Finite Group Databasebeta

← #28337370

Subgroups — #28337370

Order 13696.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#13133803C211yesmaximal, characteristic
2C2×#16812044C211yesmaximal, characteristic, complemented
3#20292631C211yesmaximal, characteristic, complemented
4C2×#16812026C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6#20292583C211yesmaximal, characteristic, complemented
7#418×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C2×#13133803—22no—
10C2×#13133803—21no—
11C2×#13133803—21no—
12C2×#13133803C414yescomplemented
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812044—22no—
17#16812029—22no—
18#16812026—22no—
19#16812025—22no—
20#16812020—21no—
21C4×C107⋊C8—21no—
22C22×D4×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#102×C107—22no—
26C23×D107—21no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428C2×C412yes—
33#13133803—24no—
34#13133803—24no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38#13133802D412yes—
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C2×C107⋊C8C2×C412yes—
42C2×C107⋊C8D412yes—
43C2×C107⋊C8D411yescharacteristic
44C2×C107⋊C8D411yescharacteristic
45C24×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107—21no—
48C2×D4×C107—21no—
49C2×D4×C107—21no—
50C2×D4×C107—21no—
51C2×D4×C107C2×C412yes—
52C22×C4×C107C2311yescharacteristic
53M16×C107—21no—
54M16×C107—21no—
55C2×C8×C107—21no—
56C4⋊C4×C107D412yes—
57C22⋊C4×C107—21no—
58C42×C107—21no—
59C23×C107—22no—
60C23×C107—41no—
61C23×C107—41no—
62C23×C107—82no—
63C23×C107#4111yescharacteristic
64C22×D107—22no—
65C22×D107—41no—
66C22×D107—41no—
67C22×D107—82no—
68D4×C107—22no—
69D4×C107—22no—
70D4×C107—41no—
71D4×C107—41no—
72C2×C4×C107—22no—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107#4111yescharacteristic
79D428—22no—
80D428—22no—
81D428—41no—
82D428—41no—
83C8×C107—41no—
84C107⋊C8—22no—
85C107⋊C8—41no—
86C107⋊C8C22⋊C412yes—
87C107⋊C8C22⋊C412yes—
88C22×C107—22no—
89C22×C107—42no—
90C22×C107—82no—
91C22×C107—82no—
92C22×C107—82no—
93C22×C107C4×D411yescharacteristic
94C22×C107#12111yescharacteristic
95C22×C107#12411yescharacteristic
96C2×D107—42no—
97C2×D107—82no—
98C2×D107—82no—
99C2×D107—82no—
100C4×C107—42no—
101C4×C107C2×C22⋊C411yescharacteristic
102C4×C107C4×D411yescharacteristic
103C4×C107#12111yescharacteristic
104C4×C107#12711yescharacteristic
105C2×C107—21no—
106C2×C107—21no—
107C2×C107—82no—
108C2×C107#39011yescharacteristic
109C2×C107#47212yes—
110D107—82no—
111#1978—1071nomaximal, Sylow
112C107#197811yescharacteristic, complemented, Sylow
113C22×D8—1071no—
114C2×#108—1071no—
115#418—1071no—
116#418D10711yescharacteristic, complemented
117C2×#100—1071no—
118C2×#100—1071no—
119#405—1071no—
120C22×D4—1071no—
121C22×D4C2×D10711yescharacteristic
122C2×D8—1074no—
123C2×D8—2141no—
124C2×D8—2141no—
125C2×D8—2142no—
126C2×M16—1071no—
127C2×M16—1071no—
128C2×M16C2×D10711yescharacteristic
129C22×C8—1071no—
130#117C2×D10711yescharacteristic
131#108—2142no—
132#102—22no—
133#102—2142no—
134#100—2142no—
135#100—2142no—
136#97—2141no—
137C4×C8—2141no—
138C24—21no—
139C24—2141no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—1072no—
146C2×D4—2141no—
147C2×D4—2141no—
148C2×D4—2141no—
149C2×D4—2141no—
150C2×D4—2141no—
151C2×D4C4×D10712yes—
152C22×C4C22×D10711yescharacteristic
153D8—2144no—
154D8—2144no—
155M16—21no—
156M16—21no—
157M16—1072no—
158M16—2141no—
159M16—2141no—
160M16—2141no—
161C2×C8—21no—
162C2×C8—1072no—
163C2×C8—1071no—
164C2×C8—1071no—
165C2×C8—1072no—
166C2×C8—2141no—
167C2×C8—2141no—
168C4⋊C4C22⋊D10712yes—
169C22⋊C4—21no—
170C42—21no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—82no—
175C23—8562no—
176C23—4281no—
177C23—4281no—
178C23—2142no—
179C23#1313381711yescharacteristic
180D4—22no—
181D4—22no—
182D4—41no—
183D4—41no—
184D4—4281no—
185D4—4281no—
186D4—2142no—
187D4—2142no—
188C2×C4—22no—
189C2×C4C2×C4×D10711yescharacteristic
190C2×C4D4×D10711yescharacteristic
191C2×C4D4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4C2×C22⋊D10711yescharacteristic
194C2×C4#1313381811yescharacteristic
195C8—41no—
196C8—1072no—
197C8—1072no—
198C8—4281no—
199C8—2142no—
200C22—22no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—8562no—
206C22—8562no—
207C22—8562no—
208C22—4282no—
209C22(C22⋊D107)⋊C411yescharacteristic
210C22#1681208111yescharacteristic
211C22#1681209511yescharacteristic
212C4—42no—
213C4C22⋊C4×D10711yescharacteristic
214C4C4×C22⋊D10711yescharacteristic
215C4#1681208211yescharacteristic
216C4#1681210511yescharacteristic
217C2—21no—
218C2—21no—
219C2—82no—
220C2—8562no—
221C2#2029252711yescharacteristic
222C2#2029289512yes—
subgroupG/H
centerC22#16812081
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#418×C107C2
socleC22×C107#121
prime radicalC22×#13133803C2
cosocleC22×C4×C107C23
hypercenter#418D107
hypocenterC107#1978
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#16812044C2semidirect
3. #20292631C2semidirect
4. C2×#16812026C2semidirect
5. C2×#16812025C2semidirect
6. #20292583C2semidirect
7. #418×C107C2semidirect
12. C2×#13133803C4semidirect
112. C107#1978semidirect
116. #418D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338032
C2×#168120442
#202926312
C2×#168120262
C2×#168120252
#202925832
#418×C1072
#1978107