Finite Group Databasebeta

← #13254750

Subgroups — #13254750

Order 1792.

520 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3249C211yesmaximal, characteristic, complemented
2#11740798C212yesmaximal
3#11729576C212yesmaximal, complemented
4#11728225C211yesmaximal, characteristic
5#11725514C211yesmaximal, characteristic, complemented
6C7×#533C2212yes—
7C2×C4×D28C2211yescharacteristic
8C7×#477—22no—
9C7×#467—22no—
10C7×#462—22no—
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×(C2×D7)⋊C8—22no—
13#89859—22no—
14#89855—22no—
15#89818—22no—
16#89816—22no—
17#89798—24no—
18#89698C2212yes—
19C2×#6377C2211yescharacteristic
20#89338—24no—
21#10030—71nomaximal, Sylow
22C22×D28D411yescharacteristic
23C22×C4×D7—21no—
24C2×C7×Q16—22no—
25C22×C7×C8D412yes—
26C7×C4⋊Q8D412yes—
27C7×C4⋊Q8D412yes—
28C7×#125—22no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C7×#117—22no—
34C2×C42×C7C2311yescharacteristic
35C2×#1143—24no—
36C2×(C2×D7)⋊C4—21no—
37C2×C4⋊Dic7D411yescharacteristic
38C22×C7⋊C8—21no—
39C4×D28—22no—
40C4×D28—22no—
41C7×#107—22no—
42C7×C4⋊C8—22no—
43C7×#103—24no—
44#6409—24no—
45C22⋊(C7⋊C8)—22no—
46(C2×D7)⋊C8—44no—
47#6383—24no—
48#6381—24no—
49C4⋊(C7⋊C8)—22no—
50#6377—22no—
51#6377D414yes—
52#3249D711yescharacteristic, complemented
53#3110—72no—
54#1955—71no—
55#1531—72no—
56#1290—71no—
57C23×D7—21no—
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C22×C4×C7C2×D411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7#4111yescharacteristic
63C22×Dic7—21no—
64C2×D28—22no—
65C2×D28—42no—
66C2×C4×D7—42no—
67C2×C4×D7—42no—
68C2×C4×D7—41no—
69C2×C4×D7—41no—
70C7×Q16—82no—
71C2×C7×C8—22no—
72C2×C7×C8—42no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C22⋊C4—42no—
80C42×C7—42no—
81C42×C7C2×D412yes—
82C42×C7C2×D412yes—
83#1143—48no—
84(C2×D7)⋊C4—42no—
85C4⋊Dic7—22no—
86C2×C7⋊C8—22no—
87C2×C7⋊C8—24no—
88C2×C7⋊C8—41no—
89C2×C7⋊C8—41no—
90#533C2×D712yes—
91C2×C4×D4—71no—
92#484—142no—
93#477—22no—
94#474—142no—
95#467—22no—
96#462—22no—
97#461—142no—
98#450—142no—
99#440—144no—
100#432—72no—
101C2×C4⋊C8C2×D711yescharacteristic
102C2×C22⋊C8—142no—
103C2×#97—71no—
104#330—144no—
105C23×C7#12111yescharacteristic
106C22×D7—22no—
107C22×D7—42no—
108C22×D7—41no—
109C22×D7—41no—
110C22×D7—42no—
111C7×Q8—44no—
112C7×Q8—42no—
113C7×Q8—42no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—41no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7#12012yes—
121C2×C4×C7#12011yescharacteristic
122C2×C4×C7#12112yes—
123C2×C4×C7#12711yescharacteristic
124C2×Dic7—42no—
125C2×Dic7—41no—
126D28—24no—
127D28—42no—
128C4×D7—82no—
129C4×D7—81no—
130C4×D7—81no—
131C7×C8—42no—
132C7⋊C8—42no—
133C7⋊C8—44no—
134C22×D4—71no—
135C23×C4—141no—
136C2×Q16—22no—
137C2×SD16—144no—
138C22×C8—141no—
139C22×C8C22⋊D712yes—
140C4⋊Q8C22⋊D712yes—
141C4⋊Q8C22⋊D712yes—
142#125—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×D4—142no—
148C4×D4—142no—
149#117—22no—
150C2×C4⋊C4—71no—
151C2×C22⋊C4—141no—
152C2×C42C22×D711yescharacteristic
153#107—22no—
154#107—144no—
155C4⋊C8—22no—
156C4⋊C8—142no—
157#103—24no—
158#103—144no—
159#102—144no—
160C22⋊C8—142no—
161C22⋊C8—284no—
162#97—74no—
163#97—142no—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7#39211yescharacteristic
167C22×C7#47312yes—
168C2×D7—44no—
169C2×D7—42no—
170C2×D7—82no—
171C2×D7—81no—
172C2×D7—81no—
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—82no—
177C4×C7—82no—
178C4×C7#45311yescharacteristic
179C4×C7#45411yescharacteristic
180Dic7—81no—
181C24—141no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×D4—142no—
185C2×D4—282no—
186C22×C4—141no—
187C22×C4—282no—
188C22×C4—282no—
189C22×C4—281no—
190C22×C4—281no—
191C22×C4D4×D711yescharacteristic
192C22×C4#115911yescharacteristic
193C22×C4C2×C22⋊D711yescharacteristic
194Q16—82no—
195SD16—288no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—142no—
199C2×C8—144no—
200C2×C8—281no—
201C2×C8—281no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—142no—
209C22⋊C4—42no—
210C22⋊C4—282no—
211C42—42no—
212C42D4×D712yes—
213C42C2×C22⋊D712yes—
214C2×C7—22no—
215C2×C7#200611yescharacteristic
216C2×C7#201011yescharacteristic
217C2×C7#202011yescharacteristic
218D7—82no—
219C23—142no—
220C23—282no—
221C23—281no—
222C23—282no—
223C23—281no—
224C23#650111yescharacteristic
225Q8—44no—
226Q8—42no—
227Q8—42no—
228D4—144no—
229D4—282no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—282no—
237C2×C4—281no—
238C2×C4—562no—
239C2×C4—561no—
240C2×C4—561no—
241C2×C4#650012yes—
242C2×C4#650212yes—
243C2×C4#650311yescharacteristic
244C2×C4#651611yescharacteristic
245C8—42no—
246C8—282no—
247C8—284no—
248C7#1003011yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—284no—
252C22—282no—
253C22—562no—
254C22—561no—
255C22—561no—
256C22#8996412yes—
257C22#8999511yescharacteristic
258C4—22no—
259C4—22no—
260C4—21no—
261C4—82no—
262C4—82no—
263C4—561no—
264C4#9001311yescharacteristic
265C4#9001711yescharacteristic
266C2—22no—
267C2—562no—
268C2#1172963311yescharacteristic
269C2#1172982811yescharacteristic
270C2#1172987211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3249C2
socleC22×C7#392
prime radicalC22×D28D4
hypercenter#3249D7
hypocenterC7#10030
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
1. C7×#3249C2semidirect
3. #11729576C2semidirect
5. #11725514C2semidirect
52. #3249D7semidirect
248. C7#10030semidirect

Maximal subgroups

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

subgroupindex
C7×#32492
#117407982
#117295762
#117282252
#117255142
#100307