Finite Group Databasebeta

← #28339734

Subgroups — #28339734

Order 13696.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16812074C211yesmaximal, characteristic, complemented
2C2×#16812067C211yesmaximal, characteristic
3#20292562C211yesmaximal, characteristic
4#20292561C212yesmaximal, complemented
5#20292551C212yesmaximal
6#20292547C211yesmaximal, characteristic, complemented
7#20292540C211yesmaximal, characteristic
8#20292478C211yesmaximal, characteristic, complemented
9#20292478C212yesmaximal, complemented
10#20292474C212yesmaximal
11#400×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14#16812074—24no—
15#16812067—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4C2212yes—
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×C4×Dic107C2212yes—
26C2×C4×Dic107C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C2×C4×Dic107C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2211yescharacteristic
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34#95×C107C2212yes—
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C2×Dic214—22no—
39C2×Dic214—24no—
40C22×Dic107C2311yescharacteristic
41C22×Dic107C2311yescharacteristic
42C22×Dic107C2312yes—
43C22×Dic107C2312yes—
44C22×Dic107C2311yescharacteristic
45C2×C4×D107—22no—
46C2×C4×D107—22no—
47C2×C4×D107Q814yescomplemented
48C4⋊Dic107—22no—
49Dic107⋊C4—24no—
50Dic107⋊C4—24no—
51Dic107⋊C4—24no—
52Dic107⋊C4—24no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—24no—
57C4×Dic107—22no—
58C4×Dic107—24no—
59C4×Dic107—24no—
60C4×Dic107D414yes—
61C22×C4×C107C2311yescharacteristic
62C22×C4×C107C2312yes—
63C22×C4×C107C2312yes—
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C23×C107C2411yescharacteristic
69C22×D107—21no—
70C22×D107—22no—
71C22×D107—22no—
72C22×D107—21no—
73C22×D107C2×Q812yes—
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107—22no—
77C2×C4×C107—21no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×Q812yes—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—21no—
93C2×Dic107—22no—
94C2×Dic107C2×D412yes—
95C2×Dic107C2×D412yes—
96C2×Dic107C2×Q812yes—
97C2×Dic107#4114yes—
98C2×Dic107#4114yes—
99C2×Dic107#4112yes—
100C4×D107—28no—
101Dic214—28no—
102C22×C107C22×D411yescharacteristic
103C22×C107C22×Q811yescharacteristic
104C22×C107C2×#4111yescharacteristic
105C22×C107C2×#4112yes—
106C22×C107C2×#4112yes—
107C2×D107—22no—
108C2×D107—24no—
109C2×D107—41no—
110C2×D107—42no—
111C2×D107—41no—
112C4×C107—22no—
113C4×C107—22no—
114C4×C107—41no—
115C4×C107—42no—
116C4×C107—42no—
117Dic107—24no—
118Dic107—24no—
119Dic107—22no—
120Dic107—41no—
121Dic107C22⋊Q814yes—
122C2×C107C2×C22⋊Q811yescharacteristic
123C2×C107#52511yescharacteristic
124C2×C107#52912yes—
125C2×C107#53211yescharacteristic
126C2×C107#53312yes—
127D107—42no—
128#2396—1071nomaximal, Sylow
129C107#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1071no—
131C2×#117—1071no—
132#400D10711yescharacteristic, complemented
133#399—1071no—
134#398—1072no—
135#396—1072no—
136#395—1071no—
137#393—1071no—
138C4×C4⋊C4—1072no—
139C4×C4⋊C4—1071no—
140C4×C22⋊C4—1072no—
141C22×Q8—1071no—
142C23×C4—1071no—
143C4×Q8—2144no—
144#117—2144no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4—1072no—
147C2×C4⋊C4—1072no—
148C2×C4⋊C4—1072no—
149C2×C4⋊C4—1072no—
150C2×C4⋊C4C2×D10711yescharacteristic
151C2×C4⋊C4C2×D10711yescharacteristic
152C2×C22⋊C4—1071no—
153C2×C22⋊C4—1071no—
154C2×C22⋊C4—1072no—
155C2×C22⋊C4—1072no—
156C2×C42—1071no—
157C2×C42—1071no—
158C2×C42—1072no—
159C2×C42—1072no—
160#95—1072no—
161#95—1072no—
162#95—1071no—
163#95C2×D10712yes—
164#95C2×D10711yescharacteristic
165#95C2×D10712yes—
166C24—1071no—
167C2×Q8—2144no—
168C2×Q8—2142no—
169C22×C4—1071no—
170C22×C4—1072no—
171C22×C4—1072no—
172C22×C4—1071no—
173C22×C4—1074no—
174C22×C4—1071no—
175C22×C4—2142no—
176C22×C4—2142no—
177C22×C4C22×D10712yes—
178C22×C4C22×D10711yescharacteristic
179C22×C4C22×D10712yes—
180C22×C4C22×D10711yescharacteristic
181C22×C4C22×D10711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—2142no—
185C4⋊C4—2144no—
186C4⋊C4—2144no—
187C4⋊C4—2144no—
188C4⋊C4—2144no—
189C22⋊C4—2142no—
190C22⋊C4—2144no—
191C22⋊C4—2144no—
192C22⋊C4—2142no—
193C42—1074no—
194C42—2142no—
195C42—2144no—
196C42—2144no—
197C23—1072no—
198C23—2141no—
199C23—2142no—
200C23—2142no—
201C23—2141no—
202C23C23×D10711yescharacteristic
203Q8—2148no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1072no—
216C2×C4—1072no—
217C2×C4—1072no—
218C2×C4—1072no—
219C2×C4—1074no—
220C2×C4—1074no—
221C2×C4—2148no—
222C2×C4—2141no—
223C2×C4—2142no—
224C2×C4—2142no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4D4×D10712yes—
230C2×C4Q8×D10712yes—
231C22—4281no—
232C22—4282no—
233C22—4281no—
234C22—2142no—
235C22—2144no—
236C22C2×D4×D10711yescharacteristic
237C22C2×Q8×D10711yescharacteristic
238C22#41×D10711yescharacteristic
239C22#41×D10712yes—
240C22#41×D10712yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—1074no—
247C4—4281no—
248C4—2144no—
249C4—2144no—
250C4—2142no—
251C2—4282no—
252C2C22⋊Q8×D10711yescharacteristic
253C2#2029313511yescharacteristic
254C2#2029316012yes—
255C2#2029317611yescharacteristic
256C2#2029318312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#400×C107C2
prime radicalC23×D107C23
hypercenter#400D107
hypocenterC107#2396
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16812074C2semidirect
4. #20292561C2semidirect
6. #20292547C2semidirect
8. #20292478C2semidirect
9. #20292478C2semidirect
11. #400×C107C2semidirect
47. C2×C4×D107Q8semidirect
129. C107#2396semidirect
132. #400D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120742
C2×#168120672
#202925622
#202925612
#202925512
#202925472
#202925402
#202924782
#202924742
#400×C1072
#2396107