Finite Group Databasebeta

← #28373509

Subgroups — #28373509

Order 13952.

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

normal subgroups only

This group has 564 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×#16812729C211yesmaximal, characteristic, complemented
2C2×#16812724C211yesmaximal, characteristic, complemented
3C22×#13182018C211yesmaximal, characteristic, complemented
4#20295597C211yesmaximal, characteristic, complemented
5#20295590C211yesmaximal, characteristic, complemented
6#20295419C212yesmaximal, complemented
7#20295414C212yesmaximal, complemented
8#20295407C212yesmaximal, complemented
9#20295397C212yesmaximal, complemented
10#20294981C211yesmaximal, characteristic, complemented
11C2×#135×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C22×D436C2211yescharacteristic
14#16812729—24no—
15#16812724—24no—
16C2×#13182019C2212yescomplemented
17C2×#13182018—24no—
18C2×#13182018C2212yes—
19C2×#13182018C2212yes—
20C2×#13182017C2212yescomplemented
21C2×#13182016C2212yes—
22C2×#13182014C2212yes—
23C2×#13182013C2211yescharacteristic
24#16812697C2212yescomplemented
25#16812695C2212yes—
26#16812688C2212yes—
27#16812686C2212yescomplemented
28#16812680—22no—
29#16812678—22no—
30#16812673—22no—
31#16812667—22no—
32#16812658C2211yescharacteristic
33#16812609C2212yescomplemented
34C8×Dic109C2212yescomplemented
35C2×#41×C109C2211yescharacteristic
36C2×#41×C109C2211yescharacteristic
37#135×C109—24no—
38C2×Q16×C109C2211yescharacteristic, complemented
39C2×SD16×C109C2211yescharacteristic, complemented
40C2×SD16×C109C2211yescharacteristic, complemented
41C2×D8×C109C2211yescharacteristic, complemented
42C22×C8×C109C2211yescharacteristic
43C23×D109—21no—
44C2×Dic218—22no—
45C2×Dic218—22no—
46C2×Dic218—21no—
47C2×Dic218C2312yes—
48C2×D436—22no—
49C2×D436—22no—
50C2×D436—21no—
51C2×D436C2312yes—
52C2×C22⋊D109—24no—
53C22×Dic109—21no—
54#13182019—24no—
55#13182018—24no—
56#13182018—24no—
57#13182017—24no—
58#13182016—24no—
59#13182014—24no—
60#13182013D414yescomplemented
61C2×#11724784C2312yes—
62C4⋊Dic109D412yes—
63#13182008—24no—
64C22⋊Dic109—24no—
65C22⋊Dic109D412yes—
66#13182006—24no—
67C4×Dic109C2312yes—
68#41×C109—22no—
69#41×C109—22no—
70#41×C109—22no—
71#41×C109—22no—
72C2×Q8×C109C2311yescharacteristic
73C2×Q8×C109C2311yescharacteristic
74C2×D4×C109—21no—
75C2×D4×C109—21no—
76C2×D4×C109C2311yescharacteristic
77C2×D4×C109C2311yescharacteristic
78C22×C4×C109—21no—
79C22×C4×C109—21no—
80C22×C4×C109C2311yescharacteristic
81Q16×C109—22no—
82SD16×C109—22no—
83SD16×C109—22no—
84D8×C109—22no—
85C2×C8×C109D414yescomplemented
86C2×C8×C109C2311yescharacteristic
87C2×C8×C109C2311yescharacteristic
88C23×C109—21no—
89C23×C109—21no—
90C23×C109C2×D411yescharacteristic
91C22×D109—22no—
92C22×D109—42no—
93C22×D109—82no—
94Q8×C109—21no—
95Q8×C109—21no—
96Q8×C109—21no—
97Q8×C109—21no—
98D4×C109—21no—
99D4×C109—21no—
100D4×C109—21no—
101D4×C109—21no—
102D4×C109—42no—
103D4×C109—42no—
104C2×C4×C109—21no—
105C2×C4×C109—21no—
106C2×C4×C109—42no—
107C2×C4×C109—42no—
108C2×C4×C109C2×D411yescharacteristic
109C2×C4×C109C2×D412yes—
110C2×C4×C109C2×D412yes—
111C2×C4×C109C2411yescharacteristic
112C22⋊D109—84no—
113C2×Dic109—22no—
114C2×Dic109—42no—
115C2×Dic109C2×D414yes—
116D436—22no—
117D436—22no—
118D436—42no—
119Dic218—22no—
120Dic218—22no—
121Dic218—42no—
122C8×C109C2×D412yes—
123C8×C109C2×D412yes—
124#11724784C2×D414yes—
125C22×C109—22no—
126C22×C109—41no—
127C22×C109—41no—
128C22×C109—42no—
129C22×C109—42no—
130C22×C109#12712yes—
131C22×C109C22×D411yescharacteristic
132C2×D109—42no—
133C2×D109—44no—
134C2×D109—84no—
135C4×C109—41no—
136C4×C109—41no—
137C4×C109#12712yes—
138C4×C109C22×D411yescharacteristic
139C4×C109C22×D411yescharacteristic
140Dic109—24no—
141Dic109—42no—
142C2×C109—22no—
143C2×C109—81no—
144C2×C109—81no—
145C2×C109C2×#12711yescharacteristic
146C2×C109#57712yes—
147D109—82no—
148#3154—1091nomaximal, Sylow
149C109#315411yescharacteristic, complemented, Sylow
150C2×#137—1091no—
151C2×AGL(1,ℤ/8)—1091no—
152C2×#135D10911yescharacteristic, complemented
153C22×SD16—1091no—
154#536—1091no—
155#535—1091no—
156#497—1092no—
157#496—1092no—
158#495—1092no—
159#492—1092no—
160#405—1091no—
161C2×#41C2×D10911yescharacteristic
162C2×#41C2×D10911yescharacteristic
163C22×Q8—1091no—
164C22×D4—1091no—
165#137—2184no—
166AGL(1,ℤ/8)—2184no—
167#135—24no—
168C2×Q16—1092no—
169C2×Q16C2×D10911yescharacteristic, complemented
170C2×SD16—1092no—
171C2×SD16—1092no—
172C2×SD16—1092no—
173C2×SD16—1092no—
174C2×SD16—2184no—
175C2×SD16C2×D10911yescharacteristic, complemented
176C2×SD16C2×D10911yescharacteristic, complemented
177C2×D8—1092no—
178C2×D8C2×D10911yescharacteristic, complemented
179C2×M16—1091no—
180C22×C8C2×D10911yescharacteristic
181C4⋊Q8—1092no—
182#127—1092no—
183#124—1092no—
184#124—1092no—
185#123—2182no—
186C22⋊Q8—2182no—
187#121—2182no—
188#120—2182no—
189#117—1091no—
190#97—1092no—
191C4×C8—1092no—
192C24—2181no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—1092no—
198C2×Q8—2182no—
199C2×Q8—2182no—
200C2×Q8—2181no—
201C2×Q8C22×D10911yescharacteristic
202C2×Q8C22×D10911yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—1092no—
206C2×D4—2184no—
207C2×D4—2181no—
208C2×D4—2182no—
209C2×D4—2182no—
210C2×D4C22×D10911yescharacteristic
211C2×D4C22×D10911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2181no—
215C22×C4C22×D10911yescharacteristic
216Q16—22no—
217Q16—2184no—
218SD16—22no—
219SD16—22no—
220SD16—2184no—
221SD16—2184no—
222SD16—2184no—
223SD16—2184no—
224D8—22no—
225D8—2184no—
226M16—1094no—
227C2×C8—1092no—
228C2×C8C22⋊D10914yescomplemented
229C2×C8C22×D10911yescharacteristic
230C2×C8C22×D10911yescharacteristic
231C4⋊C4—1092no—
232C4⋊C4—2184no—
233C22⋊C4—1092no—
234C22⋊C4—2184no—
235C22⋊C4—2184no—
236C42—1092no—
237C23—21no—
238C23—21no—
239C23—8722no—
240C23—4362no—
241C23—2182no—
242C23D4×D10911yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—4362no—
248Q8—2182no—
249Q8—2182no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—8724no—
257D4—4362no—
258D4—2182no—
259D4—2182no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—1094no—
265C2×C4—4362no—
266C2×C4—2182no—
267C2×C4D4×D10911yescharacteristic
268C2×C4D4×D10912yes—
269C2×C4C2×C22⋊D10912yes—
270C2×C4C23×D10911yescharacteristic
271C8—1094no—
272C8C2×C22⋊D10912yes—
273C8C2×C22⋊D10912yes—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—8724no—
280C22—4362no—
281C22—4364no—
282C22#1681269512yes—
283C22C2×D4×D10911yescharacteristic
284C4—41no—
285C4—41no—
286C4—4362no—
287C4—2184no—
288C4#1681269512yes—
289C4C2×D4×D10911yescharacteristic
290C4C22×C22⋊D10911yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—8722no—
295C2C2×#1681269511yescharacteristic
296C2#2029583312yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#135×C109C2
socleC22×C109C22×D4
hypercenterC2×#135D109
hypocenterC109#3154
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16812729C2semidirect
2. C2×#16812724C2semidirect
3. C22×#13182018C2semidirect
4. #20295597C2semidirect
5. #20295590C2semidirect
6. #20295419C2semidirect
7. #20295414C2semidirect
8. #20295407C2semidirect
9. #20295397C2semidirect
10. #20294981C2semidirect
11. C2×#135×C109C2semidirect
16. C2×#13182019C22semidirect
20. C2×#13182017C22semidirect
24. #16812697C22semidirect
27. #16812686C22semidirect
33. #16812609C22semidirect
34. C8×Dic109C22semidirect
38. C2×Q16×C109C22semidirect
39. C2×SD16×C109C22semidirect
40. C2×SD16×C109C22semidirect
41. C2×D8×C109C22semidirect
60. #13182013D4semidirect
85. C2×C8×C109D4semidirect
149. C109#3154semidirect
152. C2×#135D109semidirect
169. C2×Q16C2×D109semidirect
175. C2×SD16C2×D109semidirect
176. C2×SD16C2×D109semidirect
178. C2×D8C2×D109semidirect
228. C2×C8C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127292
C2×#168127242
C22×#131820182
#202955972
#202955902
#202954192
#202954142
#202954072
#202953972
#202949812
C2×#135×C1092
#3154109