Finite Group Databasebeta

← #13251550

Subgroups — #13251550

Order 1792.

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

normal subgroups only

This group has 726 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
1C7×#2310C211yesmaximal, characteristic, complemented
2#11741511C211yesmaximal, characteristic
3#11729887C211yesmaximal, characteristic, complemented
4#11727285C212yesmaximal
5#11725434C212yesmaximal, complemented
6C23×C7×Q8C2211yescharacteristic
7C2×C7×#117—21no—
8C2×#6510C412yescomplemented
9C2×#6493C2211yescharacteristic
10C7×#391—24no—
11C7×#381—21no—
12C7×#381C2212yes—
13#90003—22no—
14#90002C412yescomplemented
15C2×#6410—22no—
16#89985—21no—
17#89985—21no—
18C4×C22⋊Dic7C2211yescharacteristic
19#89979C2212yes—
20#89246—24no—
21#9466—71nomaximal, Sylow
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×Q8D412yes—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C22×C22⋊D7—21no—
31C22×D28C2×C411yescharacteristic
32C7×#117—44no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C42×C7—22no—
36C2×C22⋊Dic7—21no—
37#6510—44no—
38#6493—22no—
39#6493—22no—
40C2×(C2×D7)⋊C4—22no—
41C2×(C2×D7)⋊C4—22no—
42C2×(C2×D7)⋊C4—21no—
43C2×C4⋊Dic7C2×C411yescharacteristic
44C2×C4×Dic7C2×C412yes—
45C2×#1136—22no—
46C7×#95—24no—
47C7×#95—22no—
48C7×#95D414yes—
49#6410—44no—
50C7⋊#95—22no—
51C7⋊#95—22no—
52C7⋊#95—21no—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—22no—
55#2436—71no—
56#2310D711yescharacteristic, complemented
57#1784—71no—
58#1510—72no—
59#1284—72no—
60C24×C7C2×D411yescharacteristic
61C23×D7—21no—
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—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—24no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—24no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C412yes—
80C22×C4×C7C22⋊C412yes—
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D412yes—
84C22×C4×C7#4112yes—
85C2×C22⋊D7—44no—
86C2×C22⋊D7—41no—
87C2×C22⋊D7—41no—
88C22×Dic7—21no—
89C22×Dic7—22no—
90C2×D28—22no—
91C2×D28—42no—
92C7×C4⋊C4—44no—
93C7×C22⋊C4—44no—
94C42×C7—44no—
95C22⋊Dic7—42no—
96(C2×D7)⋊C4—44no—
97(C2×D7)⋊C4—44no—
98(C2×D7)⋊C4—42no—
99C4⋊Dic7—22no—
100C4×Dic7—42no—
101#1136—44no—
102C2×C7⋊C8—42no—
103C23×Q8C2×D711yescharacteristic
104C2×#124—72no—
105C2×#121—71no—
106C2×#117—21no—
107C2×#101—142no—
108#407—72no—
109#393—142no—
110#391—24no—
111#391—72no—
112#386—141no—
113#386—141no—
114#381—21no—
115#381C2×D712yes—
116C4×C22⋊C4—71no—
117#327—144no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—41no—
121C23×C7C2×C22⋊C411yescharacteristic
122C23×C7#12011yescharacteristic
123C23×C7#12311yescharacteristic
124C22×D7—42no—
125C22×D7—42no—
126C22×D7—41no—
127C22×D7—42no—
128C7×Q8—48no—
129C7×Q8—48no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—24no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—22no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7—44no—
147C2×C4×C7#9914yes—
148C2×C4×C7C4×D412yes—
149C2×C4×C7#12112yes—
150C22⋊D7—82no—
151C22⋊D7—82no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—41no—
156C2×Dic7—42no—
157D28—82no—
158C7⋊C8—82no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—44no—
162C22×Q8C4×D712yes—
163C22×Q8D2812yes—
164C22×D4—71no—
165C22×D4—141no—
166C23×C4C22⋊D711yescharacteristic
167C23×C4C22⋊D711yescharacteristic
168C23×C4C22×D711yescharacteristic
169C2×M16—142no—
170#124—284no—
171#121—142no—
172#121—142no—
173#117—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—71no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—141no—
178C2×C22⋊C4—142no—
179C2×C22⋊C4—142no—
180C2×C22⋊C4—141no—
181C2×C42—22no—
182C2×C42—72no—
183#101—284no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#95—24no—
187#95—22no—
188#95—142no—
189#95—142no—
190#95—141no—
191#95C22⋊D714yes—
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#38611yescharacteristic
201C22×C7C2×#9912yes—
202C22×C7#45412yes—
203C2×D7—84no—
204C2×D7—81no—
205C2×D7—81no—
206C2×D7—81no—
207C4×C7—22no—
208C4×C7—22no—
209C4×C7—24no—
210C4×C7—42no—
211C4×C7—84no—
212Dic7—82no—
213Dic7—81no—
214C24—141no—
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—142no—
226C2×D4—282no—
227C2×D4—284no—
228C2×D4—281no—
229C2×D4—281no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—141no—
239C22×C4—142no—
240C22×C4(C2×D7)⋊C412yes—
241C22×C4(C2×D7)⋊C412yes—
242C22×C4C2×C4×D711yescharacteristic
243C22×C4C2×D2811yescharacteristic
244C22×C4#115712yes—
245C22×C4C2×C22⋊D712yes—
246M16—284no—
247C2×C8—282no—
248C4⋊C4—44no—
249C4⋊C4—142no—
250C22⋊C4—44no—
251C22⋊C4—284no—
252C22⋊C4—282no—
253C22⋊C4—282no—
254C22⋊C4—284no—
255C42—44no—
256C42—282no—
257C2×C7—22no—
258C2×C7—22no—
259C2×C7—41no—
260C2×C7#188811yescharacteristic
261C2×C7#189811yescharacteristic
262C2×C7#191211yescharacteristic
263D7—161no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—282no—
268C23—282no—
269C23—282no—
270C23—281no—
271C23C2×(C2×D7)⋊C411yescharacteristic
272C23#649211yescharacteristic
273C23#651611yescharacteristic
274Q8—48no—
275Q8—48no—
276D4—562no—
277D4—562no—
278D4—562no—
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—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—281no—
301C2×C4#638014yes—
302C2×C4C4×C22⋊D712yes—
303C2×C4#649312yes—
304C8—562no—
305C7#946611yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—564no—
315C22—561no—
316C22—561no—
317C22—561no—
318C22C2×#638012yes—
319C22#8998511yescharacteristic
320C22#9001712yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—84no—
326C4—562no—
327C4—561no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—1121no—
332C2#1172929711yescharacteristic
333C2#1172941511yescharacteristic
334C2#1172941711yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2310C2
socleC22×C7#386
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#2310D7
hypocenterC7#9466
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
1. C7×#2310C2semidirect
3. #11729887C2semidirect
5. #11725434C2semidirect
8. C2×#6510C4semidirect
14. #90002C4semidirect
56. #2310D7semidirect
305. C7#9466semidirect

Maximal subgroups

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

subgroupindex
C7×#23102
#117415112
#117298872
#117272852
#117254342
#94667