Finite Group Databasebeta

← #30679373

Subgroups — #30679373

Order 16768.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#22553911C216yesmaximal, complemented
2#22553902C216yesmaximal, complemented
3#22553874C213yesmaximal, complemented
4#22553848C216yesmaximal, complemented
5#22553825C216yesmaximal, complemented
6C2×(C22⋊D131)⋊C4C213yesmaximal, complemented
7#543×C131C211yesmaximal, characteristic, complemented
8C2×#14523514C2212yescomplemented
9C22×C22⋊D131C2213yes—
10C23×Dic131C2211yescharacteristic
11C22×C4×D131C2213yescomplemented
12#16862241C22112yescomplemented
13#16862237C2216yescomplemented
14#16862235C2216yescomplemented
15#16862231C22112yescomplemented
16#16862230C22112yescomplemented
17#16862222C2216yescomplemented
18#16862221C2216yescomplemented
19#16862220C2216yes—
20#16862220C2216yes—
21#16862216C22112yescomplemented
22Q8×Dic131C2212yescomplemented
23#16862208C2216yescomplemented
24(C22⋊D131)⋊C4C22112yescomplemented
25(C22⋊D131)⋊C4C22112yescomplemented
26#16862204C2216yescomplemented
27C2×Dic131⋊C4C2213yescomplemented
28C2×(C2×D131)⋊C4C2213yescomplemented
29C2×C4×Dic131C2213yescomplemented
30C22⋊Q8×C131C2216yescomplemented
31C22⋊Q8×C131C2216yescomplemented
32C2×C22⋊C4×C131C2213yes—
33C23×D131C2313yes—
34#14523514—44no—
35C2×D524C2316yes—
36C2×D524C2316yes—
37C2×C22⋊D131—212no—
38C2×C22⋊D131C23112yes—
39C22×Dic131—42no—
40C22×Dic131C2313yes—
41C22×Dic131C2313yes—
42C22×Dic131C2313yes—
43C2×C4×D131—46no—
44C2×C4×D131C2316yes—
45C2×C4×D131C2316yes—
46C4⋊Dic131C2316yes—
47C4⋊Dic131C2316yes—
48Dic131⋊C4C23112yes—
49(C2×D131)⋊C4C23112yes—
50(C2×D131)⋊C4C23112yes—
51(C2×D131)⋊C4C23112yes—
52C4×Dic131C2316yes—
53C4×Dic131C2316yes—
54C24×C131C2311yescharacteristic
55C2×Q8×C131C2312yes—
56C2×Q8×C131C2312yes—
57C22×C4×C131C2313yes—
58C22×C4×C131C2313yes—
59C4⋊C4×C131C2316yes—
60C4⋊C4×C131C2316yes—
61C22⋊C4×C131C23112yes—
62C22×D131—26no—
63C22×D131—43no—
64C22×D131—43no—
65C22×D131C2416yes—
66D524—46no—
67D524—46no—
68C22⋊D131—224no—
69C2×Dic131—26no—
70C2×Dic131—26no—
71C2×Dic131—43no—
72C2×Dic131C2413yes—
73C2×Dic131C2416yes—
74C2×Dic131C2411yescharacteristic
75C4×D131—212no—
76C4×D131—46no—
77C23×C131—41no—
78C23×C131—41no—
79C23×C131#4114yescomplemented
80C23×C131C2413yes—
81Q8×C131—42no—
82Q8×C131—42no—
83C2×C4×C131—43no—
84C2×C4×C131—43no—
85C2×C4×C131C2416yes—
86C2×C4×C131C2416yes—
87C22×C131—23no—
88C22×C131—23no—
89C22×C131—44no—
90C22×C131C2×#4116yes—
91C22×C131C2511yescharacteristic
92C2×D131—26no—
93C2×D131—26no—
94C2×D131—26no—
95C2×D131—412no—
96C4×C131—26no—
97C4×C131—26no—
98Dic131—26no—
99Dic131—23no—
100Dic131H(5,𝔽2)12yes—
101C2×C131—26no—
102C2×C131C22×#4111yescharacteristic
103C2×C131C2×H(5,𝔽2)11yescharacteristic
104C2×C131#58511yescharacteristic
105D131—46no—
106C131#345911yescharacteristic, complemented, Sylow
107#3459—1311nomaximal, Sylow
108#543D13111yescharacteristic, complemented
109#539—1316no—
110#538—1316no—
111#534—1313no—
112#529—1316no—
113#525—1316no—
114C2×C4×D4—1313no—
115C2×#41—1312no—
116C22×D4—1313no—
117C23×C4—1311no—
118C23×C4—1313no—
119#126—13112no—
120#125—1316no—
121#124—1316no—
122#123—13112no—
123#123—13112no—
124C22⋊Q8—1316no—
125C22⋊Q8C2×D13116yescomplemented
126C22⋊Q8C2×D13116yescomplemented
127#121—1316no—
128#121—1316no—
129#121—1316no—
130#120—13112no—
131C4×Q8—1312no—
132C4×D4—1316no—
133C4×D4—13112no—
134C4×D4—13112no—
135#117—1316no—
136C2×C4⋊C4—1313no—
137C2×C22⋊C4—1313no—
138C2×C22⋊C4C2×D13113yes—
139C2×C42—1313no—
140C24—1313no—
141C24C22×D13111yescharacteristic
142#41—5244no—
143C2×Q8C22×D13112yes—
144C2×Q8C22×D13112yes—
145C2×D4—26212no—
146C2×D4—13112no—
147C2×D4—1316no—
148C2×D4—1316no—
149C22×C4—5246no—
150C22×C4—5242no—
151C22×C4—1313no—
152C22×C4—1313no—
153C22×C4—1316no—
154C22×C4—1313no—
155C22×C4—1316no—
156C22×C4C22×D13113yes—
157C22×C4C22×D13113yes—
158C4⋊C4—13112no—
159C4⋊C4—1316no—
160C4⋊C4—1316no—
161C4⋊C4C22×D13116yes—
162C4⋊C4C22×D13116yes—
163C22⋊C4—13112no—
164C22⋊C4—13112no—
165C22⋊C4—13112no—
166C22⋊C4C22×D131112yes—
167C42—1316no—
168C42—1316no—
169C23—41no—
170C23—41no—
171C23—2626no—
172C23—5243no—
173C23—5243no—
174C23—1316no—
175C23#1452351414yescomplemented
176C23C23×D13113yes—
177Q8—42no—
178Q8—42no—
179D4—26224no—
180D4—5246no—
181D4—5246no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—26212no—
185C2×C4—2626no—
186C2×C4—2626no—
187C2×C4—5246no—
188C2×C4—5243no—
189C2×C4—1316no—
190C2×C4—1313no—
191C2×C4—1311no—
192C2×C4C23×D13116yes—
193C2×C4C23×D13116yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—2626no—
198C22—2626no—
199C22—2626no—
200C22—52412no—
201C22C2×#1452351416yes—
202C22C24×D13111yescharacteristic
203C4—26no—
204C4—26no—
205C4—2626no—
206C4—2623no—
207C4—1312no—
208C2—26no—
209C2—5246no—
210C2C22×#1452351411yescharacteristic
211C2H(5,𝔽2)×D13111yescharacteristic
212C2#2255414711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D131
commutator, socle, cosocleC22×C131C25
Fitting#543×C131C2
hypercenter#543D131
hypocenterC131#3459
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #22553911C2semidirect
2. #22553902C2semidirect
3. #22553874C2semidirect
4. #22553848C2semidirect
5. #22553825C2semidirect
6. C2×(C22⋊D131)⋊C4C2semidirect
7. #543×C131C2semidirect
8. C2×#14523514C22semidirect
11. C22×C4×D131C22semidirect
12. #16862241C22semidirect
13. #16862237C22semidirect
14. #16862235C22semidirect
15. #16862231C22semidirect
16. #16862230C22semidirect
17. #16862222C22semidirect
18. #16862221C22semidirect
21. #16862216C22semidirect
22. Q8×Dic131C22semidirect
23. #16862208C22semidirect
24. (C22⋊D131)⋊C4C22semidirect
25. (C22⋊D131)⋊C4C22semidirect
26. #16862204C22semidirect
27. C2×Dic131⋊C4C22semidirect
28. C2×(C2×D131)⋊C4C22semidirect
29. C2×C4×Dic131C22semidirect
30. C22⋊Q8×C131C22semidirect
31. C22⋊Q8×C131C22semidirect
79. C23×C131#41semidirect
106. C131#3459semidirect
108. #543D131semidirect
125. C22⋊Q8C2×D131semidirect
126. C22⋊Q8C2×D131semidirect
175. C23#14523514semidirect

Maximal subgroups

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

subgroupindex
#225539112
#225539022
#225538742
#225538482
#225538252
C2×(C22⋊D131)⋊C42
#543×C1312
#3459131