Finite Group Databasebeta

← #28339749

Subgroups — #28339749

Order 13696.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic107C211yesmaximal, characteristic
3#20292567C211yesmaximal, characteristic, complemented
4#20292566C211yesmaximal, characteristic, complemented
5#20292555C212yesmaximal, complemented
6#20292541C211yesmaximal, characteristic
7#20292539C212yesmaximal, complemented
8C22⋊C4×Dic107C212yesmaximal, complemented
9C22⋊C4×Dic107C212yesmaximal, complemented
10#20292472C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic
12C22×Dic214C2211yescharacteristic
13C23×Dic107C2212yes—
14#16812074—24no—
15D4×Dic107—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×C22⋊Dic107C2212yes—
20C2×C22⋊Dic107C2212yes—
21C2×C22⋊Dic107C2212yes—
22C2×C4×Dic107C2212yes—
23C2×C4×Dic107C2211yescharacteristic
24C2×C4×Dic107C2211yescharacteristic
25C2×C4×Dic107C2211yescharacteristic
26C107⋊#95C2212yes—
27C107⋊#95C2211yescharacteristic
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—22no—
40C22×Dic107—24no—
41C22×Dic107—24no—
42C22×Dic107—24no—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C4⋊Dic107—22no—
50Dic107⋊C4—22no—
51Dic107⋊C4—22no—
52C22⋊Dic107—24no—
53C22⋊Dic107—24no—
54C22⋊Dic107—24no—
55C4×Dic107—22no—
56C4×Dic107—22no—
57C4×Dic107—24no—
58C4×Dic107D414yes—
59C24×C107C2312yes—
60C2×D4×C107—24no—
61C2×D4×C107—22no—
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2312yes—
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C22⋊C4×C107—24no—
68C22⋊C4×C107—24no—
69C23×C107—22no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107#4114yes—
76C23×C107C2411yescharacteristic
77D4×C107—28no—
78C2×C4×C107—21no—
79C2×C4×C107—22no—
80C2×C4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D412yes—
87C2×C4×C107#4112yes—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—48no—
101C2×Dic107C2×D412yes—
102C2×Dic107C2×D412yes—
103C2×Dic107#4112yes—
104C2×Dic107#4114yes—
105Dic214—28no—
106C22×C107—24no—
107C22×C107—24no—
108C22×C107—24no—
109C22×C107—42no—
110C22×C107—42no—
111C22×C107—42no—
112C22×C107—42no—
113C22×C107C22×D411yescharacteristic
114C22×C107C2×#4111yescharacteristic
115C22×C107C2×#4112yes—
116C22×C107C2×#4112yes—
117C22×C107C2×#4111yescharacteristic
118C4×C107—22no—
119C4×C107—22no—
120C4×C107—41no—
121C4×C107—42no—
122Dic107—22no—
123Dic107—22no—
124Dic107—41no—
125Dic107—42no—
126Dic107—42no—
127Dic107#12414yes—
128C2×C107—44no—
129C2×C107#52511yescharacteristic
130C2×C107#52511yescharacteristic
131C2×C107C2×#12411yescharacteristic
132C2×C107#52912yes—
133C2×C107#52912yes—
134#2398—1071nomaximal, Sylow
135C107#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—1071no—
137C2×C4×D4—1071no—
138#400—1071no—
139#400—1071no—
140#397—1072no—
141#394—1071no—
142#394D10711yescharacteristic
143#393—1072no—
144C4×C22⋊C4—1072no—
145C4×C22⋊C4—1072no—
146#376—1071no—
147C22×Q8—1071no—
148C22×D4C2×D10711yescharacteristic
149C23×C4—1072no—
150C4×Q8—2144no—
151C4×D4—2144no—
152C2×C4⋊C4—1071no—
153C2×C4⋊C4—1071no—
154C2×C4⋊C4—1071no—
155C2×C4⋊C4C2×D10711yescharacteristic
156C2×C22⋊C4—1072no—
157C2×C22⋊C4—1072no—
158C2×C22⋊C4—1072no—
159C2×C22⋊C4C2×D10712yes—
160C2×C22⋊C4C2×D10712yes—
161C2×C42—1071no—
162C2×C42—1071no—
163C2×C42—1071no—
164C2×C42—1072no—
165#95—1072no—
166#95—1071no—
167#95—1072no—
168#95—1072no—
169#95—1072no—
170#95—1072no—
171#95C2×D10711yescharacteristic
172C24C22×D10712yes—
173C2×Q8—2142no—
174C2×Q8—2142no—
175C2×Q8—2142no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—1071no—
179C22×C4—1071no—
180C22×C4—1071no—
181C22×C4—1072no—
182C22×C4—1072no—
183C22×C4—1071no—
184C22×C4—2144no—
185C22×C4—2144no—
186C22×C4—2144no—
187C22×C4C22×D10712yes—
188C22×C4C22×D10711yescharacteristic
189C22×C4C22×D10711yescharacteristic
190C22×C4C22×D10711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—2142no—
193C4⋊C4—2142no—
194C4⋊C4—2142no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—2144no—
198C22⋊C4—2144no—
199C22⋊C4—2144no—
200C42—1074no—
201C42—2142no—
202C42—2142no—
203C42—2144no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1313381714yes—
211C23C23×D10711yescharacteristic
212Q8—2148no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—1072no—
223C2×C4—1074no—
224C2×C4—1072no—
225C2×C4—1072no—
226C2×C4—4288no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2141no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4D4×D10712yes—
240C2×C4#1313381712yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D10711yescharacteristic
249C22#41×D10711yescharacteristic
250C22#41×D10712yes—
251C22C2×#1313381712yes—
252C22C2×#1313381711yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—1074no—
258C4—4281no—
259C4—4282no—
260C4—4282no—
261C4—2142no—
262C4—2142no—
263C2—44no—
264C2#2029313311yescharacteristic
265C2#2029313711yescharacteristic
266C2#124×D10711yescharacteristic
267C2#2029315412yes—
268C2#2029315812yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
prime radicalC22×D4×C107C22
hypercenter#394D107
hypocenterC107#2398
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16812074C2semidirect
3. #20292567C2semidirect
4. #20292566C2semidirect
5. #20292555C2semidirect
7. #20292539C2semidirect
8. C22⋊C4×Dic107C2semidirect
9. C22⋊C4×Dic107C2semidirect
10. #20292472C2semidirect
135. C107#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#168120742
C2×D4×Dic1072
#202925672
#202925662
#202925552
#202925412
#202925392
C22⋊C4×Dic1072
#202924722
#394×C1072
#2398107