Finite Group Databasebeta

← #13264850

Subgroups — #13264850

Order 1792.

476 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2548C211yesmaximal, characteristic
2#11729391C211yesmaximal, characteristic, complemented
3#11729378C211yesmaximal, characteristic, complemented
4#11725453C214yesmaximal, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#124—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#394C412yes—
9C7×#394C2212yes—
10C7×#390—21no—
11C7×#390—21no—
12C7×#390—21no—
13C4×C7×C4⋊C4C411yescharacteristic
14C2×#6410—21no—
15#89697—21no—
16#89696—21no—
17#89325C2212yes—
18#89323—21no—
19#89323C2212yes—
20#11882—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×C7×D4—21no—
23C22×C7×D4—21no—
24C22×C7×D4C2×C412yes—
25C7×#127—41no—
26C7×#127—41no—
27C7×#124—42no—
28C7×#124—42no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C42×C7—21no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×#1136—22no—
44C7×#95C2×C412yes—
45#6410—42no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48C4⋊(C7⋊C8)—41no—
49C4⋊(C7⋊C8)—41no—
50C4×C7⋊C8—41no—
51#2548D711yescharacteristic
52#1801—71no—
53#1461—71no—
54#1345—74no—
55C24×C7—22no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C2×C7×Q8D812yes—
59C2×C7×Q8SD1612yes—
60C2×C7×D4—42no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—42no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—82no—
69C22×C4×C7—22no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—42no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—42no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—82no—
85C7×C22⋊C4—82no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—42no—
93#1136—42no—
94#1136—42no—
95C2×C7⋊C8—42no—
96C2×C7⋊C8—42no—
97C2×#127Dic711yescharacteristic
98C2×#124—21no—
99C2×C4×Q8C2×D711yescharacteristic
100#431—141no—
101#423—141no—
102C2×#101—141no—
103#394Dic712yes—
104#394C2×D712yes—
105#390—21no—
106#390—21no—
107#390—21no—
108C4×C4⋊C4Dic711yescharacteristic
109#343—72no—
110#324—72no—
111#324—141no—
112C23×C7—42no—
113C23×C7—42no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—42no—
119C23×C7C2×C22⋊C411yescharacteristic
120C7×Q8—42no—
121C7×Q8—42no—
122C7×D4—82no—
123C7×D4—84no—
124C7×D4—84no—
125C7×D4—82no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#10212yes—
145C2×C4×C7#10212yes—
146C2×C4×C7C2×D811yescharacteristic
147C2×C4×C7C2×SD1611yescharacteristic
148C7⋊C8—82no—
149C7⋊C8—82no—
150C22×Q8C22⋊D711yescharacteristic
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C2×Dic712yes—
154C2×M16—142no—
155#127—41no—
156#127—41no—
157#124—42no—
158#124—42no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic711yescharacteristic
163C2×C4⋊C4C22⋊D711yescharacteristic
164C2×C4⋊C4C22×D711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—21no—
170C2×C42C2×Dic711yescharacteristic
171C2×C42C22⋊D711yescharacteristic
172C2×C42C22⋊D711yescharacteristic
173C4⋊C8—281no—
174C4⋊C8—281no—
175C4⋊C8—281no—
176#101—282no—
177C22⋊C8—142no—
178C4×C8—281no—
179#95C2×Dic712yes—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—82no—
183C22×C7—82no—
184C22×C7—82no—
185C22×C7—82no—
186C22×C7—82no—
187C22×C7—82no—
188C22×C7—82no—
189C22×C7C2×#9911yescharacteristic
190C22×C7C2×#10211yescharacteristic
191C22×C7#42111yescharacteristic
192C4×C7—21no—
193C4×C7—21no—
194C4×C7—22no—
195C4×C7—22no—
196C4×C7—21no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—82no—
201C4×C7#35512yes—
202C24—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8#114112yes—
206C2×Q8#114212yes—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—82no—
216C22×C4—22no—
217C22×C4C22⋊Dic711yescharacteristic
218C22×C4C22⋊Dic711yescharacteristic
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22×Dic711yescharacteristic
222C22×C4C2×C22⋊D711yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224M16—282no—
225M16—282no—
226C2×C8—282no—
227C2×C8—282no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C2×C7—21no—
245C2×C7—21no—
246C2×C7—162no—
247C2×C7#151911yescharacteristic
248C2×C7#212211yescharacteristic
249C2×C7C2×#35511yescharacteristic
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23C2×C22⋊Dic711yescharacteristic
258Q8—42no—
259Q8—42no—
260D4—84no—
261D4—82no—
262D4—84no—
263D4—82no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4#640612yes—
281C2×C4#640612yes—
282C2×C4#640811yescharacteristic
283C2×C4#640811yescharacteristic
284C2×C4C2×#114111yescharacteristic
285C2×C4C2×#114211yescharacteristic
286C8—562no—
287C8—562no—
288C7#1188211yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#640611yescharacteristic
299C22C2×#640811yescharacteristic
300C22#9000811yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4#8933712yes—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#1172942911yescharacteristic
315C2C2×#8933711yescharacteristic
316C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2548C2
socleC22×C7C2×#99
prime radicalC2×C7×#127C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2548D7
hypocenterC7#11882
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #11729391C2semidirect
3. #11729378C2semidirect
4. #11725453C2semidirect
288. C7#11882semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
#117293912
#117293782
#117254532
#118827