Finite Group Databasebeta

← #13251671

Subgroups — #13251671

Order 1792.

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

normal subgroups only

This group has 522 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×#2536C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic, complemented
3#11729391C211yesmaximal, characteristic, complemented
4#11729352C212yesmaximal
5#11725437C212yesmaximal, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C7×#398C412yes—
10C7×#394C412yes—
11C7×#388—22no—
12C7×#382C2211yescharacteristic
13C7×#380—21no—
14C2×#6410—21no—
15#89796—24no—
16#89698—22no—
17#89697C2212yes—
18#89696C2212yes—
19D4⋊(C7⋊C8)—24no—
20#89325—21no—
21#9490—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7—21no—
25C7×C22⋊Q8—44no—
26C4×C7×Q8—22no—
27C4×C7×Q8—22no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C42×C7D412yes—
39C2×C42×C7C2311yescharacteristic
40C2×#1136—22no—
41C7×#95—22no—
42C7×#95—21no—
43C7×#95C2×C412yes—
44#6410—42no—
45C22⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)—24no—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50#6377—24no—
51C4×C7⋊C8D414yes—
52#2983—71no—
53#2536D711yescharacteristic
54#1801—71no—
55#1486—72no—
56#1286—72no—
57C24×C7—21no—
58C2×C7×Q8—22no—
59C2×C7×Q8—21no—
60C2×C7×Q8—22no—
61C2×C7×Q8—21no—
62C2×C7×D4—22no—
63C2×C7×D4—42no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C7×C4⋊C4—24no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—44no—
79C7×C4⋊C4—42no—
80C7×C22⋊C4—44no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—41no—
84C42×C7—22no—
85C42×C7—22no—
86C42×C7C2×D412yes—
87C42×C7#4112yes—
88#1136—42no—
89#1136—42no—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—42no—
92C2×C22⋊Q8—21no—
93C2×C4×Q8C2×D711yescharacteristic
94C2×C4×D4C2×D711yescharacteristic
95#446—144no—
96#432—142no—
97#431—72no—
98#423—72no—
99C2×#101—141no—
100#398Dic712yes—
101#394Dic712yes—
102#388—22no—
103#382C2×D711yescharacteristic
104#380—21no—
105#343—141no—
106D4⋊C8—144no—
107C23×C7—22no—
108C23×C7—41no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7C2×C22⋊C411yescharacteristic
115C7×Q8—42no—
116C7×Q8—42no—
117C7×D4—24no—
118C7×D4—42no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—82no—
133C2×C4×C7—82no—
134C2×C4×C7C4×D412yes—
135C2×C4×C7#12011yescharacteristic
136C2×C4×C7#12112yes—
137C2×C4×C7#12311yescharacteristic
138C2×C4×C7AGL(1,ℤ/8)12yes—
139C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C7#13711yescharacteristic
141C7⋊C8—44no—
142C7⋊C8—82no—
143C22×Q8C22⋊D711yescharacteristic
144C22×D4C2×Dic711yescharacteristic
145C23×C4—21no—
146C2×M16—142no—
147C22⋊Q8—44no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×D4—22no—
151C4×D4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×Dic711yescharacteristic
155C2×C4⋊C4C2×Dic712yes—
156C2×C4⋊C4C22⋊D711yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C42C22⋊D712yes—
161C2×C42C22×D711yescharacteristic
162C4⋊C8—144no—
163C4⋊C8—142no—
164C4⋊C8—142no—
165C4⋊C8—142no—
166#101—282no—
167C22⋊C8—142no—
168#97—144no—
169C4×C8—74no—
170#95—22no—
171#95—21no—
172#95C2×Dic712yes—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—42no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—81no—
179C22×C7—81no—
180C22×C7—81no—
181C22×C7—81no—
182C22×C7#38611yescharacteristic
183C22×C7#41711yescharacteristic
184C22×C7#41811yescharacteristic
185C4×C7—22no—
186C4×C7—21no—
187C4×C7—21no—
188C4×C7—42no—
189C4×C7—42no—
190C4×C7—82no—
191C4×C7—81no—
192C4×C7#46812yes—
193C4×C7#46912yes—
194C24—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×D4—22no—
200C2×D4—42no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊Dic712yes—
208C22×C4C22⋊Dic712yes—
209C22×C4C22×Dic711yescharacteristic
210C22×C4C2×C22⋊D711yescharacteristic
211C22×C4C2×C22⋊D711yescharacteristic
212M16—282no—
213M16—282no—
214C2×C8—144no—
215C2×C8—282no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C42—22no—
226C42—22no—
227C42D4×D712yes—
228C42#115912yes—
229C2×C7—21no—
230C2×C7—21no—
231C2×C7—82no—
232C2×C7#189111yescharacteristic
233C2×C7#189511yescharacteristic
234C2×C7#191211yescharacteristic
235C23—22no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23C2×C22⋊Dic711yescharacteristic
243Q8—42no—
244Q8—42no—
245D4—24no—
246D4—42no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4#649512yes—
263C2×C4D4×Dic712yes—
264C2×C4#649811yescharacteristic
265C2×C4#650011yescharacteristic
266C2×C4#650212yes—
267C2×C4#651211yescharacteristic
268C2×C4#651411yescharacteristic
269C8—284no—
270C8—562no—
271C7#949011yescharacteristic, complemented, Sylow
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#8998711yescharacteristic
282C22#8999211yescharacteristic
283C22#9000411yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4#8981112yes—
292C4#8981412yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#1172943611yescharacteristic
297C2#1172946811yescharacteristic
298C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2536C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2536D7
hypocenterC7#9490
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #11739881C2semidirect
3. #11729391C2semidirect
5. #11725437C2semidirect
271. C7#9490semidirect

Maximal subgroups

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

subgroupindex
C7×#25362
#117398812
#117293912
#117293522
#117254372
#94907