Finite Group Databasebeta

← #13256150

Subgroups — #13256150

Order 1792.

560 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C7×#2582C211yesmaximal, characteristic
2C2×#89793C212yesmaximal
3C2×Q8⋊(C7⋊C8)C212yesmaximal
4#11727333C211yesmaximal, characteristic
5#11725440C211yesmaximal, characteristic
6C2×C4×C7×Q8C2212yes—
7C2×#6432C2211yescharacteristic
8C7×#398—22no—
9C7×#391—22no—
10C7×#389—22no—
11C7×#384C2211yescharacteristic
12C2×#6409—22no—
13#89793—24no—
14#89793—24no—
15C2×#6384—22no—
16C2×#6381—22no—
17C2×C4⋊(C7⋊C8)C2212yes—
18C2×C4×C7⋊C8C2211yescharacteristic
19Q8⋊(C7⋊C8)—24no—
20#10237—71nomaximal, Sylow
21C22×C7×Q8D412yes—
22C22×Dic14—21no—
23C4×C7×Q8—22no—
24C4×C7×Q8—22no—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4D412yes—
32C2×C42×C7—22no—
33C2×C42×C7C2311yescharacteristic
34C2×C4⋊Dic7—22no—
35C22×C7⋊C8—21no—
36C22×C7⋊C8—22no—
37#6432—22no—
38#6432—21no—
39#6432D412yes—
40C7×#95—24no—
41C7×#95—22no—
42#6409—28no—
43#6384—44no—
44#6381—28no—
45C4⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)D414yes—
47C4×C7⋊C8—22no—
48C2×#477—72no—
49#2582D711yescharacteristic
50#1956—71no—
51C2×Q8⋊C8—72no—
52#1296—71no—
53C2×C7×Q8—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8Q1614yes—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D412yes—
64C22×Dic7—41no—
65C2×Dic14—22no—
66C2×Dic14—42no—
67C2×Dic14—42no—
68C2×Dic14—41no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—82no—
77C42×C7—42no—
78C42×C7—42no—
79C42×C7C2×D411yescharacteristic
80C42×C7C2×D412yes—
81C42×C7#4111yescharacteristic
82C4⋊Dic7—24no—
83C4⋊Dic7—42no—
84C2×C7⋊C8—22no—
85C2×C7⋊C8—24no—
86C2×C7⋊C8—41no—
87C2×C7⋊C8—41no—
88C2×C7⋊C8—42no—
89C2×C7⋊C8—42no—
90C2×C4⋊Q8—71no—
91C2×C4×Q8C2×D712yes—
92#477—144no—
93#477—144no—
94C2×#107—142no—
95C2×C4⋊C8—72no—
96C2×#103—142no—
97C2×#103—142no—
98C2×C4×C8—71no—
99#398—22no—
100#391—22no—
101#389—22no—
102#384C2×D711yescharacteristic
103Q8⋊C8—144no—
104C23×C7#12011yescharacteristic
105C7×Q8—24no—
106C7×Q8—24no—
107C7×Q8—42no—
108C7×Q8—42no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—24no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—44no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7#12012yes—
120C2×C4×C7#12111yescharacteristic
121C2×C4×C7#12112yes—
122C2×C4×C7#12711yescharacteristic
123C2×C4×C7C2×Q1612yes—
124C2×C4×C7#13512yes—
125C2×C4×C7#13712yes—
126C2×C4×C7#13712yes—
127C2×Dic7—42no—
128C2×Dic7—81no—
129C2×Dic7—81no—
130Dic14—44no—
131Dic14—82no—
132C7⋊C8—42no—
133C7⋊C8—44no—
134C22×Q8—141no—
135C22×Q8C22⋊D712yes—
136C22×C8—141no—
137C22×C8—142no—
138C4⋊Q8—72no—
139C4⋊Q8—142no—
140C4⋊Q8—141no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—142no—
149C2×C4⋊C4C22⋊D712yes—
150C2×C4⋊C4C22⋊D712yes—
151C2×C42—22no—
152C2×C42C22×D711yescharacteristic
153#107—148no—
154C4⋊C8—74no—
155C4⋊C8—142no—
156#103—148no—
157#103—284no—
158C4×C8—142no—
159#95—24no—
160#95—22no—
161C22×C7#39211yescharacteristic
162C22×C7#45112yes—
163C22×C7#45212yes—
164C22×C7#45311yescharacteristic
165C22×C7#45411yescharacteristic
166C4×C7—21no—
167C4×C7—21no—
168C4×C7—44no—
169C4×C7—82no—
170C4×C7—82no—
171C4×C7#45112yes—
172C4×C7#45212yes—
173C4×C7#46712yes—
174C4×C7#47012yes—
175Dic7—82no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—142no—
182C2×Q8—282no—
183C2×Q8—282no—
184C2×Q8—281no—
185C2×Q8#114414yes—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—281no—
190C22×C4C2×C22⋊D711yescharacteristic
191C22×C4C2×C22⋊D712yes—
192C2×C8—142no—
193C2×C8—144no—
194C2×C8—281no—
195C2×C8—281no—
196C2×C8—282no—
197C2×C8—282no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C4⋊C4—144no—
207C4⋊C4—282no—
208C42—42no—
209C42—42no—
210C42D4×D711yescharacteristic
211C42D4×D712yes—
212C42#115911yescharacteristic
213C2×C7#163011yescharacteristic
214C2×C7#163311yescharacteristic
215C2×C7#163612yes—
216C2×C7#200112yes—
217C2×C7#200611yescharacteristic
218C23#651611yescharacteristic
219Q8—24no—
220Q8—24no—
221Q8—42no—
222Q8—42no—
223Q8—284no—
224Q8—562no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—282no—
236C2×C4—561no—
237C2×C4—561no—
238C2×C4#650012yes—
239C2×C4#650111yescharacteristic
240C2×C4#650212yes—
241C2×C4#650311yescharacteristic
242C2×C4#650512yes—
243C2×C4C2×#114412yes—
244C2×C4#651312yes—
245C2×C4#651412yes—
246C8—282no—
247C8—284no—
248C7#1023711yescharacteristic, complemented, Sylow
249C22#8999511yescharacteristic
250C22#9000012yes—
251C22#9001112yes—
252C22#9001311yescharacteristic
253C22#9001711yescharacteristic
254C4—21no—
255C4—21no—
256C4—44no—
257C4—82no—
258C4—82no—
259C4—562no—
260C4#8981912yes—
261C4#8982212yes—
262C4#8982312yes—
263C4#8982412yes—
264C2#1172945412yes—
265C2#1172963311yescharacteristic
266C2#1172969012yes—
267C2#1172969411yescharacteristic
268C2#1172969811yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2582C2
socle, prime radicalC23×C7#120
hypercenter#2582D7
hypocenterC7#10237
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2582

Complements

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

NH ≅ G/Ntype
248. C7#10237semidirect

Maximal subgroups

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

subgroupindex
C7×#25822
C2×#897932
C2×Q8⋊(C7⋊C8)2
#117273332
#117254402
#102377