Finite Group Databasebeta

← #13264958

Subgroups — #13264958

Order 1792.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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#11739924C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3#11729378C211yesmaximal, characteristic
4#11725302C214yesmaximal
5C2×C4×C7×Q8C2211yescharacteristic
6C2×#6432C411yescharacteristic
7C7×#423—21no—
8C2×C7×#101—21no—
9#90002—21no—
10#89758C412yes—
11#89758C2212yes—
12#89749—21no—
13C4×C4⋊Dic7C411yescharacteristic
14#89701—21no—
15#89701—21no—
16#89697—21no—
17C7×#343C2212yes—
18#89323—21no—
19#89323C2212yes—
20#11907—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×Dic14—21no—
23C2×C7×M16—22no—
24C4×C7×Q8—41no—
25C4×C7×Q8—41no—
26C2×C7×C4⋊C4C2×C411yescharacteristic
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C4⋊C4C2311yescharacteristic
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7C2×C412yes—
35C2×C4×Dic7—21no—
36#6432—41no—
37#6432—41no—
38C7×C4⋊C8—41no—
39C7×C4⋊C8—41no—
40C7×#101—42no—
41C7⋊#95—22no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95C2×C412yes—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48C4×C7⋊C8—41no—
49#2568—71no—
50#1801D711yescharacteristic
51#1461—71no—
52#1353—74no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8SD1612yes—
56C2×C7×Q8Q1612yes—
57C22×C4×C7C22⋊C411yescharacteristic
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—22no—
65C22×Dic7—22no—
66C2×Dic14—44no—
67C2×Dic14—41no—
68C2×Dic14—41no—
69C7×M16—42no—
70C7×M16—42no—
71C2×C7×C8—42no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C42×C7—21no—
76C42×C7—21no—
77C42×C7—21no—
78C42×C7—21no—
79C42×C7—21no—
80C42×C7—21no—
81C4⋊Dic7—42no—
82C4⋊Dic7—42no—
83C4⋊Dic7—42no—
84C4×Dic7—42no—
85C2×C7⋊C8—42no—
86C2×C4⋊Q8—71no—
87C2×C4×Q8C2×D711yescharacteristic
88#431—141no—
89#423—21no—
90C2×#101—21no—
91#400—72no—
92#400—72no—
93#391—141no—
94#391—141no—
95#391—141no—
96#382—141no—
97C4×C4⋊C4—71no—
98#343C2×D712yes—
99#324—72no—
100#324—141no—
101C23×C7C2×C22⋊C411yescharacteristic
102C7×Q8—42no—
103C7×Q8—42no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7#9911yescharacteristic
118C2×C4×C7#9911yescharacteristic
119C2×C4×C7#10312yes—
120C2×C4×C7#10312yes—
121C2×C4×C7C2×SD1611yescharacteristic
122C2×C4×C7C2×Q1611yescharacteristic
123C2×Dic7—42no—
124C2×Dic7—44no—
125C2×Dic7—42no—
126C2×Dic7—42no—
127C2×Dic7—42no—
128Dic14—82no—
129Dic14—82no—
130C7×C8—82no—
131C7⋊C8—82no—
132C22×Q8—141no—
133C22×Q8C22⋊D711yescharacteristic
134C2×M16—22no—
135C4⋊Q8—281no—
136C4⋊Q8—281no—
137C4×Q8—41no—
138C4×Q8—41no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—141no—
141C2×C4⋊C4—141no—
142C2×C4⋊C4C4×D711yescharacteristic
143C2×C4⋊C4D2811yescharacteristic
144C2×C4⋊C4C22×D711yescharacteristic
145C2×C42—141no—
146C2×C42C4×D711yescharacteristic
147C2×C42D2811yescharacteristic
148C2×C42C22⋊D711yescharacteristic
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—281no—
152#101—42no—
153C22⋊C8—142no—
154C4×C8—281no—
155#95—72no—
156#95—142no—
157#95—142no—
158#95—142no—
159#95—142no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7C2×#9911yescharacteristic
163C22×C7C2×#10311yescharacteristic
164C22×C7#42111yescharacteristic
165C4×C7—21no—
166C4×C7—21no—
167C4×C7—22no—
168C4×C7—22no—
169C4×C7—21no—
170C4×C7—41no—
171C4×C7—41no—
172C4×C7—41no—
173C4×C7#35512yes—
174Dic7—82no—
175Dic7—82no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—284no—
179C2×Q8—281no—
180C2×Q8—281no—
181C2×Q8#114212yes—
182C2×Q8#114412yes—
183C22×C4—142no—
184C22×C4—142no—
185C22×C4(C2×D7)⋊C411yescharacteristic
186C22×C4(C2×D7)⋊C411yescharacteristic
187C22×C4(C2×D7)⋊C411yescharacteristic
188C22×C4(C2×D7)⋊C411yescharacteristic
189C22×C4C2×C4×D711yescharacteristic
190C22×C4C2×D2811yescharacteristic
191C22×C4C2×C22⋊D711yescharacteristic
192M16—42no—
193M16—42no—
194C2×C8—42no—
195C2×C8—282no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—282no—
200C4⋊C4—282no—
201C4⋊C4—282no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—282no—
209C2×C7—21no—
210C2×C7—21no—
211C2×C7#152011yescharacteristic
212C2×C7#212311yescharacteristic
213C2×C7C2×#35511yescharacteristic
214C23C2×(C2×D7)⋊C411yescharacteristic
215Q8—42no—
216Q8—42no—
217Q8—562no—
218Q8—562no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—282no—
233C2×C4—282no—
234C2×C4—284no—
235C2×C4—282no—
236C2×C4—282no—
237C2×C4#638011yescharacteristic
238C2×C4#638011yescharacteristic
239C2×C4#638412yes—
240C2×C4#638412yes—
241C2×C4C2×#114211yescharacteristic
242C2×C4C2×#114411yescharacteristic
243C8—82no—
244C8—562no—
245C7#1190711yescharacteristic, complemented, Sylow
246C22—21no—
247C22—21no—
248C22C2×#638011yescharacteristic
249C22C2×#638411yescharacteristic
250C22#8977711yescharacteristic
251C4—21no—
252C4—21no—
253C4—21no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—562no—
260C4—562no—
261C4#8927212yes—
262C2—21no—
263C2—21no—
264C2#1172726411yescharacteristic
265C2C2×#8927211yescharacteristic
266C2#1172790111yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1801C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1801D7
hypocenterC7#11907
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
245. C7#11907semidirect

Maximal subgroups

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

subgroupindex
#117399242
C7×#18012
#117293782
#117253022
#119077