Finite Group Databasebeta

← #13254740

Subgroups — #13254740

Order 1792.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#11739889C212yesmaximal, complemented
2#11736471C211yesmaximal, characteristic
3C7×#1985C211yesmaximal, characteristic, complemented
4#11727552C212yesmaximal, complemented
5#11725514C211yesmaximal, characteristic, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7#90168C2212yes—
8C2×C7×C4⋊C8C2211yescharacteristic
9C2×C7×#97C2211yescharacteristic
10#89999—22no—
11#89887—22no—
12#89799—22no—
13#89798—24no—
14#89793—22no—
15#89739—22no—
16#89698C2212yes—
17#89618—22no—
18#89615—22no—
19C7×#336—22no—
20#89259—24no—
21#10028—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C2×#1157—22no—
24C22×C7×C8—21no—
25C22×C7×C8D412yes—
26C4×C7×Q8—22no—
27C4×C7×Q8—22no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C42×C7—21no—
31C2×C42×C7C2311yescharacteristic
32C2×#1143—24no—
33#6493—22no—
34#6493—22no—
35C2×D56—22no—
36#6438—22no—
37#6437D412yes—
38C4×D28—22no—
39C4×D28—22no—
40#6432D412yes—
41C7×C4⋊C8—22no—
42C7×#97—22no—
43#6409—24no—
44C22⋊(C7⋊C8)—22no—
45#6395—24no—
46#6392—22no—
47#6383—24no—
48#6381—24no—
49C4⋊(C7⋊C8)—22no—
50#6377D414yes—
51#3245—71no—
52#3112—72no—
53#1985D711yescharacteristic, complemented
54#1530—72no—
55#1290—71no—
56C2×C7×Q8—22no—
57C2×C7×Q8—42no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7C2×D411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7#4111yescharacteristic
63C2×C22⋊D7—42no—
64#1157—84no—
65C2×D28—22no—
66C2×D28—22no—
67C2×D28—22no—
68C2×C4×D7—42no—
69C2×Dic14—22no—
70C2×C7×C8—22no—
71C2×C7×C8—22no—
72C2×C7×C8—41no—
73C2×C7×C8—41no—
74C2×C7×C8—42no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—42no—
77C42×C7—41no—
78C42×C7—41no—
79C42×C7C2×D412yes—
80C42×C7C2×D412yes—
81#1143—48no—
82(C2×D7)⋊C4—42no—
83(C2×D7)⋊C4—42no—
84C4⋊Dic7—22no—
85C4⋊Dic7—22no—
86C2×C7⋊C8—24no—
87D56—82no—
88#532—72no—
89C2×C4×Q8C2×D711yescharacteristic
90#480—142no—
91#477—142no—
92#474—142no—
93#466—142no—
94#460—142no—
95#459—142no—
96#448—142no—
97#440—144no—
98#432—72no—
99C2×C4⋊C8C2×D711yescharacteristic
100C2×#97C2×D711yescharacteristic
101#336—22no—
102#329—144no—
103C23×C7#12111yescharacteristic
104C22×D7—42no—
105C7×Q8—24no—
106C7×Q8—42no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—41no—
112C2×C4×C7—41no—
113C2×C4×C7—41no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7#12012yes—
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7#12112yes—
119C2×C4×C7#12711yescharacteristic
120C22⋊D7—84no—
121C2×Dic7—42no—
122D28—44no—
123D28—42no—
124D28—42no—
125D28—82no—
126C4×D7—84no—
127Dic14—82no—
128C7×C8—42no—
129C7×C8—42no—
130C7⋊C8—44no—
131C2×#41—142no—
132C22×Q8D2811yescharacteristic
133C2×SD16—144no—
134C2×D8—142no—
135C22×C8—21no—
136C22×C8C22⋊D712yes—
137C4⋊Q8—72no—
138#127—72no—
139#124—142no—
140#121—142no—
141#121—142no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×D4—142no—
145C4×D4—142no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D2811yescharacteristic
148C2×C42—21no—
149C2×C42C22×D711yescharacteristic
150#107—144no—
151#107—142no—
152C4⋊C8—22no—
153C4⋊C8—142no—
154#103—144no—
155#102—144no—
156#102—144no—
157C22⋊C8—142no—
158#97—22no—
159#97—74no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7#39211yescharacteristic
163C22×C7#47212yes—
164C2×D7—82no—
165C2×D7—82no—
166C2×D7—82no—
167C4×C7—21no—
168C4×C7—22no—
169C4×C7—22no—
170C4×C7—42no—
171C4×C7—81no—
172C4×C7#45511yescharacteristic
173C4×C7#45611yescharacteristic
174Dic7—82no—
175#41—564no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—142no—
179C2×D4—142no—
180C2×D4—142no—
181C2×D4—142no—
182C2×D4—282no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—282no—
186C22×C4C2×D2811yescharacteristic
187C22×C4#115711yescharacteristic
188C22×C4C2×C22⋊D711yescharacteristic
189SD16—288no—
190D8—562no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—144no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—142no—
200C4⋊C4—142no—
201C22⋊C4—282no—
202C22⋊C4—282no—
203C42—41no—
204C42—41no—
205C42D4×D712yes—
206C42D4×D712yes—
207C2×C7—22no—
208C2×C7#200711yescharacteristic
209C2×C7#201011yescharacteristic
210C2×C7#201811yescharacteristic
211D7—162no—
212C23—282no—
213C23#649311yescharacteristic
214Q8—24no—
215Q8—42no—
216Q8—562no—
217D4—284no—
218D4—282no—
219D4—282no—
220D4—562no—
221D4—564no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—282no—
232C2×C4—564no—
233C2×C4#644512yes—
234C2×C4#644712yes—
235C2×C4#650011yescharacteristic
236C2×C4#650311yescharacteristic
237C8—42no—
238C8—42no—
239C8—284no—
240C7#1002811yescharacteristic, complemented, Sylow
241C22—21no—
242C22—21no—
243C22—562no—
244C22—562no—
245C22—562no—
246C22#8973211yescharacteristic
247C22#8991412yes—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—81no—
253C4—562no—
254C4#8983311yescharacteristic
255C4#8986711yescharacteristic
256C2—22no—
257C2—1122no—
258C2#1172744811yescharacteristic
259C2#1172774711yescharacteristic
260C2#1172776011yescharacteristic
subgroupG/H
centerC22#89732
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1985C2
socleC22×C7#392
prime radical#11736471C2
hypercenter#1985D7
hypocenterC7#10028
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. #11739889C2semidirect
3. C7×#1985C2semidirect
4. #11727552C2semidirect
5. #11725514C2semidirect
53. #1985D7semidirect
240. C7#10028semidirect

Maximal subgroups

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

subgroupindex
#117398892
#117364712
C7×#19852
#117275522
#117255142
#100287