Finite Group Databasebeta

← #13264849

Subgroups — #13264849

Order 1792.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11739921C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic, complemented
3#11729378C211yesmaximal, characteristic, complemented
4#11725299C214yesmaximal, complemented
5C2×C4×C7×Q8C2211yescharacteristic
6C2×#6510—21no—
7C2×#6437C411yescharacteristic
8C7×#423—21no—
9C2×C7×#101—21no—
10#89763C412yes—
11#89763C2212yes—
12#89760—21no—
13#89711—21no—
14#89711—21no—
15C4×C4⋊Dic7C411yescharacteristic
16#89697—21no—
17C7×#343C2212yes—
18#89323—21no—
19#89323C2212yes—
20#11882—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×D28—21no—
23C22×D28—21no—
24C22×D28C2×C412yes—
25C2×C7×M16—22no—
26C4×C7×Q8—41no—
27C4×C7×Q8—41no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7C2×C411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7D411yescharacteristic
34#6510—42no—
35#6510—42no—
36C2×(C2×D7)⋊C4—22no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—22no—
40C2×C4⋊Dic7—21no—
41C2×C4×Dic7—21no—
42#6437—41no—
43#6437—41no—
44C7×C4⋊C8—41no—
45C7×C4⋊C8—41no—
46C7×#101—42no—
47C7⋊#95C2×C412yes—
48C22⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—41no—
50C4×C7⋊C8—41no—
51#2548—71no—
52#1801D711yescharacteristic, complemented
53#1461—71no—
54#1345—74no—
55C23×D7—22no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C2×C7×Q8D812yes—
59C2×C7×Q8SD1612yes—
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22×C411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×Dic7—22no—
68C2×D28—42no—
69C2×D28—42no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28—41no—
74C2×D28—41no—
75C2×D28—42no—
76C2×D28—82no—
77C7×M16—42no—
78C7×M16—42no—
79C2×C7×C8—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89(C2×D7)⋊C4—44no—
90(C2×D7)⋊C4—42no—
91(C2×D7)⋊C4—42no—
92(C2×D7)⋊C4—82no—
93(C2×D7)⋊C4—82no—
94C4⋊Dic7—42no—
95C4×Dic7—42no—
96C2×C7⋊C8—42no—
97C2×#127—71no—
98C2×#124—141no—
99C2×C4×Q8C2×D711yescharacteristic
100#431—141no—
101#423—21no—
102C2×#101—21no—
103#394—72no—
104#394—72no—
105#390—141no—
106#390—141no—
107#390—141no—
108C4×C4⋊C4—71no—
109#343C2×D712yes—
110#324—72no—
111#324—141no—
112C23×C7C2×C22⋊C411yescharacteristic
113C22×D7—42no—
114C22×D7—42no—
115C22×D7—42no—
116C22×D7—42no—
117C22×D7—42no—
118C22×D7—42no—
119C22×D7—42no—
120C7×Q8—42no—
121C7×Q8—42no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7#9911yescharacteristic
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7#10212yes—
138C2×C4×C7#10212yes—
139C2×C4×C7C2×D811yescharacteristic
140C2×C4×C7C2×SD1611yescharacteristic
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144D28—82no—
145D28—84no—
146D28—84no—
147D28—82no—
148C7×C8—82no—
149C7⋊C8—82no—
150C22×Q8C22⋊D711yescharacteristic
151C22×D4—72no—
152C22×D4—141no—
153C22×D4—141no—
154C2×M16—22no—
155#127—281no—
156#127—281no—
157#124—282no—
158#124—282no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4C4×D711yescharacteristic
163C2×C4⋊C4D2811yescharacteristic
164C2×C4⋊C4C22×D711yescharacteristic
165C2×C22⋊C4—142no—
166C2×C22⋊C4—142no—
167C2×C22⋊C4—142no—
168C2×C22⋊C4—142no—
169C2×C42—141no—
170C2×C42C4×D711yescharacteristic
171C2×C42D2811yescharacteristic
172C2×C42C22⋊D711yescharacteristic
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—281no—
176#101—42no—
177C22⋊C8—142no—
178C4×C8—281no—
179#95—72no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7C2×#9911yescharacteristic
183C22×C7C2×#10211yescharacteristic
184C22×C7#42111yescharacteristic
185C2×D7—82no—
186C2×D7—82no—
187C2×D7—82no—
188C2×D7—82no—
189C2×D7—82no—
190C2×D7—82no—
191C2×D7—82no—
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#35512yes—
201Dic7—82no—
202C24—142no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8#114112yes—
206C2×Q8#114312yes—
207C2×D4—282no—
208C2×D4—282no—
209C2×D4—282no—
210C2×D4—282no—
211C2×D4—282no—
212C2×D4—281no—
213C2×D4—281no—
214C2×D4—282no—
215C2×D4—562no—
216C22×C4—142no—
217C22×C4(C2×D7)⋊C411yescharacteristic
218C22×C4(C2×D7)⋊C411yescharacteristic
219C22×C4(C2×D7)⋊C411yescharacteristic
220C22×C4(C2×D7)⋊C411yescharacteristic
221C22×C4C2×C4×D711yescharacteristic
222C22×C4C2×D2811yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224M16—42no—
225M16—42no—
226C2×C8—42no—
227C2×C8—282no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—282no—
232C22⋊C4—282no—
233C22⋊C4—282no—
234C22⋊C4—284no—
235C22⋊C4—562no—
236C22⋊C4—562no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—282no—
244C2×C7—21no—
245C2×C7—21no—
246C2×C7#151911yescharacteristic
247C2×C7#212211yescharacteristic
248C2×C7C2×#35511yescharacteristic
249D7—162no—
250C23—282no—
251C23—282no—
252C23—282no—
253C23—282no—
254C23—282no—
255C23—282no—
256C23—282no—
257C23C2×(C2×D7)⋊C411yescharacteristic
258Q8—42no—
259Q8—42no—
260D4—562no—
261D4—564no—
262D4—564no—
263D4—562no—
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—282no—
278C2×C4—282no—
279C2×C4—282no—
280C2×C4#638011yescharacteristic
281C2×C4#638011yescharacteristic
282C2×C4#638312yes—
283C2×C4#638312yes—
284C2×C4C2×#114111yescharacteristic
285C2×C4C2×#114311yescharacteristic
286C8—82no—
287C8—562no—
288C7#1188211yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—562no—
292C22—562no—
293C22—562no—
294C22—562no—
295C22—562no—
296C22—562no—
297C22—562no—
298C22C2×#638011yescharacteristic
299C22C2×#638311yescharacteristic
300C22#8977711yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—562no—
310C4#8927212yes—
311C2—21no—
312C2—21no—
313C2—1122no—
314C2#1172726311yescharacteristic
315C2C2×#8927211yescharacteristic
316C2#1172790011yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1801C2
socleC22×C7C2×#99
prime radicalC2×#6437C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1801D7
hypocenterC7#11882
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C7×#1801C2semidirect
3. #11729378C2semidirect
4. #11725299C2semidirect
52. #1801D7semidirect
288. C7#11882semidirect

Maximal subgroups

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

subgroupindex
#117399212
C7×#18012
#117293782
#117252992
#118827