Finite Group Databasebeta

← #13249820

Subgroups — #13249820

Order 1792.

524 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2621C211yesmaximal, characteristic
2#11739649C211yesmaximal, characteristic
3#11729527C212yesmaximal, complemented
4#11729325C211yesmaximal, characteristic
5#11725437C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C22⋊Q8—21no—
8C2×C7×#117C2211yescharacteristic
9C22×#1136—21no—
10C7×#398—21no—
11C7×#398C412yes—
12C7×#397—21no—
13C7×#397C412yes—
14C7×#391—21no—
15C7×#382C2211yescharacteristic
16#89771—24no—
17#89697C2212yes—
18#89696—21no—
19#89696C2212yes—
20#89342—24no—
21#89323—21no—
22#89320—21no—
23#9195—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C23×C4×C7D411yescharacteristic
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—22no—
28C7×C22⋊Q8—42no—
29C7×C22⋊Q8—42no—
30C7×#117—22no—
31C7×#117—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C2×C42×C7C2311yescharacteristic
43C2×#1136—22no—
44C2×#1136—22no—
45C2×#1136—42no—
46C2×#1136—42no—
47C22×C7⋊C8—22no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—21no—
51C7×#95C2×C412yes—
52C22⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—24no—
54C4⋊(C7⋊C8)—22no—
55C4⋊(C7⋊C8)—22no—
56C4⋊(C7⋊C8)D414yes—
57C4×C7⋊C8—22no—
58#2905—71no—
59#2621D711yescharacteristic
60#1546—72no—
61#1458—71no—
62#1286—72no—
63C24×C7C22⋊C411yescharacteristic
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×Q8—41no—
69C2×C7×Q8—41no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7D812yes—
80C22×C4×C7SD1612yes—
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—24no—
87C7×C4⋊C4—44no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—44no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7C2×D412yes—
97C42×C7#4112yes—
98#1136—44no—
99#1136—42no—
100#1136—42no—
101C2×C7⋊C8—24no—
102C2×C7⋊C8—44no—
103C2×C7⋊C8—42no—
104C22×M16—141no—
105C2×C4⋊Q8C2×D711yescharacteristic
106C2×C22⋊Q8—21no—
107C2×#117C2×D711yescharacteristic
108#431—72no—
109#424—144no—
110#423—72no—
111#423—141no—
112#398—21no—
113#398Dic712yes—
114#397—21no—
115#397Dic712yes—
116#391—21no—
117#382C2×D711yescharacteristic
118#332—144no—
119#324—141no—
120#323—141no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7#10212yes—
126C23×C7C2×C22⋊C411yescharacteristic
127C7×Q8—81no—
128C7×Q8—81no—
129C7×Q8—81no—
130C7×Q8—81no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
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#10212yes—
150C2×C4×C7C4×D412yes—
151C2×C4×C7#12011yescharacteristic
152C2×C4×C7#12112yes—
153C2×C4×C7#12311yescharacteristic
154C2×C4×C7C2×D811yescharacteristic
155C2×C4×C7C2×SD1611yescharacteristic
156C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C7#13711yescharacteristic
158C7⋊C8—44no—
159C7⋊C8—82no—
160C22×Q8—21no—
161C23×C4C22⋊D711yescharacteristic
162C2×M16—142no—
163C2×M16—142no—
164C2×M16—282no—
165C2×M16—282no—
166C22×C8—142no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×Dic712yes—
177C2×C4⋊C4C2×Dic711yescharacteristic
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4C2×Dic711yescharacteristic
181C2×C22⋊C4C22⋊D711yescharacteristic
182C2×C42C22⋊D711yescharacteristic
183C2×C42C22×D711yescharacteristic
184C4⋊C8—74no—
185C4⋊C8—144no—
186C4⋊C8—142no—
187C4⋊C8—142no—
188C22⋊C8—142no—
189C4×C8—142no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95C2×Dic712yes—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—22no—
197C22×C7—42no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—81no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10211yescharacteristic
203C22×C7#41711yescharacteristic
204C4×C7—22no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—82no—
210C4×C7—82no—
211C4×C7#44711yescharacteristic
212C4×C7#44811yescharacteristic
213C4×C7#48011yescharacteristic
214C4×C7#48111yescharacteristic
215C24C22⋊Dic711yescharacteristic
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4#114112yes—
229C22×C4#114212yes—
230C22×C4C22⋊Dic711yescharacteristic
231C22×C4C22⋊Dic711yescharacteristic
232C22×C4C22⋊Dic711yescharacteristic
233C22×C4C22×Dic711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235C22×C4C2×C22⋊D711yescharacteristic
236M16—284no—
237M16—282no—
238M16—282no—
239C2×C8—144no—
240C2×C8—284no—
241C2×C8—282no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—44no—
252C42—21no—
253C42—21no—
254C42D4×D712yes—
255C42#115912yes—
256C2×C7—21no—
257C2×C7—21no—
258C2×C7—42no—
259C2×C7#187411yescharacteristic
260C2×C7#190911yescharacteristic
261C2×C7#191511yescharacteristic
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23#640612yes—
267C23C2×C22⋊Dic711yescharacteristic
268Q8—81no—
269Q8—81no—
270Q8—81no—
271Q8—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4#640612yes—
291C2×C4C2×#114111yescharacteristic
292C2×C4C2×#114211yescharacteristic
293C2×C4D4×Dic712yes—
294C2×C4#649811yescharacteristic
295C2×C4#650011yescharacteristic
296C2×C4#650212yes—
297C2×C4#651211yescharacteristic
298C2×C4#651411yescharacteristic
299C8—284no—
300C8—562no—
301C7#919511yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22C2×#640611yescharacteristic
310C22#8999211yescharacteristic
311C22#9000411yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—82no—
318C4—82no—
319C4#8980511yescharacteristic
320C4#8980611yescharacteristic
321C4#8980811yescharacteristic
322C4#8981211yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2#1172943111yescharacteristic
327C2#1172961911yescharacteristic
328C2#1172962011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2621C2
socleC22×C7#386
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2621D7
hypocenterC7#9195
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #11729527C2semidirect
5. #11725437C2semidirect
301. C7#9195semidirect

Maximal subgroups

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

subgroupindex
C7×#26212
#117396492
#117295272
#117293252
#117254372
#91957