Finite Group Databasebeta

← #28348263

Subgroups — #28348263

Order 13696.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293442C212yesmaximal, complemented
2#20293441C211yesmaximal, characteristic, complemented
3#20293005C211yesmaximal, characteristic, complemented
4#20292902C211yesmaximal, characteristic, complemented
5#20292900C212yesmaximal, complemented
6#20292899C211yesmaximal, characteristic, complemented
7#20292810C212yesmaximal, complemented
8#20292808C212yesmaximal, complemented
9#20292744C211yesmaximal, characteristic, complemented
10#20292742C211yesmaximal, characteristic, complemented
11#578×C107C211yesmaximal, characteristic, complemented
12#16812156C2212yes—
13#16812140—22no—
14#16812140C2212yes—
15#16812139—22no—
16#16812139C2212yes—
17#16812139C2212yes—
18#16812138—22no—
19#16812129—22no—
20#16812129C2211yescharacteristic
21#16812128—21no—
22C2×Dic428C2211yescharacteristic
23C2×#13133808—22no—
24C2×#13133808C2212yes—
25Q16×D107—21no—
26#16812116C2212yes—
27#16812115C2211yescharacteristic
28#16812107C2212yes—
29#16812045C2211yescharacteristic
30#16812044C2211yescharacteristic
31#16812036C2211yescharacteristic
32#16812036C2212yes—
33#16812035C2211yescharacteristic
34#16812028C2212yes—
35#16812027C2212yes—
36C8×Dic107C2211yescharacteristic
372−1+4×C107C2212yes—
38#137×C107—22no—
39#137×C107C2212yes—
40#135×C107—21no—
41#135×C107C2211yescharacteristic
42C2×Q16×C107—21no—
43C2×Q16×C107C2211yescharacteristic
44#131×C107C2211yescharacteristic
45#13133818—22no—
46#13133818C2311yescharacteristic
47#13133817—22no—
48#13133817—42no—
49C2×Dic214—22no—
50C2×Dic214C2312yes—
51Q8×D107—22no—
52Dic428—21no—
53Dic428—21no—
54#13133808—22no—
55#13133808—22no—
56#13133808—22no—
57#13133808—22no—
58#13133808—22no—
59#13133808D411yescharacteristic
60#13133806—22no—
61#13133806—22no—
62#13133806D411yescharacteristic
63#13133805—22no—
64#13133805D411yescharacteristic
65#13133803D411yescharacteristic
66#13133802—22no—
67#13133802C2311yescharacteristic
68#13133802C2312yes—
69#13133801—21no—
70C2×C107⋊C8—22no—
71C2×C107⋊C8C2311yescharacteristic
72C8×D107—21no—
73Dic107⋊C4—42no—
74C4×Dic107C2311yescharacteristic
75#41×C107—22no—
76#41×C107—22no—
77#41×C107—42no—
78#41×C107C2312yes—
79#41×C107C2311yescharacteristic
80C2×Q8×C107—22no—
81C2×Q8×C107—22no—
82C2×Q8×C107C2312yes—
83Q16×C107—22no—
84Q16×C107—21no—
85Q16×C107—21no—
86Q16×C107D411yescharacteristic
87SD16×C107—22no—
88SD16×C107D412yes—
89D8×C107D411yescharacteristic
90M16×C107—21no—
91M16×C107C2311yescharacteristic
92C2×C8×C107—21no—
93C2×C8×C107C2311yescharacteristic
94Q8×C107—22no—
95Q8×C107—22no—
96Q8×C107—42no—
97Q8×C107C2×D411yescharacteristic
98Q8×C107C2×D412yes—
99D4×C107—21no—
100D4×C107—22no—
101D4×C107—42no—
102D4×C107C2×D411yescharacteristic
103D4×C107C2×D412yes—
104C2×C4×C107—21no—
105C2×C4×C107—22no—
106C2×C4×C107—22no—
107C2×C4×C107—42no—
108C2×C4×C107—42no—
109C2×C4×C107C2411yescharacteristic
110C22⋊D107—21no—
111C22⋊D107—42no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—42no—
115D428C2×D411yescharacteristic
116C4×D107—21no—
117C4×D107—42no—
118Dic214—22no—
119Dic214—22no—
120Dic214—42no—
121Dic214C2×D411yescharacteristic
122C8×C107—21no—
123C8×C107C2×D411yescharacteristic
124C8×C107C2×D411yescharacteristic
125C107⋊C8—21no—
126C107⋊C8—22no—
127C107⋊C8C2×D411yescharacteristic
128C107⋊C8C2×D411yescharacteristic
129C22×C107—22no—
130C22×C107—21no—
131C22×C107H(5,𝔽2)11yescharacteristic
132C2×D107—21no—
133C4×C107—22no—
134C4×C107—21no—
135C4×C107—42no—
136C4×C107C22×D411yescharacteristic
137C4×C107C22×D411yescharacteristic
138Dic107—22no—
139Dic107—21no—
140Dic107—42no—
141C2×C107—21no—
142C2×C107—42no—
143C2×C107—41no—
144C2×C107D4211yescharacteristic
145D107—41no—
146#3293—1071nomaximal, Sylow
147C107#329311yescharacteristic, complemented, Sylow
148#578—1072no—
149#578—1071no—
150#578D10711yescharacteristic, complemented
151#494—1071no—
152#473—1072no—
153#473—1071no—
154#473—1071no—
155#456—1072no—
156#456—1072no—
157#443—1071no—
158#443—1071no—
1592−1+4—1072no—
1602−1+4C2×D10712yes—
161#137—22no—
162#137—1072no—
163#137—1072no—
164#137—1072no—
165#137—2142no—
166#137—2142no—
167#137—2142no—
168#137C2×D10712yes—
169#135—21no—
170#135—1071no—
171#135—2142no—
172#135—2141no—
173#135C2×D10711yescharacteristic
174C2×Q16—21no—
175C2×Q16—1072no—
176C2×Q16—1071no—
177C2×Q16—2142no—
178C2×Q16—2141no—
179C2×Q16C2×D10711yescharacteristic
180#131—1072no—
181#131—1071no—
182#131C2×D10711yescharacteristic
183C4⋊Q8—1072no—
184#108—1071no—
185#108—1071no—
186#104—1071no—
187#104—1072no—
188#104—1071no—
189#101—1072no—
190#101—1072no—
191C4×C8—1071no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—1071no—
196#41—4282no—
197#41—2142no—
198#41—2142no—
199#41C22×D10711yescharacteristic
200#41C22×D10712yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—1072no—
204C2×Q8—2142no—
205C2×Q8—2142no—
206C2×Q8C22×D10712yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—1071no—
211Q16—2142no—
212Q16—2142no—
213Q16—2142no—
214Q16—2142no—
215Q16—2142no—
216Q16—2141no—
217Q16—2141no—
218Q16C22⋊D10711yescharacteristic
219SD16—22no—
220SD16—1071no—
221SD16—1071no—
222SD16—2142no—
223SD16—2142no—
224SD16—2142no—
225SD16C22⋊D10712yes—
226D8—1071no—
227D8C22⋊D10711yescharacteristic
228M16—21no—
229M16—1072no—
230M16—1071no—
231M16—2142no—
232M16—2141no—
233M16C22×D10711yescharacteristic
234C2×C8—21no—
235C2×C8—1071no—
236C2×C8—2142no—
237C2×C8—2141no—
238C2×C8C22×D10711yescharacteristic
239C4⋊C4—4282no—
240C42—1071no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1071no—
245Q8—4282no—
246Q8—2142no—
247Q8—2142no—
248Q8D4×D10711yescharacteristic
249Q8C2×C22⋊D10712yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—1071no—
254D4—4282no—
255D4—2141no—
256D4D4×D10711yescharacteristic
257D4C2×C22⋊D10712yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—4282no—
264C2×C4—4282no—
265C2×C4—2142no—
266C2×C4—2141no—
267C2×C4—2141no—
268C2×C4C23×D10711yescharacteristic
269C8—21no—
270C8—1071no—
271C8—1071no—
272C8—2142no—
273C8—2141no—
274C8C2×C22⋊D10711yescharacteristic
275C8C2×C22⋊D10711yescharacteristic
276C22—21no—
277C22—22no—
278C22—2141no—
279C22#1681215411yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—4282no—
284C4—2142no—
285C4—2141no—
286C4C2×D4×D10711yescharacteristic
287C4C22×C22⋊D10711yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—4281no—
292C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#578×C107C2
socleC2×C107D42
hypercenter#578D107
hypocenterC107#3293
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20293442C2semidirect
2. #20293441C2semidirect
3. #20293005C2semidirect
4. #20292902C2semidirect
5. #20292900C2semidirect
6. #20292899C2semidirect
7. #20292810C2semidirect
8. #20292808C2semidirect
9. #20292744C2semidirect
10. #20292742C2semidirect
11. #578×C107C2semidirect
147. C107#3293semidirect
150. #578D107semidirect

Maximal subgroups

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

subgroupindex
#202934422
#202934412
#202930052
#202929022
#202929002
#202928992
#202928102
#202928082
#202927442
#202927422
#578×C1072
#3293107