Finite Group Databasebeta

← #28334888

Subgroups — #28334888

Order 13696.

248 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×Q8×D107C211yesmaximal, characteristic
2#20292678C211yesmaximal, characteristic
3#20292649C211yesmaximal, characteristic
4#20292359C211yesmaximal, characteristic, complemented
5Q8⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292357C211yesmaximal, characteristic, complemented
7Q8⋊C8×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107C411yescharacteristic
9C2×#13133801—21no—
10Q8×Dic107C411yescharacteristic
11C4×Dic214—21no—
12C4×Dic214C2211yescharacteristic
13#16812074—21no—
14#16812074C411yescharacteristic
15C4⋊C4×D107—21no—
16C4⋊C4×D107C411yescharacteristic
17C42×D107—21no—
18C42×D107C2211yescharacteristic
19C4⋊(C107⋊C8)C2211yescharacteristic
20Dic107⋊C8—21no—
21Dic107⋊C8—21no—
22(C2×D107)⋊C8—21no—
23C4×C107⋊C8C2211yescharacteristic
24C4×Q8×C107C2211yescharacteristic
25C4⋊C8×C107C2211yescharacteristic
26C4×C8×C107C2211yescharacteristic
27C2×Dic214—21no—
28C2×Dic214C2×C411yescharacteristic
29Q8×D107—22no—
30Q8×D107—22no—
31C2×C4×D107—21no—
32C2×C4×D107—21no—
33C2×C4×D107C2×C411yescharacteristic
34C2×C4×D107D411yescharacteristic
35C2×C4×D107D411yescharacteristic
36#13133801—42no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C4⋊Dic107—21no—
40C4⋊Dic107C2×C411yescharacteristic
41Dic107⋊C4—21no—
42Dic107⋊C4—21no—
43Dic107⋊C4—21no—
44C4×Dic107—21no—
45C4×Dic107—21no—
46C4×Dic107—21no—
47C4×Dic107C2×C411yescharacteristic
48C4×Dic107D411yescharacteristic
49C4×Dic107D411yescharacteristic
50C2×Q8×C107C2×C411yescharacteristic
51C2×C8×C107—21no—
52C2×C8×C107—21no—
53C4⋊C4×C107—21no—
54C4⋊C4×C107C2×C411yescharacteristic
55C42×C107—21no—
56C42×C107C2311yescharacteristic
57C22×D107C22⋊C411yescharacteristic
58Q8×C107—21no—
59Q8×C107M1612yescomplemented
60C2×C4×C107—21no—
61C2×C4×C107—21no—
62C2×C4×C107C22×C411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×Dic107—21no—
66C2×Dic107—21no—
67C2×Dic107C22⋊C411yescharacteristic
68C2×Dic107C22⋊C411yescharacteristic
69C2×Dic107C22⋊C411yescharacteristic
70C4×D107—24no—
71C4×D107—21no—
72C4×D107—21no—
73C4×D107—41no—
74C4×D107—42no—
75C4×D107SD1612yes—
76C4×D107Q1612yes—
77Dic214—24no—
78Dic214—21no—
79Dic214M1612yescomplemented
80C8×C107—41no—
81C8×C107—41no—
82C107⋊C8—41no—
83C107⋊C8—41no—
84C22×C107C2×C22⋊C411yescharacteristic
85C2×D107—21no—
86C2×D107—21no—
87C2×D107#10312yes—
88C4×C107—22no—
89C4×C107—21no—
90C4×C107—41no—
91C4×C107C2×M1611yescharacteristic
92C4×C107C2×M1611yescharacteristic
93C4×C107C2×SD1611yescharacteristic
94C4×C107C2×Q1611yescharacteristic
95Dic107—21no—
96Dic107—22no—
97Dic107—21no—
98Dic107—41no—
99Dic107#10312yes—
100C2×C107#40711yescharacteristic
101C2×C107C2×#10311yescharacteristic
102C2×C107#42111yescharacteristic
103D107—22no—
104#1487—1071nomaximal, Sylow
105C107#148711yescharacteristic, complemented, Sylow
106C2×C4×Q8—1071no—
107#431—1071no—
108#423—1071no—
109Q8⋊C8—1071no—
110Q8⋊C8—1071no—
111Q8⋊C8—1071no—
112Q8⋊C8D10711yescharacteristic, complemented
113C22×Q8—1071no—
114C2×M16—2141no—
115C4×Q8—1071no—
116C4×Q8—1071no—
117C4×Q8—1071no—
118C4×Q8—2141no—
119C4×Q8—2141no—
120C4×Q8C2×D10711yescharacteristic
121C2×C4⋊C4—1071no—
122C2×C4⋊C4—2141no—
123C2×C42—1071no—
124C2×C42—2141no—
125C4⋊C8—1071no—
126C4⋊C8—2141no—
127C4⋊C8—2141no—
128C4⋊C8C2×D10711yescharacteristic
129C22⋊C8—2141no—
130C4×C8—1071no—
131C4×C8C2×D10711yescharacteristic
132C2×Q8—1071no—
133C2×Q8—2141no—
134C2×Q8—2142no—
135C2×Q8—2142no—
136C2×Q8C4×D10711yescharacteristic
137C22×C4—1071no—
138C22×C4—1071no—
139C22×C4—1071no—
140C22×C4—2141no—
141C22×C4—2141no—
142M16—4282no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—2141no—
146C2×C8—2141no—
147C4⋊C4—21no—
148C4⋊C4—1071no—
149C4⋊C4—2141no—
150C4⋊C4—2141no—
151C4⋊C4—2141no—
152C4⋊C4—2141no—
153C4⋊C4C4×D10711yescharacteristic
154C42—21no—
155C42—1071no—
156C42—1071no—
157C42—1071no—
158C42—2141no—
159C42—2141no—
160C42—2141no—
161C42C22×D10711yescharacteristic
162C23—1071no—
163Q8—21no—
164Q8—1072no—
165Q8—2144no—
166Q8—2141no—
167Q8#1313380112yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—1071no—
171C2×C4—1072no—
172C2×C4—1072no—
173C2×C4—1071no—
174C2×C4—1071no—
175C2×C4—4281no—
176C2×C4—4282no—
177C2×C4—2144no—
178C2×C4—2141no—
179C2×C4—2141no—
180C2×C4—2141no—
181C2×C4—2141no—
182C2×C4C2×C4×D10711yescharacteristic
183C2×C4D4×D10711yescharacteristic
184C2×C4D4×D10711yescharacteristic
185C8—41no—
186C8—41no—
187C8—4281no—
188C8—4281no—
189C22—1072no—
190C22—2141no—
191C22—2141no—
192C22C22⋊C4×D10711yescharacteristic
193C4—21no—
194C4—22no—
195C4—41no—
196C4—1072no—
197C4—4281no—
198C4—2141no—
199C4—2141no—
200C4—2142no—
201C4M16×D10711yescharacteristic
202C4C2×#1313380111yescharacteristic
203C4SD16×D10711yescharacteristic
204C4Q16×D10711yescharacteristic
205C2—2142no—
206C2#2029258911yescharacteristic
207C2#103×D10711yescharacteristic
208C2#2029264211yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
FittingQ8⋊C8×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC22×D107C22⋊C4
cosocleC42×C107C23
hypercenterQ8⋊C8D107
hypocenterC107#1487
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #20292359C2semidirect
5. Q8⋊(C107⋊C8)C2semidirect
6. #20292357C2semidirect
7. Q8⋊C8×C107C2semidirect
59. Q8×C107M16semidirect
79. Dic214M16semidirect
105. C107#1487semidirect
112. Q8⋊C8D107semidirect
167. Q8#13133801semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D1072
#202926782
#202926492
#202923592
Q8⋊(C107⋊C8)2
#202923572
Q8⋊C8×C1072
#1487107