Finite Group Databasebeta

← #13251551

Subgroups — #13251551

Order 1792.

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

normal subgroups only

This group has 726 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×#2436C211yesmaximal, characteristic
2#11741507C211yesmaximal, characteristic, complemented
3#11729887C211yesmaximal, characteristic, complemented
4#11729428C212yesmaximal
5#11725422C212yesmaximal, complemented
6C23×C7×Q8C2211yescharacteristic
7C2×C7×#124C412yescomplemented
8C2×C7×#121C2211yescharacteristic
9C2×#6489—21no—
10C7×#393—22no—
11C7×#391C412yescomplemented
12C7×#386—21no—
13C7×#386—21no—
14C4×C7×C22⋊C4C2211yescharacteristic
15#90002—24no—
16C2×#6410—22no—
17#89983—21no—
18#89983C2212yes—
19#89979C2212yes—
20#89332—24no—
21#9466—71nomaximal, Sylow
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×Q8D412yes—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C7×#124—44no—
33C7×#121—22no—
34C7×#121—22no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C42×C7C2×C412yes—
41C2×C22⋊Dic7—22no—
42#6489—44no—
43C2×C4⋊Dic7—22no—
44C2×C4×Dic7—22no—
45C2×#1136—22no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—21no—
49#6410—44no—
50C7⋊#95—24no—
51C7⋊#95—22no—
52C7⋊#95D414yes—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—22no—
55#2436D711yescharacteristic
56#2310—71no—
57#1784—71no—
58#1510—72no—
59#1284—72no—
60C24×C7—21no—
61C24×C7C2×D411yescharacteristic
62C2×C7×Q8—24no—
63C2×C7×Q8—24no—
64C2×C7×Q8—24no—
65C2×C7×Q8—24no—
66C2×C7×Q8—42no—
67C2×C7×Q8—42no—
68C2×C7×Q8—44no—
69C2×C7×Q8—44no—
70C2×C7×Q8—48no—
71C2×C7×D4—22no—
72C2×C7×D4—42no—
73C2×C7×D4—44no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—24no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7C22⋊C412yes—
86C22×C4×C7C22⋊C412yes—
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D412yes—
90C22×C4×C7#4112yes—
91C22×Dic7—24no—
92C7×C4⋊C4—22no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—44no—
96C7×C22⋊C4—42no—
97C42×C7—42no—
98C22⋊Dic7—44no—
99C4⋊Dic7—44no—
100C4×Dic7—44no—
101#1136—44no—
102C2×C7⋊C8—42no—
103C23×Q8C2×D711yescharacteristic
104C2×#124Dic712yescomplemented
105C2×#121C2×D711yescharacteristic
106C2×#117—141no—
107C2×#101—142no—
108#407—72no—
109#393—22no—
110#391—144no—
111#391Dic712yescomplemented
112#386—21no—
113#386—21no—
114#381—72no—
115#381—141no—
116C4×C22⋊C4C2×D711yescharacteristic
117#327—144no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—41no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—41no—
124C23×C7—42no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C7×Q8—48no—
129C7×Q8—48no—
130C7×D4—82no—
131C7×D4—82no—
132C7×D4—82no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—42no—
152C2×C4×C7#9914yes—
153C2×C4×C7C4×D412yes—
154C2×C4×C7#12112yes—
155C2×Dic7—44no—
156C2×Dic7—44no—
157C2×Dic7—44no—
158C7⋊C8—82no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—44no—
162C22×Q8C2×Dic712yes—
163C22×Q8C22⋊D712yes—
164C22×D4—21no—
165C22×D4C2×Dic711yescharacteristic
166C23×C4C22⋊D711yescharacteristic
167C23×C4C22⋊D711yescharacteristic
168C23×C4C22×D711yescharacteristic
169C2×M16—142no—
170#124—44no—
171#121—22no—
172#121—22no—
173#117—284no—
174C2×C4⋊C4—142no—
175C2×C4⋊C4C2×Dic711yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—142no—
181C2×C42—142no—
182C2×C42C2×Dic712yes—
183#101—284no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—74no—
190#95—144no—
191#95—142no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—44no—
200C22×C7—84no—
201C22×C7—81no—
202C22×C7—81no—
203C22×C7—81no—
204C22×C7#38611yescharacteristic
205C22×C7C2×#9912yes—
206C22×C7#45412yes—
207C4×C7—22no—
208C4×C7—24no—
209C4×C7—22no—
210C4×C7—42no—
211C4×C7—82no—
212C4×C7—81no—
213Dic7—84no—
214C24—21no—
215C24C2×C22⋊D711yescharacteristic
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—144no—
240C22×C4C22⋊Dic712yes—
241C22×C4C22⋊Dic712yes—
242C22×C4D4×D712yes—
243C22×C4#115912yes—
244C22×C4C22×Dic711yescharacteristic
245C22×C4C2×C22⋊D711yescharacteristic
246M16—284no—
247C2×C8—282no—
248C4⋊C4—22no—
249C4⋊C4—284no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—284no—
255C42—42no—
256C42—284no—
257C2×C7—22no—
258C2×C7—22no—
259C2×C7—41no—
260C2×C7—161no—
261C2×C7#188811yescharacteristic
262C2×C7#189811yescharacteristic
263C2×C7#191211yescharacteristic
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23#649811yescharacteristic
272C23#650011yescharacteristic
273C23C2×C22⋊Dic711yescharacteristic
274Q8—48no—
275Q8—48no—
276D4—82no—
277D4—82no—
278D4—82no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—284no—
299C2×C4—284no—
300C2×C4—284no—
301C2×C4#640814yes—
302C2×C4D4×Dic712yes—
303C2×C4#650212yes—
304C8—562no—
305C7#946611yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—84no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#8983412yes—
319C22C2×#640812yes—
320C22#8999211yescharacteristic
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—82no—
326C4—81no—
327C4—564no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2#1172942411yescharacteristic
333C2#1172947811yescharacteristic
334C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2436C2
socleC22×C7#386
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#2436D7
hypocenterC7#9466
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. #11741507C2semidirect
3. #11729887C2semidirect
5. #11725422C2semidirect
7. C2×C7×#124C4semidirect
11. C7×#391C4semidirect
104. C2×#124Dic7semidirect
111. #391Dic7semidirect
305. C7#9466semidirect

Maximal subgroups

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

subgroupindex
C7×#24362
#117415072
#117298872
#117294282
#117254222
#94667