Finite Group Databasebeta

← #28373487

Subgroups — #28373487

Order 13952.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812732C211yesmaximal, characteristic, complemented
2C2×#16812730C211yesmaximal, characteristic, complemented
3C2×#16812729C211yesmaximal, characteristic, complemented
4#20295572C211yesmaximal, characteristic
5C2×#16812697C211yesmaximal, characteristic, complemented
6#20295419C212yesmaximal, complemented
7#20295417C212yesmaximal, complemented
8#20295404C212yesmaximal, complemented
9#20295397C212yesmaximal, complemented
10M16×Dic109C211yesmaximal, characteristic, complemented
11C2×#137×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×Dic218C2211yescharacteristic
14#16812732—24no—
15#16812730—24no—
16#16812729—24no—
17C2×Dic436C2212yes—
18C2×#13182019C2212yes—
19C2×#13182019C2212yes—
20C2×#13182018C2212yes—
21C2×#13182017C2212yes—
22C2×#13182016C2212yes—
23C2×#13182013C2211yescharacteristic
24#16812697—22no—
25#16812697C2212yes—
26#16812697C2212yes—
27#16812697C2211yescharacteristic
28#16812695C2211yescharacteristic
29#16812688C2211yescharacteristic
30#16812686C2211yescharacteristic
31#16812671—22no—
32C4×C22⋊D109—22no—
33C2×#13182008—22no—
34C2×C4×Dic109C2211yescharacteristic
35#16812609C2212yes—
36C8×Dic109C2212yes—
37C2×#41×C109C2211yescharacteristic
38C22×Q8×C109C2211yescharacteristic
39#137×C109—24no—
40C2×Q16×C109C2212yes—
41C2×SD16×C109C2212yes—
42C2×M16×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218—22no—
46C2×Dic218—22no—
47C2×Dic218—21no—
48C2×Dic218C2311yescharacteristic
49C2×Dic218C2312yes—
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—21no—
52C2×C22⋊D109—22no—
53C22×Dic109—21no—
54C22×Dic109D412yes—
55C2×C4×D109—21no—
56Dic436—24no—
57#13182019—24no—
58#13182019—24no—
59#13182018—24no—
60#13182017—24no—
61#13182016—24no—
62#13182013D414yescomplemented
63C2×#11724784C2312yes—
64#13182008—24no—
65#13182008—24no—
66#13182008—22no—
67C22⋊Dic109—22no—
68#13182006—22no—
69C4×Dic109D412yes—
70C4×Dic109C2311yescharacteristic
71C4×Dic109C2311yescharacteristic
72#41×C109—22no—
73#41×C109—22no—
74C2×Q8×C109—21no—
75C2×Q8×C109—22no—
76C2×Q8×C109—22no—
77C2×Q8×C109C2312yes—
78C2×Q8×C109C2311yescharacteristic
79C2×D4×C109—21no—
80C2×D4×C109C2311yescharacteristic
81C22×C4×C109—21no—
82C22×C4×C109—21no—
83C22×C4×C109C2311yescharacteristic
84Q16×C109—24no—
85SD16×C109—24no—
86M16×C109D414yescomplemented
87C2×C8×C109C2312yes—
88C23×C109—21no—
89C23×C109C2×D411yescharacteristic
90C22×D109—21no—
91Q8×C109—22no—
92Q8×C109—21no—
93Q8×C109—22no—
94Q8×C109—21no—
95Q8×C109—42no—
96D4×C109—21no—
97D4×C109—21no—
98D4×C109—42no—
99C2×C4×C109—21no—
100C2×C4×C109—22no—
101C2×C4×C109—42no—
102C2×C4×C109—42no—
103C2×C4×C109C2×D411yescharacteristic
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2411yescharacteristic
107C22⋊D109—42no—
108C22⋊D109—44no—
109C2×Dic109—24no—
110C2×Dic109—21no—
111C2×Dic109—22no—
112C2×Dic109—42no—
113C2×Dic109C2×D412yes—
114C2×Dic109C2×D412yes—
115D436—21no—
116D436—21no—
117C4×D109—42no—
118Dic218—22no—
119Dic218—22no—
120Dic218—21no—
121Dic218—21no—
122Dic218—42no—
123C8×C109C2×D414yes—
124#11724784C2×D414yes—
125C22×C109—22no—
126C22×C109—41no—
127C22×C109—42no—
128C22×C109#12712yes—
129C22×C109C22×D411yescharacteristic
130C2×D109—41no—
131C2×D109—42no—
132C4×C109—41no—
133C4×C109—42no—
134C4×C109#12712yes—
135C4×C109C22×D411yescharacteristic
136C4×C109C22×D411yescharacteristic
137Dic109—22no—
138Dic109—42no—
139Dic109—41no—
140Dic109#13714yescomplemented
141C2×C109—22no—
142C2×C109—81no—
143C2×C109C2×#12711yescharacteristic
144C2×C109C2×#13712yes—
145D109—81no—
146#3152—1091nomaximal, Sylow
147C109#315211yescharacteristic, complemented, Sylow
148C2×#137—1091no—
149C2×#137—1091no—
150C2×#137—1091no—
151C2×#137D10911yescharacteristic, complemented
152#532—1091no—
153C2×C4⋊Q8—1091no—
154#497—1092no—
155#497—1092no—
156#494—1092no—
157#492—1092no—
158C4×M16—1091no—
159C2×#41—1091no—
160C2×#41C2×D10911yescharacteristic
161C22×Q8—1091no—
162C22×Q8C2×D10911yescharacteristic
163#137—24no—
164#137—2184no—
165#137—2184no—
166#137—2184no—
167C2×Q16—1092no—
168C2×Q16—1092no—
169C2×Q16—1092no—
170C2×Q16C2×D10912yes—
171C2×SD16—1092no—
172C2×SD16—1092no—
173C2×SD16—1092no—
174C2×SD16C2×D10912yes—
175C2×M16—1091no—
176C2×M16C2×D10911yescharacteristic
177C4⋊Q8—1092no—
178C4⋊Q8—1092no—
179C4⋊Q8—1091no—
180C4⋊Q8—2182no—
181#127—1091no—
182#124—1091no—
183#124—1091no—
184#121—2182no—
185C4×D4—2182no—
186C2×C4⋊C4—2182no—
187C2×C42—1091no—
188#97—1092no—
189C4×C8—1092no—
190#41—22no—
191#41—22no—
192#41—2182no—
193#41—2182no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1091no—
198C2×Q8—1092no—
199C2×Q8—2182no—
200C2×Q8—2182no—
201C2×Q8—2181no—
202C2×Q8C22×D10911yescharacteristic
203C2×Q8C22×D10912yes—
204C2×D4—21no—
205C2×D4—1091no—
206C2×D4—2182no—
207C2×D4—2181no—
208C2×D4C22×D10911yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1092no—
212C22×C4—2181no—
213C22×C4—2181no—
214C22×C4C22×D10911yescharacteristic
215Q16—24no—
216Q16—2184no—
217Q16—2184no—
218Q16—2184no—
219SD16—24no—
220SD16—2184no—
221SD16—2184no—
222SD16—2184no—
223M16—1094no—
224M16C22⋊D10914yescomplemented
225C2×C8—1092no—
226C2×C8C22×D10912yes—
227C4⋊C4—2184no—
228C4⋊C4—2184no—
229C4⋊C4—2182no—
230C22⋊C4—2182no—
231C22⋊C4—2182no—
232C42—1092no—
233C42—1091no—
234C42—1091no—
235C23—21no—
236C23—2181no—
237C23D4×D10911yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—4362no—
244Q8—2182no—
245Q8—2181no—
246Q8—2182no—
247Q8—2181no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—4364no—
252D4—4362no—
253D4—2181no—
254D4—2181no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1092no—
260C2×C4—1092no—
261C2×C4—4362no—
262C2×C4—4362no—
263C2×C4—2184no—
264C2×C4—2181no—
265C2×C4—2182no—
266C2×C4D4×D10911yescharacteristic
267C2×C4D4×D10912yes—
268C2×C4C2×C22⋊D10912yes—
269C2×C4C23×D10911yescharacteristic
270C8—1094no—
271C8C2×C22⋊D10914yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—4362no—
276C22—4361no—
277C22#1681269512yes—
278C22C2×D4×D10911yescharacteristic
279C4—41no—
280C4—42no—
281C4—1094no—
282C4—4361no—
283C4—4362no—
284C4—2182no—
285C4#1681269512yes—
286C4C2×D4×D10911yescharacteristic
287C4C22×C22⋊D10911yescharacteristic
288C2—22no—
289C2—81no—
290C2—8721no—
291C2C2×#1681269511yescharacteristic
292C2#137×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#137×C109C2
socleC22×C109C22×D4
prime radical#20295572C2
hypercenterC2×#137D109
hypocenterC109#3152
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×#16812730C2semidirect
3. C2×#16812729C2semidirect
5. C2×#16812697C2semidirect
6. #20295419C2semidirect
7. #20295417C2semidirect
8. #20295404C2semidirect
9. #20295397C2semidirect
10. M16×Dic109C2semidirect
11. C2×#137×C109C2semidirect
62. #13182013D4semidirect
86. M16×C109D4semidirect
140. Dic109#137semidirect
147. C109#3152semidirect
151. C2×#137D109semidirect
224. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×#168127302
C2×#168127292
#202955722
C2×#168126972
#202954192
#202954172
#202954042
#202953972
M16×Dic1092
C2×#137×C1092
#3152109