Finite Group Databasebeta

← #28345775

Subgroups — #28345775

Order 13696.

712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812157C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293205C211yesmaximal, characteristic, complemented
4#20292788C212yesmaximal, complemented
5#20292781C212yesmaximal, complemented
6#20292771C212yesmaximal, complemented
7#20292764C212yesmaximal, complemented
8#20292629C211yesmaximal, characteristic, complemented
9C2×#16812033C211yesmaximal, characteristic, complemented
10#20292595C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12#16812157—24no—
13#16812157—24no—
14C2×#13133819—21no—
15C2×#13133819C2212yes—
16C2×#13133818—22no—
17C2×#13133818C2211yescharacteristic
18C2×#13133818C2211yescharacteristic
19C2×Q8×D107—21no—
20C2×Q8×D107C2212yes—
21#16812129—24no—
22C2×#13133808C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133803C2211yescharacteristic
26C22×C107⋊C8C2211yescharacteristic
27#16812088—21no—
28#16812085C2212yes—
29#16812083—21no—
30#16812081—22no—
31#16812080C2212yes—
32C2×C4⋊Dic107C2211yescharacteristic
33#16812034C2212yes—
34#16812033C2212yes—
35#16812033C2212yes—
36#16812031C2212yes—
37(C2×D107)⋊C8C2212yes—
38(C2×D107)⋊C8C2212yes—
39C2×#41×C107C2211yescharacteristic
40C22×Q8×C107C2211yescharacteristic
41#137×C107—24no—
42C2×Q16×C107C2212yes—
43C2×SD16×C107C2212yes—
44C2×M16×C107C2211yescharacteristic
45#13133819—28no—
46#13133819—24no—
47#13133819—24no—
48#13133818—216no—
49#13133818—22no—
50#13133818—22no—
51#13133818—22no—
52#13133818D414yescomplemented
53C2×Dic214—22no—
54C2×Dic214D411yescharacteristic
55C2×Dic214C2311yescharacteristic
56Q8×D107—28no—
57Q8×D107—24no—
58Q8×D107—24no—
59C2×D428—22no—
60C2×D428D411yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107D412yes—
65C22×Dic107—21no—
66C2×C4×D107—22no—
67C2×C4×D107—21no—
68C2×C4×D107—22no—
69C2×C4×D107C2312yes—
70#13133808—22no—
71#13133806—22no—
72#13133805—22no—
73#13133803—22no—
74C2×C107⋊C8—22no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107—21no—
78C4⋊Dic107C2312yes—
79Dic107⋊C4—22no—
80C22⋊Dic107—22no—
81(C2×D107)⋊C4—22no—
82#41×C107—22no—
83#41×C107—22no—
84C2×Q8×C107—21no—
85C2×Q8×C107—22no—
86C2×Q8×C107D414yescomplemented
87C2×Q8×C107C2311yescharacteristic
88C2×Q8×C107C2312yes—
89C2×D4×C107—21no—
90C2×D4×C107C2311yescharacteristic
91C22×C4×C107—21no—
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94Q16×C107—24no—
95SD16×C107—24no—
96M16×C107—22no—
97C2×C8×C107C2312yes—
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—21no—
101C22×D107C2×D412yes—
102Q8×C107—24no—
103Q8×C107—22no—
104Q8×C107—21no—
105Q8×C107—21no—
106Q8×C107C2×D414yes—
107D4×C107—21no—
108D4×C107—21no—
109D4×C107—42no—
110C2×C4×C107—24no—
111C2×C4×C107—22no—
112C2×C4×C107—21no—
113C2×C4×C107—42no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D412yes—
116C2×C4×C107C2×D412yes—
117C2×C4×C107C2411yescharacteristic
118C22⋊D107—24no—
119C22⋊D107—24no—
120C22⋊D107—82no—
121C2×Dic107—21no—
122C2×Dic107—22no—
123C2×Dic107—81no—
124C2×Dic107C2×D412yes—
125D428—28no—
126D428—22no—
127D428—21no—
128D428C2×D412yes—
129C4×D107—28no—
130C4×D107—28no—
131C4×D107—24no—
132C4×D107—22no—
133C4×D107—22no—
134Dic214—28no—
135Dic214—22no—
136Dic214—21no—
137Dic214C2×D412yes—
138C8×C107—22no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—21no—
142C22×C107—21no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107#12012yes—
147C22×C107C22×D411yescharacteristic
148C2×D107—22no—
149C2×D107—22no—
150C2×D107—22no—
151C2×D107—22no—
152C2×D107—41no—
153C2×D107—41no—
154C4×C107—24no—
155C4×C107—41no—
156C4×C107#12012yes—
157C4×C107C22×D411yescharacteristic
158C4×C107C22×D411yescharacteristic
159Dic107—22no—
160Dic107—22no—
161Dic107—42no—
162C2×C107—22no—
163C2×C107—81no—
164C2×C107C2×#12011yescharacteristic
165C2×C107#57711yescharacteristic
166C2×C107#57811yescharacteristic
167D107—42no—
168D107—42no—
169#3013—1071nomaximal, Sylow
170C107#301311yescharacteristic, complemented, Sylow
171C2×2−1+4—1071no—
172C2×#137D10711yescharacteristic, complemented
173C2×#135—1071no—
174#537—1071no—
175#452—1072no—
176#451—1072no—
177#449—1072no—
178#448—1072no—
179#417—1071no—
180C2×#103—1071no—
181#408—1071no—
1822−1+4—2144no—
1832−1+4—2144no—
184C2×#41—1071no—
185C2×#41—1072no—
186C2×#41—1071no—
187C2×#41—2142no—
188C2×#41—2141no—
189C2×#41C2×D10711yescharacteristic
190C22×Q8—1072no—
191C22×Q8—2141no—
192C22×Q8C2×D10711yescharacteristic
193#137—24no—
194#135—2144no—
195C2×Q16—1071no—
196C2×Q16C2×D10712yes—
197C2×SD16—1071no—
198C2×SD16—1071no—
199C2×SD16C2×D10712yes—
200C2×D8—1071no—
201C2×M16C2×D10711yescharacteristic
202C22×C8—1071no—
203C22⋊Q8—1072no—
204C22⋊Q8—2141no—
205#121—1072no—
206#121—2141no—
207#121—2142no—
208C2×C4⋊C4—1071no—
209#103—1072no—
210#103—1072no—
211#103—1072no—
212#102—1072no—
213C22⋊C8—1072no—
214C22⋊C8—1072no—
215#41—22no—
216#41—22no—
217#41—1074no—
218#41—21416no—
219#41—2144no—
220#41—2148no—
221#41—2144no—
222#41—2142no—
223#41—2142no—
224#41—2142no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8—1071no—
228C2×Q8—1071no—
229C2×Q8—2142no—
230C2×Q8—2144no—
231C2×Q8—2148no—
232C2×Q8—2144no—
233C2×Q8C22⋊D10714yescomplemented
234C2×Q8C22×D10711yescharacteristic
235C2×Q8C22×D10712yes—
236C2×D4—21no—
237C2×D4—1072no—
238C2×D4—1071no—
239C2×D4—1071no—
240C2×D4—2142no—
241C2×D4—2142no—
242C2×D4—2141no—
243C2×D4C22×D10711yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—1072no—
247C22×C4—2142no—
248C22×C4—2142no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252Q16—24no—
253Q16—2142no—
254SD16—24no—
255SD16—2142no—
256SD16—2142no—
257D8—2142no—
258M16—22no—
259C2×C8—1071no—
260C2×C8—1071no—
261C2×C8—2142no—
262C2×C8C22×D10712yes—
263C4⋊C4—1072no—
264C4⋊C4—2142no—
265C4⋊C4—2141no—
266C22⋊C4—2142no—
267C22⋊C4—2142no—
268C23—21no—
269C23—1072no—
270C23—2141no—
271C23D4×D10711yescharacteristic
272Q8—24no—
273Q8—22no—
274Q8—21no—
275Q8—21no—
276Q8—1072no—
277Q8—2148no—
278Q8—2142no—
279Q8—2141no—
280Q8C2×C22⋊D10714yes—
281D4—21no—
282D4—21no—
283D4—42no—
284D4—8562no—
285D4—1072no—
286D4—2148no—
287D4—2144no—
288D4—2144no—
289D4—2142no—
290D4—2141no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—8561no—
296C2×C4—1072no—
297C2×C4—2148no—
298C2×C4—2148no—
299C2×C4—2142no—
300C2×C4—2141no—
301C2×C4—2144no—
302C2×C4—2142no—
303C2×C4—2142no—
304C2×C4D4×D10711yescharacteristic
305C2×C4D4×D10712yes—
306C2×C4C2×C22⋊D10712yes—
307C2×C4C23×D10711yescharacteristic
308C8—22no—
309C8—2141no—
310C8—2141no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—4281no—
317C22—4281no—
318C22—2142no—
319C22—2142no—
320C22—2142no—
321C22—2142no—
322C22#1681207712yes—
323C22C2×D4×D10711yescharacteristic
324C4—24no—
325C4—41no—
326C4—4282no—
327C4—2142no—
328C4—2142no—
329C4#1681207712yes—
330C4C2×D4×D10711yescharacteristic
331C4C22×C22⋊D10711yescharacteristic
332C2—22no—
333C2—81no—
334C2—4282no—
335C2—4282no—
336C2C2×#1681207711yescharacteristic
337C2#2029343611yescharacteristic
338C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
hypercenterC2×#137D107
hypocenterC107#3013
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16812157C2semidirect
2. C2×#16812129C2semidirect
3. #20293205C2semidirect
4. #20292788C2semidirect
5. #20292781C2semidirect
6. #20292771C2semidirect
7. #20292764C2semidirect
8. #20292629C2semidirect
9. C2×#16812033C2semidirect
10. #20292595C2semidirect
11. C2×#137×C107C2semidirect
52. #13133818D4semidirect
86. C2×Q8×C107D4semidirect
170. C107#3013semidirect
172. C2×#137D107semidirect
233. C2×Q8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121572
C2×#168121292
#202932052
#202927882
#202927812
#202927712
#202927642
#202926292
C2×#168120332
#202925952
C2×#137×C1072
#3013107