Finite Group Databasebeta

← #28335523

Subgroups — #28335523

Order 13696.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293297C211yesmaximal, characteristic, complemented
2#20292824C211yesmaximal, characteristic, complemented
3#20292819C211yesmaximal, characteristic, complemented
4#20292371C211yesmaximal, characteristic, complemented
5#20292359C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7#493×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9C2×#13133806—21no—
10C2×#13133803—21no—
11#16812105—21no—
12#16812081—22no—
13#16812080—21no—
14C4×Dic214C2211yescharacteristic
15C4×D428C2211yescharacteristic
16D4×Dic107—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812030—21no—
20#16812030—21no—
21C2×D8×C107—22no—
22#127×C107C2211yescharacteristic
23#127×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214D411yescharacteristic
28C2×D428D411yescharacteristic
29C2×C22⋊D107—22no—
30C2×C22⋊D107—22no—
31C22×Dic107—22no—
32C2×C4×D107—21no—
33#13133806—42no—
34#13133803—42no—
35C2×C107⋊C8—21no—
36C2×C107⋊C8—21no—
37C4⋊Dic107D411yescharacteristic
38C4⋊Dic107D411yescharacteristic
39Dic107⋊C4—21no—
40C22⋊Dic107—22no—
41(C2×D107)⋊C4—21no—
42C4×Dic107—21no—
43C2×D4×C107—21no—
44C2×D4×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107—21no—
48C2×D4×C107D412yes—
49D8×C107—44no—
50C2×C8×C107—21no—
51C42×C107C2311yescharacteristic
52C23×C107—22no—
53C23×C107—41no—
54C22×D107—21no—
55D4×C107—42no—
56D4×C107—42no—
57D4×C107—41no—
58D4×C107—41no—
59D4×C107—41no—
60D4×C107—41no—
61D4×C107—41no—
62D4×C107—41no—
63D4×C107—41no—
64D4×C107—41no—
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C22⋊D107—44no—
69C22⋊D107—82no—
70C2×Dic107—21no—
71C2×Dic107—21no—
72C2×Dic107—21no—
73C2×Dic107—44no—
74D428—21no—
75D428—21no—
76C4×D107—41no—
77C4×D107—41no—
78Dic214—21no—
79Dic214D812yescomplemented
80C8×C107—22no—
81C107⋊C8—41no—
82C107⋊C8—41no—
83C22×C107—42no—
84C22×C107—42no—
85C22×C107—42no—
86C22×C107—81no—
87C22×C107—81no—
88C22×C107—81no—
89C22×C107#12011yescharacteristic
90C2×D107—41no—
91C2×D107—41no—
92C2×D107—41no—
93C4×C107—21no—
94C4×C107C2×D811yescharacteristic
95C4×C107#13511yescharacteristic
96C4×C107AGL(1,ℤ/8)11yescharacteristic
97C4×C107AGL(1,ℤ/8)11yescharacteristic
98Dic107—22no—
99Dic107—41no—
100Dic107—41no—
101C2×C107—82no—
102C2×C107—161no—
103C2×C107#44711yescharacteristic
104C2×C107#44911yescharacteristic
105C2×C107#45311yescharacteristic
106D107—81no—
107#1621—1071nomaximal, Sylow
108C107#162111yescharacteristic, complemented, Sylow
109#550—1071no—
110#493D10711yescharacteristic, complemented
111#460—1071no—
112#459—1071no—
113#331—1071no—
114Q8⋊C8—1071no—
115D4⋊C8—1071no—
116C2×#41—2141no—
117C2×SD16—2141no—
118C2×D8—22no—
119C2×D8—2141no—
120#127—2141no—
121#127C2×D10711yescharacteristic
122#127C2×D10711yescharacteristic
123#121—2141no—
124#121—2142no—
125C4×Q8—1071no—
126C4×D4—1071no—
127C4×D4—2141no—
128C4⋊C8—1071no—
129C4⋊C8—1071no—
130#102—2141no—
131#102—2141no—
132C4×C8C2×D10711yescharacteristic
133#41—4282no—
134#41—4282no—
135C2×Q8—1071no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—1071no—
142C2×D4—2142no—
143C2×D4—2142no—
144C2×D4C22⋊D10712yes—
145C22×C4—2142no—
146C22×C4—2141no—
147SD16—4282no—
148D8—44no—
149D8—4282no—
150C2×C8—21no—
151C2×C8—2141no—
152C2×C8—2141no—
153C4⋊C4—1071no—
154C4⋊C4—1071no—
155C4⋊C4—2141no—
156C22⋊C4—2142no—
157C22⋊C4—2141no—
158C42—2141no—
159C42C22×D10711yescharacteristic
160C23—22no—
161C23—41no—
162C23—2141no—
163Q8—1072no—
164Q8—2141no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—8562no—
176D4—4284no—
177D4—2141no—
178D4—2141no—
179C2×C4—4284no—
180C2×C4—4281no—
181C2×C4—4281no—
182C2×C4—2141no—
183C2×C4—2141no—
184C2×C4—2141no—
185C2×C4D4×D10711yescharacteristic
186C2×C4D4×D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C8—22no—
189C8—4281no—
190C8—4281no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—4281no—
198C22—4281no—
199C22—4281no—
200C22#1681207711yescharacteristic
201C4—21no—
202C4—4281no—
203C4—4281no—
204C4—2142no—
205C4D8×D10711yescharacteristic
206C4#1681212811yescharacteristic
207C4#1681213311yescharacteristic
208C4#1681213311yescharacteristic
209C2—82no—
210C2—161no—
211C2—8561no—
212C2#2029275911yescharacteristic
213C2#2029276911yescharacteristic
214C2#2029279411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#493×C107C2
socleC22×C107#120
hypercenter#493D107
hypocenterC107#1621
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #20293297C2semidirect
2. #20292824C2semidirect
3. #20292819C2semidirect
4. #20292371C2semidirect
5. #20292359C2semidirect
6. #20292356C2semidirect
7. #493×C107C2semidirect
79. Dic214D8semidirect
108. C107#1621semidirect
110. #493D107semidirect

Maximal subgroups

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

subgroupindex
#202932972
#202928242
#202928192
#202923712
#202923592
#202923562
#493×C1072
#1621107