Finite Group Databasebeta

← #28344810

Subgroups — #28344810

Order 13696.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C2×#16812107C211yesmaximal, characteristic, complemented
2C2×#16812089C211yesmaximal, characteristic
3C2×#16812085C211yesmaximal, characteristic
4C2×C42×D107C211yesmaximal, characteristic
5#20292533C211yesmaximal, characteristic, complemented
6#20292524C214yesmaximal, complemented
7#20292496C211yesmaximal, characteristic, complemented
8#20292486C214yesmaximal
9#391×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×C4×D107C2211yescharacteristic
12C22×C4×D107C2212yes—
13#16812107—24no—
14#16812089—24no—
15#16812085—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4⋊Dic107C2212yes—
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2214yes—
20C2×Dic107⋊C4C2214yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×C4×Dic107C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C42×D107—24no—
26C107⋊#95C2214yes—
27C22×Q8×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2214yes—
31C23×D107C2311yescharacteristic
32C2×Dic214—22no—
33C2×Dic214—22no—
34C2×Dic214—22no—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C22×Dic107C2311yescharacteristic
38C22×Dic107C2312yes—
39C2×C4×D107—22no—
40C2×C4×D107—22no—
41C2×C4×D107—24no—
42C2×C4×D107—24no—
43C2×C4×D107D414yes—
44C2×C4×D107Q818yescomplemented
45C4⋊Dic107—22no—
46C4⋊Dic107—24no—
47Dic107⋊C4—24no—
48Dic107⋊C4—28no—
49Dic107⋊C4—28no—
50(C2×D107)⋊C4—24no—
51(C2×D107)⋊C4—24no—
52C4×Dic107—24no—
53C4×Dic107D414yes—
54C2×Q8×C107—22no—
55C2×Q8×C107—22no—
56C2×Q8×C107—22no—
57C22×C4×C107C2312yes—
58C22×C4×C107C2312yes—
59C22×C4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C42×C107—22no—
63C23×C107C2411yescharacteristic
64C22×D107—21no—
65C22×D107—22no—
66C22×D107—21no—
67C22×D107C2×D412yes—
68C22×D107C2×Q814yes—
69Q8×C107—44no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—24no—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×Q814yes—
80C2×C4×C107#4112yes—
81C2×Dic107—21no—
82C2×Dic107—21no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—24no—
87C2×Dic107—22no—
88C2×Dic107C2×D412yes—
89C2×Dic107C2×D414yes—
90C2×Dic107C2×Q814yes—
91C2×Dic107#4114yes—
92C4×D107—28no—
93C4×D107—216no—
94Dic214—44no—
95C22×C107C22×D411yescharacteristic
96C22×C107C22×D411yescharacteristic
97C22×C107C22×Q812yes—
98C22×C107C2×#4112yes—
99C22×C107C2×#4111yescharacteristic
100C2×D107—22no—
101C2×D107—24no—
102C2×D107—22no—
103C2×D107—24no—
104C2×D107C4⋊Q814yescomplemented
105C4×C107—24no—
106C4×C107—42no—
107C4×C107—42no—
108C4×C107#12114yes—
109Dic107—22no—
110Dic107—42no—
111Dic107—42no—
112Dic107C22⋊Q818yescomplemented
113C2×C107C2×#12111yescharacteristic
114C2×C107C2×C22⋊Q812yes—
115C2×C107C2×C4⋊Q811yescharacteristic
116C2×C107#53211yescharacteristic
117C2×C107#53312yes—
118D107—24no—
119#2857—1071nomaximal, Sylow
120C107#285711yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—1071no—
122C2×C22⋊Q8—1071no—
123C2×C22⋊Q8—1071no—
124C22×C42—1071no—
125#391—1071no—
126#391D10711yescharacteristic, complemented
127#389—1074no—
128#384—1071no—
129#380—1074no—
130C22×Q8—1071no—
131C22×Q8C2×D10711yescharacteristic
132C23×C4—1071no—
133C23×C4—1072no—
134C4⋊Q8—2144no—
135C22⋊Q8—2144no—
136C22⋊Q8—2144no—
137C2×C4⋊C4—1072no—
138C2×C4⋊C4—1071no—
139C2×C4⋊C4—1072no—
140C2×C4⋊C4—1074no—
141C2×C4⋊C4—1074no—
142C2×C4⋊C4C2×D10711yescharacteristic
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C42—1071no—
146C2×C42—1072no—
147C2×C42—2144no—
148C2×C42C2×D10711yescharacteristic
149#95—1074no—
150#95C2×D10714yes—
151C24—1071no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—2142no—
156C2×Q8—2142no—
157C2×Q8—2142no—
158C22×C4—1072no—
159C22×C4—1071no—
160C22×C4—1074no—
161C22×C4—1078no—
162C22×C4—1072no—
163C22×C4—1072no—
164C22×C4—2144no—
165C22×C4—2144no—
166C22×C4—2142no—
167C22×C4—2142no—
168C22×C4C22×D10712yes—
169C22×C4C22×D10712yes—
170C22×C4C22×D10712yes—
171C22×C4C22×D10711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—2144no—
174C4⋊C4—2144no—
175C4⋊C4—2148no—
176C4⋊C4—2148no—
177C4⋊C4—2142no—
178C22⋊C4—2144no—
179C22⋊C4—2144no—
180C42—22no—
181C42—1074no—
182C42—2144no—
183C23—1072no—
184C23—1074no—
185C23—2141no—
186C23—2142no—
187C23—2141no—
188C23C23×D10711yescharacteristic
189Q8—44no—
190Q8—4284no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—1074no—
199C2×C4—1074no—
200C2×C4—1074no—
201C2×C4—1072no—
202C2×C4—21416no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2144no—
207C2×C4—2142no—
208C2×C4—2141no—
209C2×C4—2141no—
210C2×C4—2148no—
211C2×C4D4×D10712yes—
212C2×C4D4×D10712yes—
213C2×C4Q8×D10714yes—
214C2×C4#1313381812yes—
215C22—1074no—
216C22—2144no—
217C22—2142no—
218C22—2144no—
219C22—2142no—
220C22C2×D4×D10711yescharacteristic
221C22C2×D4×D10711yescharacteristic
222C22C2×Q8×D10712yes—
223C22#41×D10712yes—
224C22C2×#1313381811yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—1078no—
229C4—4282no—
230C4—4282no—
231C4—2142no—
232C4#1681208214yes—
233C2—2144no—
234C2C2×#1681208211yescharacteristic
235C2C22⋊Q8×D10712yes—
236C2C4⋊Q8×D10711yescharacteristic
237C2#2029317611yescharacteristic
238C2#2029318112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
prime radicalC23×D107C23
hypercenter#391D107
hypocenterC107#2857
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16812107C2semidirect
5. #20292533C2semidirect
6. #20292524C2semidirect
7. #20292496C2semidirect
9. #391×C107C2semidirect
44. C2×C4×D107Q8semidirect
104. C2×D107C4⋊Q8semidirect
112. Dic107C22⋊Q8semidirect
120. C107#2857semidirect
126. #391D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121072
C2×#168120892
C2×#168120852
C2×C42×D1072
#202925332
#202925242
#202924962
#202924862
#391×C1072
#2857107