Finite Group Databasebeta

← #28346177

Subgroups — #28346177

Order 13696.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#20293425C212yesmaximal, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20292899C212yesmaximal, complemented
4#20292894C212yesmaximal, complemented
5#20292887C214yesmaximal, complemented
6C2×#16812045C211yesmaximal, characteristic, complemented
7#20292611C212yesmaximal, complemented
8C2×#135×C107C211yesmaximal, characteristic, complemented
9C2×#13133818C2211yescharacteristic
10#16812140C2214yes—
11#16812139—22no—
12#16812135C2214yes—
13#16812134—22no—
14#16812129—24no—
15#16812115—21no—
16#16812115—21no—
17#16812115C2212yes—
18#16812115C2212yes—
19C2×#13133802C2211yescharacteristic
20C2×#13133802C2212yes—
21C2×#13133801—21no—
22C2×C8×D107—21no—
23#16812045C2212yescomplemented
24#16812045C2212yescomplemented
25#16812028C2214yescomplemented
26#16812026C2214yescomplemented
27C2×#41×C107C2212yes—
28#135×C107—21no—
29#135×C107—21no—
30#135×C107—21no—
31#135×C107—21no—
32C2×Q16×C107C2211yescharacteristic, complemented
33C2×SD16×C107C2212yescomplemented
34C2×D8×C107C2211yescharacteristic, complemented
35C22×C8×C107C2211yescharacteristic
36#13133818—21no—
37#13133818—21no—
38#13133818D412yes—
39#13133818C2312yes—
40C2×Dic214D411yescharacteristic
41C2×D428D411yescharacteristic
42C2×C22⋊D107—21no—
43C2×C4×D107—21no—
44#13133808—24no—
45#13133806—24no—
46#13133805—24no—
47#13133803—24no—
48#13133802—21no—
49#13133802—22no—
50#13133802C2314yes—
51#13133802C2312yes—
52#13133801—22no—
53#13133801—22no—
54C2×C107⋊C8—21no—
55C2×C107⋊C8—22no—
56C8×D107—22no—
57C8×D107—22no—
58#41×C107—22no—
59#41×C107—22no—
60#41×C107—22no—
61#41×C107—22no—
62C2×Q8×C107C2312yes—
63C2×D4×C107—22no—
64C2×D4×C107C2312yes—
65C22×C4×C107—22no—
66C22×C4×C107C2311yescharacteristic
67Q16×C107—21no—
68Q16×C107—21no—
69SD16×C107—22no—
70SD16×C107—22no—
71D8×C107—21no—
72D8×C107—21no—
73C2×C8×C107D411yescharacteristic
74C2×C8×C107D411yescharacteristic
75C2×C8×C107D411yescharacteristic
76C2×C8×C107D411yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C23×C107—22no—
80C23×C107#4111yescharacteristic
81C22×D107—21no—
82Q8×C107—22no—
83Q8×C107—22no—
84D4×C107—22no—
85D4×C107—22no—
86D4×C107—42no—
87D4×C107—42no—
88C2×C4×C107—22no—
89C2×C4×C107—42no—
90C2×C4×C107—42no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107#4111yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—22no—
98C22⋊D107—22no—
99C2×Dic107—21no—
100D428—21no—
101D428C2×D412yes—
102C4×D107—22no—
103C4×D107—22no—
104Dic214—21no—
105Dic214C2×D412yes—
106C8×C107C2×D411yescharacteristic
107C8×C107C2×D411yescharacteristic
108C8×C107C2×D411yescharacteristic
109C8×C107C2×D411yescharacteristic
110C107⋊C8—22no—
111C107⋊C8—24no—
112C22×C107—41no—
113C22×C107—42no—
114C22×C107—82no—
115C22×C107#12111yescharacteristic
116C22×C107#12111yescharacteristic
117C22×C107C2×#4111yescharacteristic
118C2×D107—22no—
119C2×D107—42no—
120C4×C107—42no—
121C4×C107#12111yescharacteristic
122C4×C107#12111yescharacteristic
123C4×C107C22×D411yescharacteristic
124C4×C107C22×D411yescharacteristic
125Dic107—22no—
126C2×C107—21no—
127C2×C107—21no—
128C2×C107—21no—
129C2×C107—82no—
130C2×C107C2×#12111yescharacteristic
131D107—42no—
132#3067—1071nomaximal, Sylow
133C107#306711yescharacteristic, complemented, Sylow
134#575—1072no—
135C2×#135D10711yescharacteristic, complemented
136C2×#131—1071no—
137#473—1072no—
138#472—1072no—
139#471—1074no—
140C2×#108—1071no—
141#413—1072no—
142C2×#41—1071no—
143C2×#41C2×D10712yes—
144#137—1074no—
145#137—2142no—
146AGL(1,ℤ/8)—1074no—
147AGL(1,ℤ/8)—2142no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—2144no—
153C2×Q16C2×D10711yescharacteristic, complemented
154C2×SD16C2×D10712yescomplemented
155C2×D8C2×D10711yescharacteristic, complemented
156#131—1072no—
157#131—1072no—
158#131—2141no—
159#131—2141no—
160C2×M16—1071no—
161C2×M16—1072no—
162C2×M16—2141no—
163C22×C8—2141no—
164C22×C8C2×D10711yescharacteristic
165#108—1072no—
166#108—1072no—
167#101—1074no—
168#100—1074no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—1072no—
174#41—1072no—
175#41—2141no—
176#41—2141no—
177C2×Q8—1071no—
178C2×Q8C22×D10712yes—
179C2×D4—22no—
180C2×D4—1071no—
181C2×D4—2141no—
182C2×D4C22×D10712yes—
183C22×C4—22no—
184C22×C4—2141no—
185C22×C4C22×D10711yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—2144no—
189SD16—22no—
190SD16—22no—
191SD16—2144no—
192SD16—2144no—
193D8—21no—
194D8—21no—
195D8—2144no—
196M16—1072no—
197M16—1074no—
198M16—2142no—
199M16—2141no—
200M16—2142no—
201M16—2142no—
202C2×C8—2142no—
203C2×C8—2141no—
204C2×C8—2142no—
205C2×C8—2142no—
206C2×C8C22⋊D10711yescharacteristic
207C2×C8C22⋊D10711yescharacteristic
208C2×C8C22⋊D10711yescharacteristic
209C2×C8C22⋊D10711yescharacteristic
210C2×C8C22×D10711yescharacteristic
211C2×C8C22×D10711yescharacteristic
212C23—22no—
213C23—2141no—
214C23#1313381711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—1072no—
218Q8—2141no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—1072no—
224D4—2142no—
225D4—2142no—
226D4—2141no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—2141no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4D4×D10711yescharacteristic
234C2×C4D4×D10711yescharacteristic
235C2×C4C2×C22⋊D10711yescharacteristic
236C2×C4C2×C22⋊D10711yescharacteristic
237C2×C4#1313381711yescharacteristic
238C2×C4C23×D10711yescharacteristic
239C8—2144no—
240C8—2142no—
241C8C2×C22⋊D10711yescharacteristic
242C8C2×C22⋊D10711yescharacteristic
243C8C2×C22⋊D10711yescharacteristic
244C8C2×C22⋊D10711yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—4282no—
249C22—2142no—
250C22#1681208011yescharacteristic
251C22#1681208011yescharacteristic
252C22C2×#1313381711yescharacteristic
253C4—42no—
254C4—2142no—
255C4#1681208011yescharacteristic
256C4#1681208011yescharacteristic
257C4C2×D4×D10711yescharacteristic
258C4C22×C22⋊D10711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—4282no—
264C2C2×#1681208011yescharacteristic
subgroupG/H
centerC4#16812080
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#135×C107C2
socleC2×C107C2×#121
hypercenterC2×#135D107
hypocenterC107#3067
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. C2×#16812115C2semidirect
3. #20292899C2semidirect
4. #20292894C2semidirect
5. #20292887C2semidirect
6. C2×#16812045C2semidirect
7. #20292611C2semidirect
8. C2×#135×C107C2semidirect
23. #16812045C22semidirect
24. #16812045C22semidirect
25. #16812028C22semidirect
26. #16812026C22semidirect
32. C2×Q16×C107C22semidirect
33. C2×SD16×C107C22semidirect
34. C2×D8×C107C22semidirect
133. C107#3067semidirect
135. C2×#135D107semidirect
153. C2×Q16C2×D107semidirect
154. C2×SD16C2×D107semidirect
155. C2×D8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
C2×#168121152
#202928992
#202928942
#202928872
C2×#168120452
#202926112
C2×#135×C1072
#3067107