Finite Group Databasebeta

← #28337497

Subgroups — #28337497

Order 13696.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16812140C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic
3C2×#16812037C211yesmaximal, characteristic
4#20292636C211yesmaximal, characteristic, complemented
5C2×#16812028C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—22no—
10#16812140—22no—
11C2×#13133808—21no—
12C2×#13133805—21no—
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812107—21no—
16#16812098—21no—
17#16812092—22no—
18#16812085—22no—
19#16812068C2211yescharacteristic
20#16812037—22no—
21#16812033—22no—
22#16812028—22no—
23C22×Q8×C107C2211yescharacteristic
24C22×Q8×C107C2211yescharacteristic
25C4⋊Q8×C107—22no—
26C4⋊Q8×C107—22no—
27C2×C4⋊C4×C107—22no—
28C2×C42×C107C2211yescharacteristic
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33C2×Dic214D411yescharacteristic
34C2×D428D411yescharacteristic
35C2×C22⋊D107—21no—
36C2×C4×D107—21no—
37#13133808—24no—
38#13133805—24no—
39#13133802—21no—
40#13133802—21no—
41#13133802—21no—
42#13133802D412yes—
43C2×C107⋊C8—21no—
44C2×C107⋊C8—21no—
45C4⋊Dic107D412yes—
46Dic107⋊C4—24no—
47C22⋊Dic107—21no—
48(C2×D107)⋊C4—24no—
49C4×Dic107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—21no—
55C2×Q8×C107—21no—
56C2×Q8×C107—21no—
57C2×Q8×C107—21no—
58C2×Q8×C107—21no—
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107D412yes—
62C22×C4×C107—21no—
63C22×C4×C107—21no—
64C22×C4×C107D412yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—28no—
67C42×C107—21no—
68C42×C107D412yes—
69C23×C107C2×D411yescharacteristic
70C22×D107—21no—
71Q8×C107—22no—
72Q8×C107—22no—
73Q8×C107—41no—
74Q8×C107—41no—
75Q8×C107—41no—
76Q8×C107—41no—
77Q8×C107—41no—
78Q8×C107—41no—
79C2×C4×C107—24no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—41no—
86C2×C4×C107—41no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107#4111yescharacteristic
93C22⋊D107—41no—
94C22⋊D107—41no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97D428—21no—
98D428—21no—
99C4×D107—41no—
100C4×D107—41no—
101Dic214—21no—
102Dic214—21no—
103C107⋊C8—22no—
104C107⋊C8—41no—
105C22×C107—22no—
106C22×C107#12011yescharacteristic
107C22×C107#12011yescharacteristic
108C22×C107#12111yescharacteristic
109C2×D107—41no—
110C2×D107—42no—
111C4×C107—24no—
112C4×C107—42no—
113C4×C107—42no—
114C4×C107#12011yescharacteristic
115C4×C107#12111yescharacteristic
116C4×C107#12111yescharacteristic
117C4×C107#12711yescharacteristic
118Dic107—41no—
119Dic107—42no—
120C2×C107—21no—
121C2×C107—21no—
122C2×C107#39211yescharacteristic
123C2×C107#45612yes—
124D107—81no—
125#2005—1071nomaximal, Sylow
126C107#200511yescharacteristic, complemented, Sylow
127C2×#137—1071no—
128#536—1071no—
129C2×C4⋊Q8D10711yescharacteristic, complemented
130C2×#104—1071no—
131#419—1071no—
132C2×#101—1071no—
133#339—1071no—
134C2×#41—1071no—
135C22×Q8C2×D10711yescharacteristic
136C22×Q8C2×D10711yescharacteristic
137#137—2142no—
138#137—2142no—
139C2×Q16—2141no—
140C2×SD16—2141no—
141C2×M16—1071no—
142C2×M16—1071no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—2141no—
146#124—2141no—
147#123—2142no—
148C22⋊Q8—2142no—
149#117—1071no—
150C2×C4⋊C4—22no—
151C2×C42C2×D10711yescharacteristic
152#104—2142no—
153#103—2142no—
154#101—2142no—
155#41—2141no—
156#41—2141no—
157#41—2141no—
158#41—2141no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—1071no—
171C2×Q8C22⋊D10712yes—
172C2×D4—1071no—
173C2×D4—2141no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—2141no—
177C22×C4C22⋊D10712yes—
178C22×C4C22×D10711yescharacteristic
179Q16—2144no—
180SD16—2144no—
181M16—1072no—
182M16—2141no—
183M16—2141no—
184M16—2141no—
185C2×C8—2141no—
186C2×C8—2141no—
187C4⋊C4—28no—
188C4⋊C4—1072no—
189C4⋊C4—2144no—
190C22⋊C4—2144no—
191C22⋊C4—2141no—
192C42—21no—
193C42—2141no—
194C42C22⋊D10712yes—
195C23—2141no—
196C23C2×C22⋊D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—2141no—
206Q8—2141no—
207D4—4281no—
208D4—4281no—
209D4—2141no—
210D4—2141no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—4281no—
220C2×C4—4281no—
221C2×C4—2142no—
222C2×C4—2141no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4C2×C22⋊D10711yescharacteristic
227C2×C4C2×C22⋊D10711yescharacteristic
228C2×C4#1313381711yescharacteristic
229C8—4281no—
230C8—2142no—
231C22—22no—
232C22—4281no—
233C22—4282no—
234C22#1681207711yescharacteristic
235C22#1681207811yescharacteristic
236C22#1681208111yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—4281no—
241C4—4282no—
242C4#1681207711yescharacteristic
243C4#1681208011yescharacteristic
244C4#1681208111yescharacteristic
245C4#1681210511yescharacteristic
246C2—21no—
247C2—21no—
248C2—8561no—
249C2#2029253511yescharacteristic
250C2#2029280912yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×C4⋊Q8×C107C2
socleC22×C107#120
prime radicalC2×#13133818C22
hypercenterC2×C4⋊Q8D107
hypocenterC107#2005
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #20292636C2semidirect
5. C2×#16812028C2semidirect
6. #20292387C2semidirect
7. C2×C4⋊Q8×C107C2semidirect
126. C107#2005semidirect
129. C2×C4⋊Q8D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202932012
C2×#168120372
#202926362
C2×#168120282
#202923872
C2×C4⋊Q8×C1072
#2005107