Finite Group Databasebeta

← #13251636

Subgroups — #13251636

Order 1792.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2580C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic
3#11729376C212yesmaximal, complemented
4#11729371C211yesmaximal, characteristic
5#11725437C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#399C412yes—
9C7×#393C412yes—
10C7×#390—21no—
11C7×#389—21no—
12C7×#385—22no—
13C7×#382C2211yescharacteristic
14C2×#6407—21no—
15C2×C22⋊(C7⋊C8)—22no—
16D4×C7⋊C8—24no—
17#89697C2212yes—
18#89696C2212yes—
19Q8⋊(C7⋊C8)—24no—
20#89325—21no—
21#9483—71nomaximal, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C22×C7×D4D411yescharacteristic
24C23×C4×C7D412yes—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C4×C7×D4—22no—
28C4×C7×D4—22no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4C2×C412yes—
37C2×C42×C7—21no—
38C2×C42×C7C2311yescharacteristic
39C2×#1136—22no—
40C22×C7⋊C8—24no—
41C7×#95—22no—
42C7×#95—21no—
43C7×#95C2×C412yes—
44#6407—42no—
45C22⋊(C7⋊C8)—24no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4×C7⋊C8D414yes—
51#2972—71no—
52#2580D711yescharacteristic
53#1800—71no—
54#1487—72no—
55#1286—72no—
56C24×C7C22⋊C412yes—
57C2×C7×Q8—22no—
58C2×C7×Q8—42no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4SD1612yes—
62C2×C7×D4Q1612yes—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—82no—
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7C22×C411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—82no—
79C7×C22⋊C4—24no—
80C7×C22⋊C4—44no—
81C7×C22⋊C4—82no—
82C42×C7—42no—
83C42×C7C2×D412yes—
84C42×C7#4112yes—
85#1136—42no—
86#1136—42no—
87C2×C7⋊C8—24no—
88C2×C7⋊C8—42no—
89C2×C7⋊C8—84no—
90C2×C4×Q8C2×D711yescharacteristic
91C2×C4×D4C2×D711yescharacteristic
92C8×D4—144no—
93#431—72no—
94#423—72no—
95C2×#100—141no—
96C2×C22⋊C8—142no—
97#399Dic712yes—
98#393Dic712yes—
99#390—21no—
100#389—21no—
101#385—22no—
102#382C2×D711yescharacteristic
103#343—141no—
104Q8⋊C8—144no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—42no—
108C23×C7—42no—
109C23×C7#10314yes—
110C23×C7C2×C22⋊C411yescharacteristic
111C7×Q8—24no—
112C7×Q8—42no—
113C7×D4—24no—
114C7×D4—24no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—44no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7—82no—
130C2×C4×C7C4×D412yes—
131C2×C4×C7#12011yescharacteristic
132C2×C4×C7#12112yes—
133C2×C4×C7#12311yescharacteristic
134C2×C4×C7C2×SD1611yescharacteristic
135C2×C4×C7C2×Q1611yescharacteristic
136C2×C4×C7#13512yes—
137C7⋊C8—44no—
138C7⋊C8—82no—
139C22×Q8C2×Dic711yescharacteristic
140C22×D4C22⋊D711yescharacteristic
141C23×C4C22⋊D712yes—
142C2×M16—142no—
143C22×C8—144no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×D4—22no—
147C4×D4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C2×Dic711yescharacteristic
152C2×C4⋊C4C22⋊D711yescharacteristic
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C2×Dic712yes—
156C2×C42—21no—
157C2×C42C22×D711yescharacteristic
158C4⋊C8—142no—
159C4⋊C8—142no—
160C4⋊C8—142no—
161#100—282no—
162C22⋊C8—144no—
163C22⋊C8—142no—
164C4×C8—74no—
165#95—22no—
166#95—21no—
167#95C2×Dic712yes—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—24no—
171C22×C7—44no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—82no—
175C22×C7#38611yescharacteristic
176C22×C7C2×#10311yescharacteristic
177C22×C7#41611yescharacteristic
178C4×C7—22no—
179C4×C7—21no—
180C4×C7—21no—
181C4×C7—42no—
182C4×C7—42no—
183C4×C7—82no—
184C4×C7—81no—
185C4×C7#46512yes—
186C4×C7#46712yes—
187C24C22⋊Dic712yes—
188C2×Q8—22no—
189C2×Q8—42no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4#114312yes—
193C2×D4#114412yes—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—82no—
200C22×C4C22⋊Dic712yes—
201C22×C4C22×Dic711yescharacteristic
202C22×C4C2×C22⋊D711yescharacteristic
203C22×C4C2×C22⋊D711yescharacteristic
204M16—282no—
205M16—282no—
206C2×C8—144no—
207C2×C8—282no—
208C2×C8—564no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C22⋊C4—82no—
218C42—42no—
219C42D4×D712yes—
220C42#115912yes—
221C2×C7—21no—
222C2×C7—21no—
223C2×C7—44no—
224C2×C7#189111yescharacteristic
225C2×C7#189211yescharacteristic
226C2×C7#191111yescharacteristic
227C23—22no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23#640914yes—
232C23C2×C22⋊Dic711yescharacteristic
233Q8—24no—
234Q8—42no—
235D4—24no—
236D4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4D4×Dic712yes—
253C2×C4#649811yescharacteristic
254C2×C4#650011yescharacteristic
255C2×C4#650212yes—
256C2×C4C2×#114311yescharacteristic
257C2×C4C2×#114411yescharacteristic
258C2×C4#651312yes—
259C8—284no—
260C8—562no—
261C7#948311yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#8999211yescharacteristic
270C22C2×#640911yescharacteristic
271C22#9000511yescharacteristic
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—42no—
277C4—82no—
278C4—81no—
279C4#8982012yes—
280C4#8982312yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#1172945011yescharacteristic
285C2#1172946811yescharacteristic
286C2#1172960711yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2580C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2580D7
hypocenterC7#9483
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #11729376C2semidirect
5. #11725437C2semidirect
261. C7#9483semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117397462
#117293762
#117293712
#117254372
#94837