Finite Group Databasebeta

← #13254752

Subgroups — #13254752

Order 1792.

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

normal subgroups only

This group has 520 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#11739760C212yesmaximal
2#11736482C211yesmaximal, characteristic, complemented
3C7×#1955C211yesmaximal, characteristic
4#11727555C212yesmaximal, complemented
5#11725514C211yesmaximal, characteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7#90161C2212yes—
8C2×C7×C4⋊C8C2211yescharacteristic
9C2×C7×C22⋊C8—22no—
10C2×C7×#97C2211yescharacteristic
11#89990—22no—
12#89888—22no—
13#89791—22no—
14#89790—24no—
15#89779—22no—
16#89742—22no—
17#89698C2212yes—
18#89622—22no—
19#89621—22no—
20#89260—24no—
21#10030—71nomaximal, Sylow
22C22×C7×D4D411yescharacteristic
23C23×C4×C7—21no—
24C22×C7×C8—21no—
25C22×C7×C8D412yes—
26C4×C7×D4—22no—
27C4×C7×D4—22no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C22⋊C4—21no—
30C2×C42×C7C2311yescharacteristic
31C2×#1142—24no—
32#6489—22no—
33#6487—22no—
34#6487—22no—
35C2×Dic28—22no—
36#6433—22no—
37#6432D412yes—
38#6432D412yes—
39C4×Dic14—22no—
40C4×Dic14—22no—
41C7×C4⋊C8—22no—
42C7×C22⋊C8—44no—
43C7×#97—22no—
44#6406—24no—
45C22⋊(C7⋊C8)—22no—
46#6392—22no—
47#6390—24no—
48#6384—24no—
49#6381—24no—
50C4⋊(C7⋊C8)—22no—
51#6377D414yes—
52#3249—71no—
53#3110—72no—
54#1955D711yescharacteristic
55#1531—72no—
56#1290—71no—
57C24×C7—21no—
58C2×C7×D4—22no—
59C2×C7×D4—42no—
60C22×C4×C7—21no—
61C22×C4×C7—42no—
62C22×C4×C7—42no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7#4111yescharacteristic
68C2×Dic14—22no—
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×C22⋊C4—42no—
77C42×C7C2×D412yes—
78C42×C7C2×D412yes—
79C22⋊Dic7—42no—
80#1142—48no—
81C4⋊Dic7—22no—
82C4⋊Dic7—22no—
83C4⋊Dic7—22no—
84C4⋊Dic7—22no—
85Dic7⋊C4—42no—
86Dic7⋊C4—42no—
87C4×Dic7—42no—
88C2×C7⋊C8—24no—
89Dic28—82no—
90#533—72no—
91C2×C4×D4C2×D711yescharacteristic
92#484—142no—
93#477—142no—
94#474—142no—
95#467—142no—
96#462—142no—
97#461—142no—
98#450—142no—
99#440—144no—
100#432—72no—
101C2×C4⋊C8C2×D711yescharacteristic
102C2×C22⋊C8—22no—
103C2×#97C2×D711yescharacteristic
104#330—144no—
105C23×C7—22no—
106C23×C7—42no—
107C23×C7—41no—
108C23×C7—41no—
109C23×C7—42no—
110C23×C7#12111yescharacteristic
111C7×D4—24no—
112C7×D4—42no—
113C2×C4×C7—22no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—41no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—82no—
120C2×C4×C7—81no—
121C2×C4×C7—81no—
122C2×C4×C7#12012yes—
123C2×C4×C7#12011yescharacteristic
124C2×C4×C7#12112yes—
125C2×C4×C7#12711yescharacteristic
126C2×Dic7—42no—
127C2×Dic7—42no—
128Dic14—44no—
129Dic14—42no—
130Dic14—42no—
131C7×C8—42no—
132C7×C8—42no—
133C7⋊C8—44no—
134C22×D4D2811yescharacteristic
135C23×C4—21no—
136C2×Q16—142no—
137C2×SD16—144no—
138C22×C8—21no—
139C22×C8C22⋊D712yes—
140C4⋊Q8—72no—
141C4⋊Q8—72no—
142#125—142no—
143C22⋊Q8—142no—
144C22⋊Q8—142no—
145C4×Q8—142no—
146C4×Q8—142no—
147C4×D4—22no—
148C4×D4—22no—
149#117—142no—
150C2×C4⋊C4D2811yescharacteristic
151C2×C22⋊C4—21no—
152C2×C42C22×D711yescharacteristic
153#107—144no—
154#107—142no—
155C4⋊C8—22no—
156C4⋊C8—142no—
157#103—144no—
158#103—144no—
159#102—144no—
160C22⋊C8—44no—
161C22⋊C8—142no—
162#97—22no—
163#97—74no—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7—44no—
167C22×C7—42no—
168C22×C7—82no—
169C22×C7—81no—
170C22×C7—81no—
171C22×C7#39211yescharacteristic
172C22×C7#47312yes—
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—81no—
177C4×C7#45311yescharacteristic
178C4×C7#45411yescharacteristic
179Dic7—82no—
180Dic7—82no—
181C24—21no—
182C2×Q8—142no—
183C2×Q8—142no—
184C2×D4—22no—
185C2×D4—42no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4C2×D2811yescharacteristic
192C22×C4#115711yescharacteristic
193C22×C4C2×C22⋊D711yescharacteristic
194Q16—562no—
195SD16—288no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—144no—
202C4⋊C4—22no—
203C4⋊C4—142no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—142no—
207C4⋊C4—282no—
208C4⋊C4—282no—
209C22⋊C4—42no—
210C22⋊C4—282no—
211C42—282no—
212C42D4×D712yes—
213C42D4×D712yes—
214C2×C7—22no—
215C2×C7—82no—
216C2×C7#200611yescharacteristic
217C2×C7#201011yescharacteristic
218C2×C7#202011yescharacteristic
219C23—22no—
220C23—42no—
221C23—41no—
222C23—42no—
223C23—41no—
224C23#649311yescharacteristic
225Q8—284no—
226Q8—282no—
227Q8—282no—
228D4—24no—
229D4—42no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—282no—
240C2×C4—282no—
241C2×C4#644512yes—
242C2×C4#644712yes—
243C2×C4#650011yescharacteristic
244C2×C4#650311yescharacteristic
245C8—42no—
246C8—42no—
247C8—284no—
248C7#1003011yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—44no—
252C22—42no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22#8973211yescharacteristic
257C22#8991512yes—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—81no—
262C4—562no—
263C4—562no—
264C4#8983411yescharacteristic
265C4#8984911yescharacteristic
266C2—22no—
267C2—82no—
268C2#1172741811yescharacteristic
269C2#1172774711yescharacteristic
270C2#1172776211yescharacteristic
subgroupG/H
centerC22#89732
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1955C2
socleC22×C7#392
prime radicalC22×C7×D4D4
hypercenter#1955D7
hypocenterC7#10030
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #11736482C2semidirect
4. #11727555C2semidirect
5. #11725514C2semidirect
248. C7#10030semidirect

Maximal subgroups

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

subgroupindex
#117397602
#117364822
C7×#19552
#117275552
#117255142
#100307