Finite Group Databasebeta

← #28337499

Subgroups — #28337499

Order 13696.

362 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812142C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic
3C2×#16812107C211yesmaximal, characteristic, complemented
4C2×#16812035C211yesmaximal, characteristic
5#20292637C211yesmaximal, characteristic, complemented
6#20292388C211yesmaximal, characteristic, complemented
7C2×#101×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10#16812142—22no—
11#16812142—22no—
12C2×Dic428—21no—
13C2×#13133807—21no—
14#16812107—22no—
15#16812107—21no—
16#16812107—22no—
17#16812098—21no—
18#16812092—22no—
19#16812085—22no—
20#16812068C2211yescharacteristic
21C2×Dic107⋊C4—22no—
22C2×C4×Dic107C2211yescharacteristic
23#16812035—22no—
24#16812034—22no—
25C22×Q8×C107C2211yescharacteristic
26C2×M16×C107C2211yescharacteristic
27C2×M16×C107C2211yescharacteristic
28#101×C107—22no—
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33C2×Dic214—21no—
34C2×Dic214—21no—
35C2×Dic214—21no—
36C2×Dic214—21no—
37C2×Dic214—21no—
38C2×Dic214D411yescharacteristic
39C2×Dic214D412yes—
40C2×D428D411yescharacteristic
41C2×C22⋊D107—21no—
42C22×Dic107—21no—
43C22×Dic107—21no—
44C2×C4×D107—21no—
45Dic428—24no—
46#13133807—24no—
47C4⋊Dic107D412yes—
48Dic107⋊C4—28no—
49Dic107⋊C4—24no—
50C22⋊Dic107—21no—
51(C2×D107)⋊C4—24no—
52C4×Dic107—21no—
53C4×Dic107—21no—
54C4×Dic107D412yes—
55C2×Q8×C107—21no—
56C2×Q8×C107—21no—
57C2×Q8×C107—21no—
58C2×Q8×C107—21no—
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C22×C4×C107D412yes—
62C22×C4×C107C2311yescharacteristic
63M16×C107—21no—
64M16×C107—21no—
65M16×C107—21no—
66M16×C107D412yes—
67C2×C8×C107—21no—
68C2×C8×C107—21no—
69C23×C107C2×D411yescharacteristic
70C22×D107—21no—
71Q8×C107—41no—
72Q8×C107—41no—
73Q8×C107—41no—
74Q8×C107—41no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107#4111yescharacteristic
84C22⋊D107—41no—
85C22⋊D107—41no—
86C2×Dic107—24no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—41no—
92C2×Dic107—41no—
93D428—21no—
94D428—21no—
95C4×D107—41no—
96C4×D107—41no—
97Dic214—22no—
98Dic214—21no—
99Dic214—22no—
100Dic214—21no—
101Dic214—41no—
102Dic214—41no—
103C8×C107—22no—
104C8×C107—41no—
105C22×C107—22no—
106C22×C107#12011yescharacteristic
107C22×C107#12011yescharacteristic
108C22×C107#12111yescharacteristic
109C2×D107—41no—
110C2×D107—42no—
111C4×C107—42no—
112C4×C107#12011yescharacteristic
113C4×C107#12111yescharacteristic
114C4×C107#12111yescharacteristic
115C4×C107#12711yescharacteristic
116Dic107—24no—
117Dic107—42no—
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⋊Q8—1071no—
130C2×#104—1071no—
131#419—1071no—
132C2×#101D10711yescharacteristic, complemented
133#339—1071no—
134C2×#41—1071no—
135C22×Q8—1071no—
136C22×Q8C2×D10711yescharacteristic
137#137—2142no—
138#137—2142no—
139C2×Q16—2141no—
140C2×SD16—2141no—
141C2×M16C2×D10711yescharacteristic
142C2×M16C2×D10711yescharacteristic
143C4⋊Q8—2142no—
144C4⋊Q8—2142no—
145C4⋊Q8—2141no—
146#124—2141no—
147#123—2142no—
148C22⋊Q8—2142no—
149#117—1071no—
150C2×C4⋊C4—2142no—
151C2×C42—1071no—
152#104—2142no—
153#103—2142no—
154#101—22no—
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—1071no—
166C2×Q8—1072no—
167C2×Q8—2141no—
168C2×Q8—2141no—
169C2×Q8—2141no—
170C2×Q8—2141no—
171C2×Q8—2141no—
172C2×D4—1071no—
173C2×D4—2141no—
174C22×C4—2141no—
175C22×C4—2141no—
176C22×C4—2141no—
177C22×C4D42812yes—
178C22×C4C22×D10711yescharacteristic
179Q16—2144no—
180SD16—2144no—
181M16—21no—
182M16—21no—
183M16—21no—
184M16C22⋊D10712yes—
185C2×C8—21no—
186C2×C8—21no—
187C4⋊C4—1072no—
188C4⋊C4—2148no—
189C4⋊C4—2144no—
190C22⋊C4—2144no—
191C22⋊C4—2141no—
192C42—1072no—
193C42—2141no—
194C42—2141no—
195C23—2141no—
196C23C2×D42811yescharacteristic
197Q8—41no—
198Q8—41no—
199Q8—41no—
200Q8—41no—
201Q8—4281no—
202Q8—4281no—
203Q8—2142no—
204Q8—2141no—
205Q8—2142no—
206Q8—2141no—
207D4—4281no—
208D4—4281no—
209D4—2141no—
210D4—2141no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—4281no—
215C2×C4—4281no—
216C2×C4—4281no—
217C2×C4—4281no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4—2144no—
222C2×C4—2142no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4D4×D10711yescharacteristic
227C2×C4C2×C22⋊D10711yescharacteristic
228C2×C4#1313381811yescharacteristic
229C8—22no—
230C8—41no—
231C22—22no—
232C22—4281no—
233C22—4282no—
234C22#1681207911yescharacteristic
235C22#1681207911yescharacteristic
236C22#1681208311yescharacteristic
237C4—42no—
238C4—4281no—
239C4—4282no—
240C4—4282no—
241C4—2144no—
242C4#1681207711yescharacteristic
243C4#1681208211yescharacteristic
244C4#1681208211yescharacteristic
245C4#1681210511yescharacteristic
246C2—21no—
247C2—21no—
248C2—8561no—
249C2#2029253611yescharacteristic
250C2#2029281212yes—
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#101×C107C2
socleC22×C107#120
prime radicalC2×#13133818C22
hypercenterC2×#101D107
hypocenterC107#2005
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
3. C2×#16812107C2semidirect
5. #20292637C2semidirect
6. #20292388C2semidirect
7. C2×#101×C107C2semidirect
126. C107#2005semidirect
132. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
#202932012
C2×#168121072
C2×#168120352
#202926372
#202923882
C2×#101×C1072
#2005107