Finite Group Databasebeta

← #13256042

Subgroups — #13256042

Order 1792.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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#11740614C212yesmaximal
2#11735924C211yesmaximal, characteristic, complemented
3C7×#1961C211yesmaximal, characteristic
4#11727300C212yesmaximal, complemented
5#11725467C211yesmaximal, characteristic, complemented
6C2×C7×#127C2211yescharacteristic
7#90161C2212yes—
8C2×C7×#102—22no—
9C2×C7×#97C2211yescharacteristic
10C7×#382C2211yescharacteristic
11#89990—24no—
12#89878—22no—
13#89848—22no—
14#89847—24no—
15#89842—22no—
16#89697C2212yes—
17#89474—22no—
18#89474—22no—
19#89247—24no—
20#10220—71nomaximal, Sylow
21C22×C7×D4—21no—
22C22×C7×D4—22no—
23C22×C7×C8D412yes—
24C7×#127—22no—
25C7×#127—21no—
26C7×#127D412yes—
27C2×C7×C4⋊C4—21no—
28C2×C42×C7C2311yescharacteristic
29C2×#1142—28no—
30#6489—22no—
31#6487—22no—
32#6487—22no—
33#6433—22no—
34#6432D412yes—
35#6432D412yes—
36C4×Dic14—22no—
37C4×Dic14—22no—
38C7×#102—44no—
39C7×#97—22no—
40C7×#95—22no—
41#6406—28no—
42C22⋊(C7⋊C8)D414yes—
43#6392—24no—
44#6391—22no—
45#6391—22no—
46C4⋊(C7⋊C8)—22no—
47C4×C7⋊C8—22no—
48#3208—72no—
49#3162—71no—
50#1961D711yescharacteristic
51#1518—72no—
52#1281—71no—
53C24×C7—41no—
54C2×C7×D4—24no—
55C2×C7×D4—22no—
56C2×C7×D4—41no—
57C2×C7×D4—42no—
58C2×C7×D4—42no—
59C2×C7×D4—42no—
60C2×C7×D4—84no—
61C22×C4×C7—41no—
62C22×C4×C7C2×D412yes—
63C22×C4×C7C2×D411yescharacteristic
64C2×Dic14—22no—
65C2×Dic14—22no—
66C2×C7×C8—22no—
67C2×C7×C8—22no—
68C2×C7×C8SD1614yescomplemented
69C7×C4⋊C4—42no—
70C42×C7C2×D411yescharacteristic
71C42×C7C2×D412yes—
72C42×C7#4111yescharacteristic
73C22⋊Dic7—42no—
74#1142—88no—
75C4⋊Dic7—22no—
76C4⋊Dic7—22no—
77C4⋊Dic7—22no—
78C4⋊Dic7—22no—
79Dic7⋊C4—42no—
80Dic7⋊C4—42no—
81C4×Dic7—42no—
82C2×C7⋊C8—24no—
83#533—72no—
84C2×#127C2×D711yescharacteristic
85#501—142no—
86#501—142no—
87#492—142no—
88#486—142no—
89#460—144no—
90#450—144no—
91#431—72no—
92#427—142no—
93C2×#102—22no—
94C2×#97C2×D711yescharacteristic
95#382C2×D711yescharacteristic
96#328—144no—
97C23×C7—42no—
98C23×C7—84no—
99C23×C7—81no—
100C23×C7—81no—
101C23×C7#12711yescharacteristic
102C7×D4—48no—
103C7×D4—44no—
104C7×D4—84no—
105C7×D4—82no—
106C2×C4×C7—42no—
107C2×C4×C7—82no—
108C2×C4×C7—81no—
109C2×C4×C7#12012yes—
110C2×C4×C7#12011yescharacteristic
111C2×C4×C7#12112yes—
112C2×C4×C7#12111yescharacteristic
113C2×C4×C7C2×SD1612yes—
114C2×C4×C7#13712yes—
115C2×Dic7—42no—
116C2×Dic7—42no—
117Dic14—44no—
118Dic14—82no—
119C7×C8—24no—
120C7⋊C8—44no—
121C22×D4—22no—
122C22×D4—21no—
123C2×SD16—148no—
124C22×C8C22⋊D712yes—
125C4⋊Q8—72no—
126C4⋊Q8—72no—
127#127—21no—
128#127—22no—
129#127D2812yes—
130#125—142no—
131C22⋊Q8—142no—
132C22⋊Q8—142no—
133C4×Q8—142no—
134C4×Q8—142no—
135#117—142no—
136C2×C4⋊C4—21no—
137C2×C42C22×D711yescharacteristic
138#107—144no—
139#106—142no—
140#106—142no—
141C4⋊C8—142no—
142#102—44no—
143#102—148no—
144C22⋊C8—74no—
145#97—22no—
146C4×C8—142no—
147#95—22no—
148C22×C7—22no—
149C22×C7—84no—
150C22×C7—82no—
151C22×C7—162no—
152C22×C7—162no—
153C22×C7#39211yescharacteristic
154C22×C7#49211yescharacteristic
155C22×C7#49711yescharacteristic
156C4×C7—22no—
157C4×C7—22no—
158C4×C7—161no—
159C4×C7#44812yes—
160C4×C7#46112yes—
161Dic7—82no—
162Dic7—82no—
163C24—41no—
164C2×Q8—142no—
165C2×Q8—142no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—84no—
173C22×C4—41no—
174C22×C4D4×D712yes—
175C22×C4C2×C22⋊D711yescharacteristic
176SD16—568no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—144no—
180C2×C8#114214yescomplemented
181C4⋊C4—42no—
182C4⋊C4—142no—
183C4⋊C4—142no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—282no—
187C4⋊C4—282no—
188C22⋊C4—282no—
189C42—282no—
190C42C2×D2811yescharacteristic
191C42#115711yescharacteristic
192C42D4×D712yes—
193C2×C7—21no—
194C2×C7—21no—
195C2×C7—162no—
196C2×C7#200812yes—
197C2×C7#201311yescharacteristic
198C23—42no—
199C23—84no—
200C23—81no—
201C23—81no—
202C23#650311yescharacteristic
203Q8—284no—
204Q8—562no—
205D4—48no—
206D4—44no—
207D4—84no—
208D4—82no—
209C2×C4—42no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—282no—
213C2×C4—282no—
214C2×C4#644512yes—
215C2×C4#644712yes—
216C2×C4#649311yescharacteristic
217C2×C4C2×#114212yes—
218C2×C4#650011yescharacteristic
219C2×C4#651412yes—
220C8—24no—
221C8—284no—
222C7#1022011yescharacteristic, complemented, Sylow
223C22—22no—
224C22—82no—
225C22—84no—
226C22—162no—
227C22—162no—
228C22#8973211yescharacteristic
229C22#8983211yescharacteristic
230C22#8984811yescharacteristic
231C4—22no—
232C4—22no—
233C4—161no—
234C4—562no—
235C4—562no—
236C4#8979112yes—
237C4#8981212yes—
238C2—21no—
239C2—21no—
240C2—162no—
241C2#1172741511yescharacteristic
242C2#1172808512yes—
subgroupG/H
centerC22#89732
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1961C2
socleC22×C7#392
prime radicalC2×C7×#127C22
hypercenter#1961D7
hypocenterC7#10220
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #11735924C2semidirect
4. #11727300C2semidirect
5. #11725467C2semidirect
68. C2×C7×C8SD16semidirect
180. C2×C8#1142semidirect
222. C7#10220semidirect

Maximal subgroups

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

subgroupindex
#117406142
#117359242
C7×#19612
#117273002
#117254672
#102207