2#2029262511yescharacteristic
218C2#104×D10711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107C2×C22⋊C4
prime radical#121×C107C22
cosocleC22×C4×C107C23
hypercenter#327D107
hypocenterC107#1509
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20292681C2semidirect
3. #20292478C2semidirect
4. #20292365C2semidirect
5. #20292363C2semidirect
9. #16812100C4semidirect
10. #16812096C4semidirect
13. #16812074C4semidirect
15. D4×Dic107C4semidirect
110. C107#1509semidirect

Maximal subgroups

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

subgroupindex
#202931582
#202926812
#202924782
#202923652
#202923632
#202923612
#327×C1072
#1509107