Finite Group Databasebeta

← #28334871

Subgroups — #28334871

Order 13696.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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#20293159C211yesmaximal, characteristic
2C4⋊C8×D107C211yesmaximal, characteristic, complemented
3C4×#13133801C211yesmaximal, characteristic, complemented
4#20292359C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×#13133801—21no—
9C2×C8×D107—21no—
10#16812102—21no—
11#16812100C411yescharacteristic
12#16812096C411yescharacteristic
13#16812091—21no—
14#16812089C411yescharacteristic
15#16812080C411yescharacteristic
16C4×Dic214C2211yescharacteristic
17C4×C22⋊D107—21no—
18#16812068—21no—
19C42×D107C2211yescharacteristic
20C4⋊(C107⋊C8)C2211yescharacteristic
21Dic107⋊C8—21no—
22#16812019—21no—
23C4×C107⋊C8C2211yescharacteristic
24C4×D4×C107C2211yescharacteristic
25C4⋊C8×C107C2211yescharacteristic
26C4×C8×C107C2211yescharacteristic
27C2×Dic214C2×C411yescharacteristic
28C2×C22⋊D107—21no—
29C2×C4×D107C2×C411yescharacteristic
30C2×C4×D107D411yescharacteristic
31C2×C4×D107D411yescharacteristic
32#13133801—24no—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C8×D107—42no—
36C4⋊Dic107—21no—
37C4⋊Dic107—21no—
38C4⋊Dic107C2×C411yescharacteristic
39Dic107⋊C4—21no—
40Dic107⋊C4—21no—
41C22⋊Dic107—21no—
42C22⋊Dic107—21no—
43C22⋊Dic107—21no—
44(C2×D107)⋊C4—21no—
45C4×Dic107—21no—
46C4×Dic107C2×C411yescharacteristic
47C4×Dic107D411yescharacteristic
48C4×Dic107D411yescharacteristic
49C2×D4×C107C2×C411yescharacteristic
50C22×C4×C107—21no—
51C2×C8×C107—21no—
52C2×C8×C107—21no—
53C4⋊C4×C107C2×C411yescharacteristic
54C22⋊C4×C107—21no—
55C42×C107C2311yescharacteristic
56C23×C107—21no—
57C22×D107C22⋊C411yescharacteristic
58D4×C107—21no—
59D4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107C22×C411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C22⋊D107—42no—
67C2×Dic107—21no—
68C2×Dic107—21no—
69C2×Dic107C22⋊C411yescharacteristic
70C2×Dic107C22⋊C411yescharacteristic
71C2×Dic107C22⋊C411yescharacteristic
72C4×D107—22no—
73C4×D107—21no—
74C4×D107—21no—
75C4×D107—21no—
76C4×D107—21no—
77Dic214—21no—
78Dic214—21no—
79C8×C107—22no—
80C8×C107—41no—
81C107⋊C8—22no—
82C107⋊C8—41no—
83C22×C107—41no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107C2×C22⋊C411yescharacteristic
87C2×D107—21no—
88C2×D107—21no—
89C2×D107#10412yes—
90C4×C107—21no—
91C4×C107—41no—
92C4×C107#13111yescharacteristic
93C4×C107#13111yescharacteristic
94C4×C107AGL(1,ℤ/8)11yescharacteristic
95C4×C107#13711yescharacteristic
96Dic107—21no—
97Dic107—21no—
98Dic107—21no—
99Dic107—41no—
100Dic107—41no—
101C2×C107—81no—
102C2×C107#40811yescharacteristic
103C2×C107#41711yescharacteristic
104C2×C107C2×#10411yescharacteristic
105D107—22no—
106#1483—1071nomaximal, Sylow
107C107#148311yescharacteristic, complemented, Sylow
108#529—1071no—
109C2×C4⋊C8—1071no—
110C4×M16—1071no—
111Q8⋊C8—1071no—
112Q8⋊C8—1071no—
113D4⋊C8—1071no—
114D4⋊C8D10711yescharacteristic, complemented
115C2×M16—2141no—
116C22×C8—2141no—
117#126—2141no—
118#125—1071no—
119#124—1071no—
120#123—2141no—
121C22⋊Q8—1071no—
122#121—1071no—
123C4×Q8—1071no—
124C4×D4—2141no—
125C4×D4C2×D10711yescharacteristic
126#117—2141no—
127C2×C42—1071no—
128C4⋊C8—1071no—
129C4⋊C8—2141no—
130C4⋊C8C2×D10711yescharacteristic
131#97—2141no—
132C4×C8—1071no—
133C4×C8C2×D10711yescharacteristic
134C2×Q8—1071no—
135C2×D4—2141no—
136C2×D4C4×D10711yescharacteristic
137C22×C4—21no—
138C22×C4—1071no—
139C22×C4—1071no—
140C22×C4—1071no—
141M16—2144no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—4282no—
145C2×C8—2141no—
146C2×C8—2141no—
147C4⋊C4—1071no—
148C4⋊C4—2141no—
149C4⋊C4—2141no—
150C4⋊C4—2141no—
151C4⋊C4—2141no—
152C4⋊C4C4×D10711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—2141no—
155C22⋊C4—2141no—
156C22⋊C4—2141no—
157C22⋊C4—2141no—
158C42—1071no—
159C42—1071no—
160C42—1071no—
161C42—2141no—
162C42C22×D10711yescharacteristic
163C23—21no—
164C23—1071no—
165Q8—2141no—
166Q8—2141no—
167D4—21no—
168D4—21no—
169D4—4282no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1071no—
174C2×C4—1071no—
175C2×C4—1071no—
176C2×C4—2142no—
177C2×C4—2141no—
178C2×C4—2141no—
179C2×C4—2141no—
180C2×C4—2141no—
181C2×C4—2141no—
182C2×C4—2141no—
183C2×C4C2×C4×D10711yescharacteristic
184C2×C4D4×D10711yescharacteristic
185C2×C4D4×D10711yescharacteristic
186C8—22no—
187C8—41no—
188C8—4281no—
189C8—2142no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—1072no—
194C22—2141no—
195C22—2141no—
196C22C22⋊C4×D10711yescharacteristic
197C4—21no—
198C4—41no—
199C4—4281no—
200C4—4281no—
201C4—2141no—
202C4—2141no—
203C4—2141no—
204C4#1681211511yescharacteristic
205C4#1681211711yescharacteristic
206C4#1681213311yescharacteristic
207C4#1681213811yescharacteristic
208C2—81no—
209C2—2142no—
210C2#2029259211yescharacteristic
211C2#2029262511yescharacteristic
212C2#104×D10711yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
FittingD4⋊C8×C107C2
socleC22×C107C2×C22⋊C4
prime radical#16812080C4
cosocleC42×C107C23
hypercenterD4⋊C8D107
hypocenterC107#1483
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×D107C2semidirect
3. C4×#13133801C2semidirect
4. #20292359C2semidirect
5. #20292357C2semidirect
6. D4⋊(C107⋊C8)C2semidirect
7. D4⋊C8×C107C2semidirect
107. C107#1483semidirect
114. D4⋊C8D107semidirect

Maximal subgroups

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

subgroupindex
#202931592
C4⋊C8×D1072
C4×#131338012
#202923592
#202923572
D4⋊(C107⋊C8)2
D4⋊C8×C1072
#1483107