Finite Group Databasebeta

← #13248175

Subgroups — #13248175

Order 1792.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2379C211yesmaximal, characteristic
2#11729480C211yesmaximal, characteristic
3C2×#89335C211yesmaximal, characteristic
4#11727302C212yesmaximal, complemented
5#11725436C212yesmaximal, complemented
6C2×C7×#124C2211yescharacteristic
7C7×#391—21no—
8C7×#388—22no—
9C7×#386—21no—
10C7×#385—22no—
11C7×#382C2212yes—
12C2×#6409—21no—
13C2×#6406—21no—
14#89707C2211yescharacteristic
15C2×#6377C2211yescharacteristic
16C2×#6377C2212yes—
17#89335C414yescomplemented
18#89324—22no—
19#89324—22no—
20#8914—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×C7×D4D411yescharacteristic
23C23×C4×C7—21no—
24C7×#124—22no—
25C7×#124C2×C412yes—
26C7×#124Q812yes—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4D412yes—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—21no—
33C2×C42×C7—21no—
34C2×C42×C7C2311yescharacteristic
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—22no—
37C22×C7⋊C8—21no—
38C22×C7⋊C8—22no—
39C7×#95—22no—
40C7×#95—22no—
41C7×#95—22no—
42#6409—42no—
43#6406—42no—
44#6377—22no—
45#6377—22no—
46#6377C2×C414yescomplemented
47#2379D711yescharacteristic
48#1959—71no—
49#1836—72no—
50C2×#329—71no—
51#1295—72no—
52C24×C7—21no—
53C2×C7×Q8—21no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8C4⋊C412yes—
57C2×C7×D4—21no—
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4C4⋊C412yes—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—81no—
65C22×C4×C7—82no—
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D412yes—
68C22×Dic7—41no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—81no—
72C7×C22⋊C4—24no—
73C7×C22⋊C4—24no—
74C7×C22⋊C4—81no—
75C7×C22⋊C4—82no—
76C42×C7—81no—
77C42×C7C22×C411yescharacteristic
78C42×C7C2×Q811yescharacteristic
79C42×C7#4112yes—
80C4⋊Dic7—41no—
81C4⋊Dic7—41no—
82C4⋊Dic7—44no—
83C2×C7⋊C8—22no—
84C2×C7⋊C8—24no—
85C2×C7⋊C8—42no—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—41no—
88C2×C7⋊C8—41no—
89C2×#124C2×D711yescharacteristic
90C2×#103—141no—
91C2×#102—141no—
92C2×#97—72no—
93C2×#97—71no—
94#391—21no—
95#388—22no—
96#386—21no—
97#385—22no—
98#384—71no—
99#382C2×D712yes—
100#340—142no—
101#340—142no—
102#329—74no—
103C23×C7—22no—
104C23×C7—41no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—41no—
108C23×C7#12011yescharacteristic
109C7×Q8—42no—
110C7×Q8—81no—
111C7×D4—42no—
112C7×D4—81no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—84no—
127C2×C4×C7C2×C4⋊C411yescharacteristic
128C2×C4×C7C4×D412yes—
129C2×C4×C7C22⋊Q812yes—
130C2×C4×C7#12311yescharacteristic
131C2×Dic7—81no—
132C2×Dic7—82no—
133C7⋊C8—42no—
134C7⋊C8—44no—
135C22×Q8C22⋊D711yescharacteristic
136C22×D4C22⋊D711yescharacteristic
137C23×C4—21no—
138C22×C8—142no—
139C22×C8—141no—
140#124—22no—
141#124Dic1412yes—
142#124C4×D712yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—142no—
145C2×C4⋊C4—141no—
146C2×C4⋊C4C22⋊D712yes—
147C2×C4⋊C4C22⋊D712yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C42—21no—
152C2×C42C22×D711yescharacteristic
153#103—282no—
154#102—282no—
155#97—74no—
156#97—142no—
157#97—142no—
158#95—22no—
159#95—22no—
160#95—22no—
161C22×C7—42no—
162C22×C7—44no—
163C22×C7—81no—
164C22×C7—81no—
165C22×C7—82no—
166C22×C7#38511yescharacteristic
167C22×C7#45311yescharacteristic
168C22×C7#45412yes—
169C22×C7#45511yescharacteristic
170C22×C7#45512yes—
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—22no—
174C4×C7—22no—
175C4×C7—42no—
176C4×C7—82no—
177C4×C7—82no—
178Dic7—161no—
179C24—21no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8Dic7⋊C412yes—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4Dic7⋊C412yes—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—81no—
192C22×C4—82no—
193C22×C4—281no—
194C22×C4C2×C22⋊D711yescharacteristic
195C22×C4C2×C22⋊D712yes—
196C2×C8—142no—
197C2×C8—144no—
198C2×C8—282no—
199C2×C8—281no—
200C2×C8—282no—
201C2×C8—281no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C4⋊C4—281no—
206C4⋊C4—281no—
207C4⋊C4—284no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—81no—
211C22⋊C4—82no—
212C42—81no—
213C42C2×Dic1411yescharacteristic
214C42C2×C4×D711yescharacteristic
215C42#115712yes—
216C2×C7—82no—
217C2×C7#165711yescharacteristic
218C2×C7#165911yescharacteristic
219C2×C7#166212yes—
220C2×C7#185611yescharacteristic
221C2×C7#186012yes—
222C23—22no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—42no—
227C23#651611yescharacteristic
228Q8—42no—
229Q8—81no—
230D4—42no—
231D4—81no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—84no—
246C2×C4—561no—
247C2×C4—562no—
248C2×C4C2×Dic7⋊C411yescharacteristic
249C2×C4#648712yes—
250C2×C4C4×C22⋊D712yes—
251C2×C4#649211yescharacteristic
252C8—284no—
253C8—282no—
254C7#891411yescharacteristic, complemented, Sylow
255C22—44no—
256C22—42no—
257C22—81no—
258C22—82no—
259C22—81no—
260C22#8998211yescharacteristic
261C22#9001311yescharacteristic
262C22#9001412yes—
263C22#9001411yescharacteristic
264C22#9001712yes—
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—82no—
271C4—82no—
272C4—1121no—
273C2—82no—
274C2#1172940111yescharacteristic
275C2#1172941912yes—
276C2#1172970311yescharacteristic
277C2#1172970812yes—
278C2#1172970911yescharacteristic
subgroupG/H
centerC23#6516
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2379C2
socleC23×C7#120
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2379D7
hypocenterC7#8914
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
4. #11727302C2semidirect
5. #11725436C2semidirect
17. #89335C4semidirect
46. #6377C2×C4semidirect
254. C7#8914semidirect

Maximal subgroups

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

subgroupindex
C7×#23792
#117294802
C2×#893352
#117273022
#117254362
#89147