Finite Group Databasebeta

← #20290601

Subgroups — #20290601

Order 6784.

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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×#13133287C211yesmaximal, characteristic, complemented
2#16811633C211yesmaximal, characteristic, complemented
3#16811620C212yesmaximal
4#16811506C212yesmaximal, complemented
5#131×D53C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7#13133296C2212yes—
8#13133287—21no—
9#13133287—21no—
10#13133286—21no—
11#13133285—21no—
12#13133282—22no—
13#13133281—22no—
14#41×D53C2211yescharacteristic
15#13133225—21no—
16#13133225C411yescharacteristic, complemented
17#13133224C411yescharacteristic, complemented
18#13133223—21no—
19#13133223C411yescharacteristic, complemented
20M16×D53—21no—
21M16×D53C2211yescharacteristic
22C2×C8×D53—21no—
23C2×C8×D53C2211yescharacteristic
24#131×C53C411yescharacteristic, complemented
25#41×C53C2×C411yescharacteristic
26#11719949D411yescharacteristic
27Q8×D53Q811yescharacteristic
28#11719947—21no—
29#11719947D411yescharacteristic
30D4×D53—21no—
31D4×D53Q811yescharacteristic
32#11719945—21no—
33#11719945C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×C4×D53C2311yescharacteristic
36C22⋊#6157—22no—
37C2×#88324—22no—
38C4⋊#6157—22no—
39C4×#6157—22no—
40#11719936—22no—
41#11719936—22no—
42M16×C53—21no—
43M16×C53C2×C411yescharacteristic
44C2×C8×C53—21no—
45C2×C8×C53C2×C411yescharacteristic
46#11719917—21no—
47#11719917C2×C411yescharacteristic
48C2×#88322—21no—
49C2×#88322C2×C411yescharacteristic
50#11719912—21no—
51#11719912—21no—
52#11719912—21no—
53C8×D53—21no—
54C8×D53—21no—
55C8×D53—21no—
56C8×D53D411yescharacteristic
57C8×D53D411yescharacteristic
58C8×D53D411yescharacteristic
59C8×D53D411yescharacteristic
60C22×D53—21no—
61C22×D53#4111yescharacteristic
62C2×#6157—42no—
63Q8×C53C4⋊C411yescharacteristic
64D4×C53—21no—
65D4×C53C4⋊C411yescharacteristic
66C2×C4×C53—21no—
67C2×C4×C53C22×C411yescharacteristic
68C22⋊D53—21no—
69C22⋊D53—21no—
70C22⋊D53—21no—
71C2×Dic53—21no—
72C2×Dic53#4111yescharacteristic
73D212—21no—
74D212C4⋊C411yescharacteristic
75C4×D53—21no—
76C4×D53—21no—
77C4×D53—21no—
78C4×D53C2×D411yescharacteristic
79C4×D53C2×D411yescharacteristic
80C4×D53C2×D411yescharacteristic
81C4×D53C2×Q811yescharacteristic
82Dic106—21no—
83Dic106C4⋊C411yescharacteristic
84#88324—42no—
85C8×C53—21no—
86C8×C53C22⋊C411yescharacteristic
87C8×C53C22⋊C411yescharacteristic
88#88322—21no—
89#88322C22⋊C411yescharacteristic
90#88322C22⋊C411yescharacteristic
91C22×C53—21no—
92C22×C53#11711yescharacteristic
93C2×D53—21no—
94C2×D53—41no—
95C2×D53—42no—
96C2×D53#12111yescharacteristic
97C2×D53C22⋊Q811yescharacteristic
98#6157—82no—
99C4×C53—21no—
100C4×C53C2×C22⋊C411yescharacteristic
101C4×C53C2×C4⋊C411yescharacteristic
102Dic53—21no—
103Dic53#12111yescharacteristic
104Dic53C22⋊Q811yescharacteristic
105#1814—531nomaximal, Sylow
106C2×C53—21no—
107C2×C53—41no—
108C2×C53#38011yescharacteristic
109D53—21no—
110D53—21no—
111D53—41no—
112C2×#131—531no—
113C2×#108—531no—
114#427—531no—
115#421—532no—
116#344—532no—
117C53#181411yescharacteristic, complemented, Sylow
118C2×#41—531no—
119#131—531no—
120#131—531no—
121#131—531no—
122#131—1061no—
123#131—1061no—
124#131#615711yescharacteristic, complemented
125C2×M16—531no—
126C2×M16—532no—
127C2×M16—1061no—
128C22×C8—531no—
129C22×C8—1061no—
130#117—532no—
131#108—1061no—
132#108—1061no—
133#107—1061no—
134#106—1061no—
135#104—1062no—
136#104—1062no—
137#41—531no—
138#41—531no—
139#41—531no—
140#41—1061no—
141#41—1061no—
142#41C2×#615711yescharacteristic
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C22×C4—531no—
147C22×C4—1061no—
148M16—21no—
149M16—531no—
150M16—1061no—
151M16—1061no—
152M16—1061no—
153M16—1061no—
154M16—1062no—
155M16—1062no—
156M16C2×#615711yescharacteristic
157C2×C8—21no—
158C2×C8—531no—
159C2×C8—531no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8—531no—
163C2×C8—1061no—
164C2×C8—1061no—
165C2×C8—1061no—
166C2×C8—1061no—
167C2×C8—1062no—
168C2×C8C2×#615711yescharacteristic
169C4⋊C4—1062no—
170C22⋊C4—1062no—
171C42—1062no—
172C23—531no—
173C23—1061no—
174Q8—531no—
175Q8—1061no—
176Q8C4⋊#615711yescharacteristic
177D4—21no—
178D4—531no—
179D4—1061no—
180D4—1061no—
181D4—1061no—
182D4—1061no—
183D4C4⋊#615711yescharacteristic
184C2×C4—21no—
185C2×C4—531no—
186C2×C4—531no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—2122no—
195C2×C4C22×#615711yescharacteristic
196C8—21no—
197C8—531no—
198C8—531no—
199C8—1061no—
200C8—2122no—
201C8C22⋊#615711yescharacteristic
202C8C22⋊#615711yescharacteristic
203C22—21no—
204C22—531no—
205C22—531no—
206C22—1061no—
207C22—2121no—
208C22—2122no—
209C22#1313329611yescharacteristic
210C4—21no—
211C4—531no—
212C4—531no—
213C4—1061no—
214C4—4242no—
215C4C2×C22⋊#615711yescharacteristic
216C4C2×C4⋊#615711yescharacteristic
217C2—21no—
218C2—41no—
219C2—1061no—
220C2—1061no—
221C2—2121no—
222C2#1681156511yescharacteristic
subgroupG/H
centerC2#16811565
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#131×C53C4
socleC2×C53#380
hypercenter#131#6157
hypocenterC53#1814
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131

Complements

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

NH ≅ G/Ntype
1. C2×#13133287C2semidirect
2. #16811633C2semidirect
4. #16811506C2semidirect
16. #13133225C4semidirect
17. #13133224C4semidirect
19. #13133223C4semidirect
24. #131×C53C4semidirect
117. C53#1814semidirect
124. #131#6157semidirect