Finite Group Databasebeta

← #13264322

Subgroups — #13264322

Order 1792.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2611C211yesmaximal, characteristic
2#11729341C211yesmaximal, characteristic
3#11729326C211yesmaximal, characteristic
4#11725458C212yesmaximal, complemented
5#11725456C212yesmaximal, complemented
6C2×C7×#125C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C22×#1136—21no—
9C7×#400C2211yescharacteristic
10C7×#399C411yescharacteristic
11C7×#398C2211yescharacteristic
12C7×#397C411yescharacteristic
13C7×#389—21no—
14C7×#388—21no—
15C7×#387—21no—
16C7×#385—21no—
17C7×#376C411yescharacteristic
18#89698—21no—
19#89325—21no—
20#89325C2212yes—
21#89323C2212yes—
22#89320—21no—
23#11767—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C7×#125—41no—
26C7×#125—41no—
27C7×#117—41no—
28C7×#117—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4C2311yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7—21no—
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×#1136—24no—
45C2×#1136—22no—
46C2×#1136—41no—
47C2×#1136—42no—
48C22×C7⋊C8—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2611D711yescharacteristic
60#1742—71no—
61#1464—71no—
62#1353—72no—
63#1346—72no—
64C24×C7C22⋊C411yescharacteristic
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—42no—
99#1136—24no—
100#1136—44no—
101#1136—42no—
102C2×C7⋊C8—44no—
103C2×C7⋊C8—42no—
104C2×C7⋊C8—42no—
105C22×M16—141no—
106C2×#125Dic711yescharacteristic
107C2×#117C2×D711yescharacteristic
108#432—141no—
109#400C2×D711yescharacteristic
110#399Dic711yescharacteristic
111#398C2×D711yescharacteristic
112#397Dic711yescharacteristic
113#389—21no—
114#388—21no—
115#387—21no—
116#385—21no—
117#376Dic711yescharacteristic
118#343—72no—
119#343—141no—
120#324—72no—
121#323—141no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7#10412yes—
127C23×C7C2×C22⋊C411yescharacteristic
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7#9911yescharacteristic
157C2×C4×C7#9911yescharacteristic
158C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C7#13711yescharacteristic
160C7⋊C8—44no—
161C7⋊C8—82no—
162C23×C4C22⋊D711yescharacteristic
163C2×M16—144no—
164C2×M16—142no—
165C2×M16—281no—
166C2×M16—282no—
167C22×C8—281no—
168#125—41no—
169#125—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×Dic711yescharacteristic
177C2×C4⋊C4C2×Dic711yescharacteristic
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C4⋊C4C22×D711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4C2×Dic711yescharacteristic
183C2×C22⋊C4C22⋊D711yescharacteristic
184C2×C42—21no—
185C2×C42C2×Dic711yescharacteristic
186C2×C42C22⋊D711yescharacteristic
187C4⋊C8—281no—
188C22⋊C8—142no—
189#97—281no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C2×Dic711yescharacteristic
196#95C2×Dic711yescharacteristic
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—22no—
200C22×C7—42no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—81no—
204C22×C7C2×#9911yescharacteristic
205C22×C7#41711yescharacteristic
206C22×C7C2×#10411yescharacteristic
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—42no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—81no—
217C4×C7—81no—
218C4×C7—81no—
219C24C22⋊Dic711yescharacteristic
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C22⋊Dic711yescharacteristic
230C22×C4C22⋊Dic711yescharacteristic
231C22×C4C22⋊Dic711yescharacteristic
232C22×C4C22×Dic711yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235M16—144no—
236M16—284no—
237M16—282no—
238C2×C8—284no—
239C2×C8—282no—
240C2×C8—282no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—42no—
260C2×C7—21no—
261C2×C7—21no—
262C2×C7—42no—
263C2×C7#150411yescharacteristic
264C2×C7#212311yescharacteristic
265C2×C7#213211yescharacteristic
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23#641112yes—
271C23C2×C22⋊Dic711yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4#640811yescharacteristic
301C2×C4#640811yescharacteristic
302C2×C4#651211yescharacteristic
303C2×C4#651411yescharacteristic
304C8—284no—
305C8—562no—
306C7#1176711yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#640811yescharacteristic
315C22#9000411yescharacteristic
316C22C2×#641111yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#1172946211yescharacteristic
333C2#1172957411yescharacteristic
334C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2611C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2611D7
hypocenterC7#11767
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
4. #11725458C2semidirect
5. #11725456C2semidirect
306. C7#11767semidirect

Maximal subgroups

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

subgroupindex
C7×#26112
#117293412
#117293262
#117254582
#117254562
#117677