Finite Group Databasebeta

← #13256087

Subgroups — #13256087

Order 1792.

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

normal subgroups only

This group has 466 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#11740668C212yesmaximal
2#11735924C211yesmaximal, characteristic
3C7×#1960C211yesmaximal, characteristic
4#11727300C212yesmaximal
5#11725467C211yesmaximal, characteristic
6C2×C7×C4⋊Q8C2211yescharacteristic
7#90161C2212yes—
8C2×C7×#103—22no—
9C2×C7×#97C2211yescharacteristic
10C7×#382C2211yescharacteristic
11#89878—22no—
12#89866—24no—
13#89865—22no—
14#89851—22no—
15#89743—24no—
16#89697C2212yes—
17#89474—22no—
18#89474—22no—
19#89247—24no—
20#10227—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×C8D412yes—
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—21no—
25C7×C4⋊Q8D412yes—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—21no—
28C2×C42×C7C2311yescharacteristic
29#6489—22no—
30#6487—22no—
31#6487—22no—
32#6433—22no—
33#6432D412yes—
34#6432D412yes—
35C4×Dic14—22no—
36C4×Dic14—22no—
37C7×#103—44no—
38C7×#97—22no—
39C7×#95—22no—
40C22⋊(C7⋊C8)D414yes—
41#6392—24no—
42#6391—22no—
43#6391—22no—
44#6384—28no—
45#6382—28no—
46C4⋊(C7⋊C8)—22no—
47C4×C7⋊C8—22no—
48#3229—72no—
49#3162—71no—
50#1960D711yescharacteristic
51#1518—72no—
52#1281—71no—
53C2×C7×Q8—22no—
54C2×C7×Q8—41no—
55C2×C7×Q8—42no—
56C2×C7×Q8—42no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7C2×D412yes—
60C22×C4×C7C2×D411yescharacteristic
61C2×Dic14—22no—
62C2×Dic14—22no—
63C2×C7×C8—22no—
64C2×C7×C8—22no—
65C2×C7×C8SD1614yes—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C42×C7C2×D411yescharacteristic
70C42×C7C2×D412yes—
71C42×C7#4111yescharacteristic
72C22⋊Dic7—42no—
73C4⋊Dic7—22no—
74C4⋊Dic7—22no—
75C4⋊Dic7—22no—
76C4⋊Dic7—22no—
77Dic7⋊C4—42no—
78Dic7⋊C4—42no—
79C4×Dic7—42no—
80C2×C7⋊C8—24no—
81#533—72no—
82C2×C4⋊Q8C2×D711yescharacteristic
83#501—142no—
84#501—142no—
85#498—142no—
86#487—142no—
87#483—144no—
88#475—144no—
89#431—72no—
90#427—142no—
91C2×#103—22no—
92C2×#97C2×D711yescharacteristic
93#382C2×D711yescharacteristic
94#328—144no—
95C23×C7#12711yescharacteristic
96C7×Q8—44no—
97C7×Q8—82no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—81no—
101C2×C4×C7—81no—
102C2×C4×C7—82no—
103C2×C4×C7—81no—
104C2×C4×C7#12012yes—
105C2×C4×C7#12011yescharacteristic
106C2×C4×C7#12112yes—
107C2×C4×C7#12111yescharacteristic
108C2×C4×C7C2×SD1612yes—
109C2×C4×C7#13712yes—
110C2×Dic7—42no—
111C2×Dic7—42no—
112Dic14—44no—
113Dic14—82no—
114C7×C8—24no—
115C7⋊C8—44no—
116C22×Q8—21no—
117C22×C8C22⋊D712yes—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8—72no—
121C4⋊Q8—72no—
122C4⋊Q8D2812yes—
123#125—142no—
124C22⋊Q8—142no—
125C22⋊Q8—142no—
126C4×Q8—142no—
127C4×Q8—142no—
128#117—142no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C42C22×D711yescharacteristic
132#107—144no—
133#106—148no—
134#106—142no—
135#106—142no—
136C4⋊C8—142no—
137#103—44no—
138#103—148no—
139C22⋊C8—74no—
140#97—22no—
141C4×C8—142no—
142#95—22no—
143C22×C7—22no—
144C22×C7#39211yescharacteristic
145C22×C7#49211yescharacteristic
146C22×C7#49711yescharacteristic
147C4×C7—22no—
148C4×C7—22no—
149C4×C7—82no—
150C4×C7—161no—
151C4×C7#44812yes—
152C4×C7#46112yes—
153Dic7—82no—
154Dic7—82no—
155C2×Q8—22no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—142no—
160C2×Q8—142no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4D4×D712yes—
164C22×C4C2×C22⋊D711yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—144no—
168C2×C8#114214yes—
169C4⋊C4—24no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C22⋊C4—282no—
179C42—282no—
180C42C2×D2811yescharacteristic
181C42#115711yescharacteristic
182C42D4×D712yes—
183C2×C7—21no—
184C2×C7—21no—
185C2×C7#200912yes—
186C2×C7#201311yescharacteristic
187C23#650311yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—284no—
191Q8—562no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—282no—
199C2×C4—282no—
200C2×C4#644512yes—
201C2×C4#644712yes—
202C2×C4#649311yescharacteristic
203C2×C4C2×#114212yes—
204C2×C4#650011yescharacteristic
205C2×C4#651412yes—
206C8—24no—
207C8—284no—
208C7#1022711yescharacteristic, complemented, Sylow
209C22—22no—
210C22#8973211yescharacteristic
211C22#8983211yescharacteristic
212C22#8984811yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—161no—
217C4—562no—
218C4—562no—
219C4#8979112yes—
220C4#8981212yes—
221C2—21no—
222C2—21no—
223C2#1172741511yescharacteristic
224C2#1172810512yes—
subgroupG/H
centerC22#89732
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1960C2
socleC22×C7#392
prime radicalC23×C7#127
hypercenter#1960D7
hypocenterC7#10227
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
208. C7#10227semidirect

Maximal subgroups

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

subgroupindex
#117406682
#117359242
C7×#19602
#117273002
#117254672
#102277