Finite Group Databasebeta

← #13264496

Subgroups — #13264496

Order 1792.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2619C211yesmaximal, characteristic
2#11729343C211yesmaximal, characteristic, complemented
3#11729326C211yesmaximal, characteristic, complemented
4#11725453C214yesmaximal, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#121—21no—
7C2×C7×#117C2211yescharacteristic
8C7×#394—21no—
9C7×#394C2212yes—
10C7×#392—21no—
11C7×#392C412yes—
12C7×#390—21no—
13C7×#382C411yescharacteristic
14#89979—21no—
15#89698—21no—
16#89325—21no—
17#89325C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11806—71nomaximal, Sylow
21C22×C7×D4—22no—
22C22×C7×D4—21no—
23C22×C7×D4—21no—
24C22×C7×D4C2×C412yes—
25C23×C4×C7D411yescharacteristic
26C7×#127—41no—
27C7×#127—41no—
28C7×#121—44no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C4⋊C4C2311yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C42×C7C2×C411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×#1136—22no—
42C7×#95—21no—
43C7×#95C2×C412yes—
44C22⋊(C7⋊C8)—22no—
45C22⋊(C7⋊C8)—41no—
46C22⋊(C7⋊C8)—41no—
47C4⋊(C7⋊C8)—41no—
48#6377—41no—
49#2619D711yescharacteristic
50#1822—71no—
51#1464—71no—
52#1345—74no—
53C24×C7—22no—
54C24×C7C22⋊C411yescharacteristic
55C2×C7×D4—42no—
56C2×C7×D4—42no—
57C2×C7×D4—42no—
58C2×C7×D4—42no—
59C2×C7×D4—44no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4—82no—
63C2×C7×D4—82no—
64C2×C7×D4—82no—
65C2×C7×D4—82no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—41no—
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—42no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—82no—
84C7×C22⋊C4—82no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89#1136—42no—
90#1136—42no—
91C2×C7⋊C8—42no—
92C2×C7⋊C8—42no—
93C2×#127Dic711yescharacteristic
94C2×#121—21no—
95C2×#117C2×D711yescharacteristic
96#432—141no—
97#407—141no—
98#394—21no—
99#394C2×D712yes—
100#392—21no—
101#392Dic712yes—
102#390—21no—
103#382Dic711yescharacteristic
104#343—72no—
105#343—141no—
106#324—72no—
107#323—141no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—42no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7C2×C22⋊C411yescharacteristic
121C7×D4—82no—
122C7×D4—82no—
123C7×D4—84no—
124C7×D4—84no—
125C7×D4—88no—
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—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7AGL(1,ℤ/8)12yes—
146C7⋊C8—82no—
147C7⋊C8—82no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×Dic712yes—
152C23×C4C22⋊D711yescharacteristic
153C2×M16—142no—
154#127—41no—
155#127—41no—
156#121—44no—
157#117—41no—
158#117—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C22⋊D711yescharacteristic
161C2×C4⋊C4C22×D711yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4C2×Dic711yescharacteristic
166C2×C22⋊C4C22⋊D711yescharacteristic
167C2×C42C2×Dic711yescharacteristic
168C2×C42C22⋊D711yescharacteristic
169C4⋊C8—281no—
170C22⋊C8—142no—
171C22⋊C8—281no—
172C22⋊C8—281no—
173#97—281no—
174#95—21no—
175#95C2×Dic712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—42no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—81no—
183C22×C7—82no—
184C22×C7—82no—
185C22×C7—82no—
186C22×C7—82no—
187C22×C7—82no—
188C22×C7—82no—
189C22×C7—82no—
190C22×C7C2×#9911yescharacteristic
191C22×C7#41811yescharacteristic
192C22×C7#42111yescharacteristic
193C4×C7—22no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—82no—
201C24—22no—
202C24C22⋊Dic711yescharacteristic
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—44no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—82no—
211C2×D4—82no—
212C2×D4—82no—
213C2×D4—82no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22×Dic711yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224C22×C4C2×C22⋊D711yescharacteristic
225M16—282no—
226M16—282no—
227C2×C8—282no—
228C2×C8—282no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—44no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C2×C7—21no—
242C2×C7—21no—
243C2×C7—81no—
244C2×C7—162no—
245C2×C7#151011yescharacteristic
246C2×C7#212211yescharacteristic
247C2×C7#213411yescharacteristic
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23C2×C22⋊Dic711yescharacteristic
261D4—82no—
262D4—82no—
263D4—84no—
264D4—84no—
265D4—88no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4#640811yescharacteristic
284C2×C4#640811yescharacteristic
285C2×C4#649512yes—
286C8—562no—
287C8—562no—
288C7#1180611yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22#8998711yescharacteristic
304C22C2×#640811yescharacteristic
305C22#9000811yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—162no—
318C2#1172942811yescharacteristic
319C2#1172961311yescharacteristic
320C2#1172961611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
Fitting, prime radicalC7×#2619C2
socleC22×C7C2×#99
cosocleC2×C7×C4⋊C4C23
hypercenter#2619D7
hypocenterC7#11806
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
2. #11729343C2semidirect
3. #11729326C2semidirect
4. #11725453C2semidirect
288. C7#11806semidirect

Maximal subgroups

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

subgroupindex
C7×#26192
#117293432
#117293262
#117254532
#118067