Finite Group Databasebeta

← #13251656

Subgroups — #13251656

Order 1792.

518 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 119 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#11739749C211yesmaximal, 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—
15#89979—22no—
16#89784—24no—
17#89697C2212yes—
18#89696C2212yes—
19Q8⋊(C7⋊C8)—24no—
20#89325—21no—
21#9487—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—24no—
40C2×#1136—22no—
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#2975—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—21no—
61C2×C7×D4—22no—
62C2×C7×D4—21no—
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—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—82no—
80C7×C22⋊C4—24no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—82no—
83C42×C7—42no—
84C42×C7C2×D412yes—
85C42×C7#4112yes—
86#1136—42no—
87#1136—42no—
88#1136—84no—
89C2×C7⋊C8—24no—
90C2×C7⋊C8—42no—
91C2×C4×Q8C2×D711yescharacteristic
92C2×C4×D4C2×D711yescharacteristic
93#436—144no—
94#431—72no—
95#423—72no—
96C2×#100—141no—
97#407—142no—
98#399Dic712yes—
99#393Dic712yes—
100#390—21no—
101#389—21no—
102#385—22no—
103#382C2×D711yescharacteristic
104#343—141no—
105Q8⋊C8—144no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—22no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7C2×C22⋊C411yescharacteristic
112C7×Q8—24no—
113C7×Q8—42no—
114C7×D4—42no—
115C7×D4—42no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—44no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—82no—
131C2×C4×C7C4×D412yes—
132C2×C4×C7#12011yescharacteristic
133C2×C4×C7#12112yes—
134C2×C4×C7#12311yescharacteristic
135C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C7#13712yes—
137C2×C4×C7#13711yescharacteristic
138C7⋊C8—44no—
139C7⋊C8—82no—
140C22×Q8C2×Dic711yescharacteristic
141C22×D4C22⋊D711yescharacteristic
142C23×C4C22⋊D712yes—
143C2×M16—144no—
144C2×M16—142no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C2×Dic711yescharacteristic
153C2×C4⋊C4C22⋊D711yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4C2×Dic712yes—
157C2×C42—21no—
158C2×C42C22×D711yescharacteristic
159C4⋊C8—142no—
160C4⋊C8—142no—
161C4⋊C8—142no—
162#100—282no—
163C22⋊C8—144no—
164C22⋊C8—142no—
165C4×C8—74no—
166#95—22no—
167#95—21no—
168#95C2×Dic712yes—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—44no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—82no—
176C22×C7#38611yescharacteristic
177C22×C7#41711yescharacteristic
178C22×C7#41911yescharacteristic
179C4×C7—22no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—42no—
183C4×C7—42no—
184C4×C7—82no—
185C4×C7—81no—
186C4×C7#46812yes—
187C4×C7#47012yes—
188C24C22⋊Dic712yes—
189C2×Q8—22no—
190C2×Q8—42no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—82no—
201C22×C4C22⋊Dic712yes—
202C22×C4C22×Dic711yescharacteristic
203C22×C4C2×C22⋊D711yescharacteristic
204C22×C4C2×C22⋊D711yescharacteristic
205M16—282no—
206M16—282no—
207M16—564no—
208C2×C8—144no—
209C2×C8—282no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—82no—
217C22⋊C4—24no—
218C22⋊C4—44no—
219C22⋊C4—82no—
220C42—42no—
221C42D4×D712yes—
222C42#115912yes—
223C2×C7—21no—
224C2×C7—21no—
225C2×C7—82no—
226C2×C7#189111yescharacteristic
227C2×C7#189411yescharacteristic
228C2×C7#191111yescharacteristic
229C23—22no—
230C23—22no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23C2×C22⋊Dic711yescharacteristic
235Q8—24no—
236Q8—42no—
237D4—42no—
238D4—42no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4D4×Dic712yes—
255C2×C4#649811yescharacteristic
256C2×C4#650011yescharacteristic
257C2×C4#650212yes—
258C2×C4#650512yes—
259C2×C4#651211yescharacteristic
260C2×C4#651411yescharacteristic
261C8—284no—
262C8—562no—
263C7#948711yescharacteristic, complemented, Sylow
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#8999211yescharacteristic
272C22#8999711yescharacteristic
273C22#9000411yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4#8982112yes—
282C4#8982412yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#1172945111yescharacteristic
287C2#1172946811yescharacteristic
288C2#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#9487
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
263. C7#9487semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117397492
#117293762
#117293712
#117254372
#94877