Finite Group Databasebeta

← #28348389

Subgroups — #28348389

Order 13696.

398 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293342C211yesmaximal, characteristic, complemented
2#20292977C211yesmaximal, characteristic, complemented
3#20292965C212yesmaximal, complemented
4#20292917C211yesmaximal, characteristic, complemented
5#20292858C212yesmaximal, complemented
6#20292829C211yesmaximal, characteristic, complemented
7#20292772C212yesmaximal, complemented
8C4×#13133803C211yesmaximal, characteristic, complemented
9D4×C107⋊C8C211yesmaximal, characteristic, complemented
10#20292623C212yesmaximal, complemented
11#547×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2212yes—
13C2×#13133803C2211yescharacteristic
14C22×C107⋊C8C2212yes—
15#16812109C2211yescharacteristic
16#16812095—22no—
17#16812083C2212yes—
18#16812080—21no—
19C4×D428C2211yescharacteristic
20#16812068C2212yes—
21#16812067—21no—
22#16812042C2211yescharacteristic
23#16812040C2212yes—
24C4⋊(C107⋊C8)C2211yescharacteristic
25#16812033C2212yes—
26#16812033C2212yes—
27#16812033C2212yes—
28#16812030C2211yescharacteristic
29#16812030C2212yes—
30#16812029C2211yescharacteristic
31C22⋊(C107⋊C8)C2212yes—
32C4×C107⋊C8C2211yescharacteristic
33C2×#41×C107C2212yes—
34C4⋊Q8×C107C2211yescharacteristic
35#123×C107—22no—
36C22⋊Q8×C107C2212yes—
37#121×C107—21no—
38C4×D4×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40C2×C4⋊C4×C107—21no—
41C2×Dic214—21no—
42C2×D428C2311yescharacteristic
43C2×C22⋊D107—22no—
44C2×C4×D107—21no—
45#13133805—42no—
46#13133803—22no—
47C2×C107⋊C8—42no—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2312yes—
51C4⋊Dic107—21no—
52C4⋊Dic107C2312yes—
53C4⋊Dic107C2311yescharacteristic
54C22⋊Dic107—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—21no—
57C4×Dic107—22no—
58#41×C107—24no—
59#41×C107—42no—
60C2×Q8×C107C2312yes—
61C2×D4×C107—22no—
62C2×D4×C107D411yescharacteristic
63C2×D4×C107C2311yescharacteristic
64C22×C4×C107—21no—
65C22×C4×C107—22no—
66C22×C4×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—22no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107D411yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C22⋊C4×C107—22no—
73C22⋊C4×C107—21no—
74C22⋊C4×C107D412yes—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C23×C107C2×D412yes—
78C22×D107—21no—
79Q8×C107—22no—
80Q8×C107—22no—
81D4×C107—22no—
82D4×C107—21no—
83D4×C107—44no—
84D4×C107#4112yes—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107—44no—
90C2×C4×C107—42no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—82no—
96C2×Dic107—21no—
97C2×Dic107—22no—
98D428—21no—
99D428—21no—
100C4×D107—42no—
101Dic214—81no—
102C107⋊C8—21no—
103C107⋊C8—21no—
104C107⋊C8—22no—
105C22×C107—22no—
106C22×C107—22no—
107C22×C107—22no—
108C22×C107—22no—
109C22×C107—41no—
110C22×C107—41no—
111C22×C107C22×D411yescharacteristic
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—22no—
116C4×C107—21no—
117C4×C107—42no—
118C4×C107—41no—
119C4×C107#13512yes—
120C4×C107C2×#4111yescharacteristic
121C4×C107H(5,𝔽2)11yescharacteristic
122Dic107—41no—
123Dic107—42no—
124C2×C107—42no—
125C2×C107—42no—
126C2×C107#54611yescharacteristic
127C2×C107C2×#13511yescharacteristic
128C2×C107#57711yescharacteristic
129D107—81no—
130#3303—1071nomaximal, Sylow
131C107#330311yescharacteristic, complemented, Sylow
132#555—1071no—
133#547D10711yescharacteristic, complemented
134#487—1071no—
135#485—1072no—
136#476—1071no—
137#465—1072no—
138#461—1071no—
139#449—1072no—
140C4×D8—1071no—
141C8×D4—1071no—
142#416—1072no—
143C2×#41C2×D10712yes—
144C2×SD16—1072no—
145C2×D8—1071no—
146C22×C8—1072no—
147C4⋊Q8—1071no—
148C4⋊Q8C2×D10711yescharacteristic
149#124—2142no—
150#123—22no—
151C22⋊Q8C2×D10712yes—
152#121—21no—
153#121—1072no—
154#121—2141no—
155C4×D4—1071no—
156C4×D4C2×D10711yescharacteristic
157C4×D4C2×D10711yescharacteristic
158#117—1072no—
159#117—2141no—
160C2×C4⋊C4—21no—
161#107—1071no—
162#106—1072no—
163C4⋊C8—1071no—
164#103—1072no—
165#103—1072no—
166#103—1072no—
167#102—1071no—
168#102—1072no—
169#102—1071no—
170C22⋊C8—1072no—
171C4×C8—1071no—
172#41—24no—
173#41—42no—
174C2×Q8—2141no—
175C2×Q8C22×D10712yes—
176C2×D4—22no—
177C2×D4—1071no—
178C2×D4—2142no—
179C2×D4C22⋊D10711yescharacteristic
180C2×D4C22×D10711yescharacteristic
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2141no—
184C22×C4C22×D10712yes—
185SD16—4282no—
186D8—2142no—
187C2×C8—1071no—
188C2×C8—1071no—
189C2×C8—1072no—
190C2×C8—4282no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—1072no—
195C4⋊C4—1071no—
196C4⋊C4—2141no—
197C4⋊C4C22⋊D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C22⋊C4—2141no—
204C22⋊C4C22⋊D10712yes—
205C42—2142no—
206C42C22×D10711yescharacteristic
207C23—21no—
208C23—2141no—
209C23C2×C22⋊D10712yes—
210Q8—22no—
211Q8—22no—
212Q8—8561no—
213D4—22no—
214D4—21no—
215D4—44no—
216D4—8562no—
217D4—2141no—
218D4—2141no—
219D4#1313381712yes—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—4282no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4C2×C22⋊D10712yes—
230C2×C4C2×C22⋊D10711yescharacteristic
231C2×C4C2×C22⋊D10711yescharacteristic
232C2×C4C23×D10711yescharacteristic
233C8—2141no—
234C8—2141no—
235C8—2142no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—4281no—
243C22—4281no—
244C22—4281no—
245C22C22×C22⋊D10711yescharacteristic
246C4—21no—
247C4—22no—
248C4—41no—
249C4—42no—
250C4—4281no—
251C4—4282no—
252C4#1681212912yes—
253C4C2×#1313381711yescharacteristic
254C4#1681215411yescharacteristic
255C2—42no—
256C2—42no—
257C2—8561no—
258C2#2029327111yescharacteristic
259C2C2×#1681212911yescharacteristic
260C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20292977C2semidirect
3. #20292965C2semidirect
4. #20292917C2semidirect
5. #20292858C2semidirect
6. #20292829C2semidirect
7. #20292772C2semidirect
8. C4×#13133803C2semidirect
9. D4×C107⋊C8C2semidirect
10. #20292623C2semidirect
11. #547×C107C2semidirect
131. C107#3303semidirect
133. #547D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202929772
#202929652
#202929172
#202928582
#202928292
#202927722
C4×#131338032
D4×C107⋊C82
#202926232
#547×C1072
#3303107