Finite Group Databasebeta

← #13248174

Subgroups — #13248174

Order 1792.

468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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#11740402C211yesmaximal, characteristic
2C7×#1959C211yesmaximal, characteristic
3C2×#89335C211yesmaximal, characteristic
4#11727329C212yesmaximal, complemented
5#11725465C212yesmaximal, complemented
6C2×C7×#124C2211yescharacteristic
7C2×C7×#103—21no—
8C2×C7×#102—21no—
9C2×C7×#97C2211yescharacteristic
10C7×#384C2211yescharacteristic
11#90002—21no—
12#89992—21no—
13#89723—22no—
14#89722—22no—
15#89708C2212yes—
16C2×#6377C2212yes—
17#89335C414yescomplemented
18#89324—22no—
19#89324—22no—
20#8914—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×C7×D4D411yescharacteristic
23C23×Dic7—21no—
24C22×C7×C8—21no—
25C7×#124—22no—
26C7×#124C2×C412yes—
27C7×#124Q812yes—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C42×C7C2311yescharacteristic
32C2×C22⋊Dic7—22no—
33C2×C22⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7D412yes—
36C2×C4⋊Dic7D412yes—
37C2×C4×Dic7—21no—
38C22×C7⋊C8—22no—
39C7×#103—42no—
40C7×#102—42no—
41C7×#97—22no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45#6377—22no—
46#6377C2×C414yescomplemented
47#2379—71no—
48#1959D711yescharacteristic
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—41no—
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D412yes—
65C22×Dic7—22no—
66C22×Dic7—22no—
67C22×Dic7—81no—
68C22×Dic7—82no—
69C2×C7×C8—22no—
70C2×C7×C8—41no—
71C2×C7×C8—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—44no—
75C7×C22⋊C4—24no—
76C7×C22⋊C4—24no—
77C42×C7C22×C411yescharacteristic
78C42×C7C2×Q811yescharacteristic
79C42×C7#4112yes—
80C22⋊Dic7—81no—
81C22⋊Dic7—82no—
82C4⋊Dic7—42no—
83C4⋊Dic7—42no—
84C4⋊Dic7—81no—
85C4×Dic7—81no—
86C2×C7⋊C8—24no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C2×#124C2×D711yescharacteristic
90C2×#103—21no—
91C2×#102—21no—
92C2×#97—72no—
93C2×#97C2×D711yescharacteristic
94#391—141no—
95#388—142no—
96#386—141no—
97#385—142no—
98#384C2×D711yescharacteristic
99#382—72no—
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—81no—
121C2×C4×C7—82no—
122C2×C4×C7C2×C4⋊C411yescharacteristic
123C2×C4×C7C4×D412yes—
124C2×C4×C7C22⋊Q812yes—
125C2×C4×C7#12311yescharacteristic
126C2×Dic7—42no—
127C2×Dic7—42no—
128C2×Dic7—42no—
129C2×Dic7—42no—
130C2×Dic7—42no—
131C2×Dic7—42no—
132C2×Dic7—84no—
133C7×C8—42no—
134C7⋊C8—44no—
135C22×Q8C22⋊D711yescharacteristic
136C22×D4C22⋊D711yescharacteristic
137C23×C4—141no—
138C22×C8—21no—
139C22×C8—142no—
140#124—22no—
141#124Dic1412yes—
142#124C4×D712yes—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4—141no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—142no—
150C2×C22⋊C4—141no—
151C2×C42—141no—
152C2×C42C22×D711yescharacteristic
153#103—42no—
154#102—42no—
155#97—22no—
156#97—74no—
157#97—142no—
158#95—142no—
159#95—142no—
160#95—142no—
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#45512yes—
170C22×C7#45511yescharacteristic
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—22no—
174C4×C7—22no—
175C4×C7—42no—
176C4×C7—161no—
177Dic7—82no—
178Dic7—82no—
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—41no—
190C22×C4—142no—
191C22×C4—142no—
192C22×C4—562no—
193C22×C4—561no—
194C22×C4D4×D712yes—
195C22×C4C2×C22⋊D711yescharacteristic
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—144no—
200C2×C8—282no—
201C2×C8—282no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—282no—
206C4⋊C4—282no—
207C4⋊C4—561no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—562no—
211C22⋊C4—561no—
212C42—561no—
213C42C2×Dic1411yescharacteristic
214C42C2×C4×D711yescharacteristic
215C42#115912yes—
216C2×C7—82no—
217C2×C7#165711yescharacteristic
218C2×C7#165911yescharacteristic
219C2×C7#166212yes—
220C2×C7#185611yescharacteristic
221C2×C7#186012yes—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23#650011yescharacteristic
228Q8—42no—
229Q8—81no—
230D4—42no—
231D4—81no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—282no—
242C2×C4—282no—
243C2×C4—282no—
244C2×C4—282no—
245C2×C4—282no—
246C2×C4—282no—
247C2×C4—564no—
248C2×C4#644112yes—
249C2×C4(C22⋊D7)⋊C412yes—
250C2×C4C2×Dic7⋊C411yescharacteristic
251C2×C4#649811yescharacteristic
252C8—42no—
253C8—284no—
254C7#891411yescharacteristic, complemented, Sylow
255C22—42no—
256C22—44no—
257C22—82no—
258C22—81no—
259C22—81no—
260C22#8971811yescharacteristic
261C22#8983311yescharacteristic
262C22#8984911yescharacteristic
263C22#8997312yes—
264C22#8997412yes—
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—161no—
271C4—562no—
272C4—562no—
273C2—82no—
274C2#1172739611yescharacteristic
275C2#1172805212yes—
276C2#1172900012yes—
277C2#1172908211yescharacteristic
278C2#1172908311yescharacteristic
subgroupG/H
centerC23#6500
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1959C2
socleC23×C7#120
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#1959D7
hypocenterC7#8914
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
4. #11727329C2semidirect
5. #11725465C2semidirect
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
#117404022
C7×#19592
C2×#893352
#117273292
#117254652
#89147