Finite Group Databasebeta

← #30676416

Subgroups — #30676416

Order 16768.

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×#16862279C211yesmaximal, characteristic, complemented
2C2×#16862277C211yesmaximal, characteristic, complemented
3C2×#16862276C211yesmaximal, characteristic, complemented
4#22553858C211yesmaximal, characteristic
5C2×#16862244C211yesmaximal, characteristic, complemented
6#22553705C212yesmaximal, complemented
7#22553703C212yesmaximal, complemented
8#22553690C212yesmaximal, complemented
9#22553683C212yesmaximal, complemented
10M16×Dic131C211yesmaximal, characteristic, complemented
11C2×#137×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×Dic262C2211yescharacteristic
14#16862279—24no—
15#16862277—24no—
16#16862276—24no—
17C2×Dic524C2212yes—
18C2×#14523503C2212yes—
19C2×#14523503C2212yes—
20C2×#14523502C2212yes—
21C2×#14523501C2212yes—
22C2×#14523500C2212yes—
23C2×#14523497C2211yescharacteristic
24#16862244—22no—
25#16862244C2211yescharacteristic
26#16862244C2212yes—
27#16862244C2212yes—
28#16862242C2211yescharacteristic
29#16862235C2211yescharacteristic
30#16862233C2211yescharacteristic
31#16862218—22no—
32C4×C22⋊D131—22no—
33C2×Dic131⋊C4—22no—
34C2×C4×Dic131C2211yescharacteristic
35#16862156C2212yes—
36C8×Dic131C2212yes—
37C2×#41×C131C2211yescharacteristic
38C22×Q8×C131C2211yescharacteristic
39#137×C131—24no—
40C2×Q16×C131C2212yes—
41C2×SD16×C131C2212yes—
42C2×M16×C131C2211yescharacteristic
43#14523513—22no—
44#14523513—22no—
45C2×Dic262—21no—
46C2×Dic262—22no—
47C2×Dic262—22no—
48C2×Dic262C2311yescharacteristic
49C2×Dic262C2312yes—
50C2×D524C2311yescharacteristic
51C2×C22⋊D131—22no—
52C2×C22⋊D131—21no—
53C22×Dic131—21no—
54C22×Dic131D412yes—
55C2×C4×D131—21no—
56Dic524—24no—
57#14523503—24no—
58#14523503—24no—
59#14523502—24no—
60#14523501—24no—
61#14523500—24no—
62#14523497D414yescomplemented
63C2×C131⋊C8C2312yes—
64Dic131⋊C4—22no—
65Dic131⋊C4—24no—
66Dic131⋊C4—24no—
67C22⋊Dic131—22no—
68(C2×D131)⋊C4—22no—
69C4×Dic131D412yes—
70C4×Dic131C2311yescharacteristic
71C4×Dic131C2311yescharacteristic
72#41×C131—22no—
73#41×C131—22no—
74C2×Q8×C131—21no—
75C2×Q8×C131—22no—
76C2×Q8×C131—22no—
77C2×Q8×C131C2311yescharacteristic
78C2×Q8×C131C2312yes—
79C2×D4×C131—21no—
80C2×D4×C131C2311yescharacteristic
81C22×C4×C131—21no—
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84Q16×C131—24no—
85SD16×C131—24no—
86M16×C131D414yescomplemented
87C2×C8×C131C2312yes—
88C22×D131—21no—
89Dic262—22no—
90Dic262—22no—
91Dic262—21no—
92Dic262—21no—
93Dic262—42no—
94D524—21no—
95D524—21no—
96C22⋊D131—42no—
97C22⋊D131—44no—
98C2×Dic131—24no—
99C2×Dic131—21no—
100C2×Dic131—22no—
101C2×Dic131—42no—
102C2×Dic131C2×D412yes—
103C2×Dic131C2×D412yes—
104C4×D131—42no—
105C131⋊C8C2×D414yes—
106C23×C131—21no—
107C23×C131C2×D411yescharacteristic
108Q8×C131—22no—
109Q8×C131—21no—
110Q8×C131—22no—
111Q8×C131—21no—
112Q8×C131—42no—
113D4×C131—21no—
114D4×C131—21no—
115D4×C131—42no—
116C2×C4×C131—21no—
117C2×C4×C131—22no—
118C2×C4×C131—42no—
119C2×C4×C131—42no—
120C2×C4×C131C2×D411yescharacteristic
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2411yescharacteristic
124C8×C131C2×D414yes—
125C22×C131—22no—
126C22×C131—41no—
127C22×C131—42no—
128C22×C131#12712yes—
129C22×C131C22×D411yescharacteristic
130C2×D131—41no—
131C2×D131—42no—
132C4×C131—41no—
133C4×C131—42no—
134C4×C131#12712yes—
135C4×C131C22×D411yescharacteristic
136C4×C131C22×D411yescharacteristic
137Dic131—22no—
138Dic131—41no—
139Dic131—42no—
140Dic131#13714yescomplemented
141C2×C131—22no—
142C2×C131—81no—
143C2×C131C2×#12711yescharacteristic
144C2×C131C2×#13712yes—
145D131—81no—
146C131#315211yescharacteristic, complemented, Sylow
147#3152—1311nomaximal, Sylow
148C2×#137—1311no—
149C2×#137—1311no—
150C2×#137—1311no—
151C2×#137D13111yescharacteristic, complemented
152#532—1311no—
153C2×C4⋊Q8—1311no—
154#497—1312no—
155#497—1312no—
156#494—1312no—
157#492—1312no—
158C4×M16—1311no—
159C2×#41—1311no—
160C2×#41C2×D13111yescharacteristic
161C22×Q8—1311no—
162C22×Q8C2×D13111yescharacteristic
163#137—24no—
164#137—2624no—
165#137—2624no—
166#137—2624no—
167C2×Q16—1312no—
168C2×Q16—1312no—
169C2×Q16—1312no—
170C2×Q16C2×D13112yes—
171C2×SD16—1312no—
172C2×SD16—1312no—
173C2×SD16—1312no—
174C2×SD16C2×D13112yes—
175C2×M16—1311no—
176C2×M16C2×D13111yescharacteristic
177C4⋊Q8—2622no—
178C4⋊Q8—1312no—
179C4⋊Q8—1312no—
180C4⋊Q8—1311no—
181#127—1311no—
182#124—1311no—
183#124—1311no—
184#121—2622no—
185C4×D4—2622no—
186C2×C4⋊C4—2622no—
187C2×C42—1311no—
188#97—1312no—
189C4×C8—1312no—
190#41—22no—
191#41—22no—
192#41—2622no—
193#41—2622no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—2621no—
198C2×Q8—2622no—
199C2×Q8—2622no—
200C2×Q8—1311no—
201C2×Q8—1312no—
202C2×Q8C22×D13111yescharacteristic
203C2×Q8C22×D13112yes—
204C2×D4—21no—
205C2×D4—2621no—
206C2×D4—2622no—
207C2×D4—1311no—
208C2×D4C22×D13111yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2621no—
212C22×C4—2621no—
213C22×C4—1312no—
214C22×C4C22×D13111yescharacteristic
215Q16—24no—
216Q16—2624no—
217Q16—2624no—
218Q16—2624no—
219SD16—24no—
220SD16—2624no—
221SD16—2624no—
222SD16—2624no—
223M16—1314no—
224M16C22⋊D13114yescomplemented
225C2×C8—1312no—
226C2×C8C22×D13112yes—
227C4⋊C4—2624no—
228C4⋊C4—2624no—
229C4⋊C4—2622no—
230C22⋊C4—2622no—
231C22⋊C4—2622no—
232C42—1312no—
233C42—1311no—
234C42—1311no—
235C23—21no—
236C23—2621no—
237C23D4×D13111yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—2622no—
244Q8—2621no—
245Q8—2622no—
246Q8—2621no—
247Q8—5242no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—2621no—
252D4—2621no—
253D4—5242no—
254D4—5244no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—2624no—
260C2×C4—2621no—
261C2×C4—2622no—
262C2×C4—5242no—
263C2×C4—5242no—
264C2×C4—1312no—
265C2×C4—1312no—
266C2×C4D4×D13111yescharacteristic
267C2×C4D4×D13112yes—
268C2×C4C2×C22⋊D13112yes—
269C2×C4C23×D13111yescharacteristic
270C8—1314no—
271C8C2×C22⋊D13114yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—5242no—
276C22—5241no—
277C22#1686224212yes—
278C22C2×D4×D13111yescharacteristic
279C4—41no—
280C4—42no—
281C4—2622no—
282C4—5241no—
283C4—5242no—
284C4—1314no—
285C4#1686224212yes—
286C4C2×D4×D13111yescharacteristic
287C4C22×C22⋊D13111yescharacteristic
288C2—22no—
289C2—81no—
290C2—10481no—
291C2C2×#1686224211yescharacteristic
292C2#137×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#137×C131C2
socleC22×C131C22×D4
prime radical#22553858C2
hypercenterC2×#137D131
hypocenterC131#3152
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16862279C2semidirect
2. C2×#16862277C2semidirect
3. C2×#16862276C2semidirect
5. C2×#16862244C2semidirect
6. #22553705C2semidirect
7. #22553703C2semidirect
8. #22553690C2semidirect
9. #22553683C2semidirect
10. M16×Dic131C2semidirect
11. C2×#137×C131C2semidirect
62. #14523497D4semidirect
86. M16×C131D4semidirect
140. Dic131#137semidirect
146. C131#3152semidirect
151. C2×#137D131semidirect
224. M16C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622792
C2×#168622772
C2×#168622762
#225538582
C2×#168622442
#225537052
#225537032
#225536902
#225536832
M16×Dic1312
C2×#137×C1312
#3152131