Finite Group Databasebeta

← #13263435

Subgroups — #13263435

Order 1792.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#11729379C211yesmaximal, characteristic, complemented
3#11729378C211yesmaximal, characteristic, complemented
4#11725446C214yesmaximal, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#124C412yes—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#394—21no—
9C7×#394—21no—
10C7×#390—21no—
11C7×#390—21no—
12C7×#390C2212yes—
13C4×C7×C4⋊C4C411yescharacteristic
14#89698—21no—
15#89697—21no—
16#89323—21no—
17#89323—21no—
18#89323C2212yes—
19#89323C2212yes—
20#11567—71nomaximal, Sylow
21C22×C7×Q8C2×C411yescharacteristic
22C22×C7×D4—21no—
23C22×C7×D4—21no—
24C22×C7×D4C2×C412yes—
25C7×#127—41no—
26C7×#127—41no—
27C7×#124—44no—
28C4×C7×Q8—41no—
29C4×C7×Q8—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4D411yescharacteristic
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⋊C4—22no—
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7C2×C412yes—
40C2×C42×C7D411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C7×#95—21no—
43C22⋊(C7⋊C8)—22no—
44C22⋊(C7⋊C8)—22no—
45C4⋊(C7⋊C8)—41no—
46C4⋊(C7⋊C8)—41no—
47#6377—41no—
48C4×C7⋊C8—41no—
49#2548D711yescharacteristic
50#1467—71no—
51#1461—71no—
52#1318—74no—
53C24×C7—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—41no—
57C2×C7×D4—24no—
58C2×C7×D4—42no—
59C2×C7×D4—42no—
60C2×C7×D4—42no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C2×C7×D4—82no—
66C22×C4×C7—22no—
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C7×C4⋊C4—41no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C22⋊C4—44no—
79C7×C22⋊C4—44no—
80C7×C22⋊C4—82no—
81C7×C22⋊C4—82no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—42no—
89C2×C7⋊C8—42no—
90C2×C7⋊C8—42no—
91C2×#127Dic711yescharacteristic
92C2×#124Dic712yes—
93C2×C4×Q8C2×D711yescharacteristic
94#432—141no—
95#431—141no—
96#394—21no—
97#394—21no—
98#390—21no—
99#390—21no—
100#390C2×D712yes—
101C4×C4⋊C4Dic711yescharacteristic
102#324—72no—
103#324—72no—
104#324—141no—
105#324—141no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7—42no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7—42no—
112C23×C7—42no—
113C23×C7C2×C22⋊C411yescharacteristic
114C7×Q8—44no—
115C7×D4—84no—
116C7×D4—82no—
117C7×D4—84no—
118C7×D4—82no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7#9911yescharacteristic
138C2×C4×C7#10012yes—
139C2×C4×C7#13112yes—
140C7⋊C8—82no—
141C7⋊C8—82no—
142C22×Q8C2×Dic711yescharacteristic
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C2×Dic712yes—
146#127—41no—
147#127—41no—
148#124—44no—
149C4×Q8—41no—
150C4×Q8—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4C22⋊D711yescharacteristic
153C2×C4⋊C4C22⋊D711yescharacteristic
154C2×C4⋊C4C22×D711yescharacteristic
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C42C2×Dic711yescharacteristic
160C2×C42C2×Dic712yes—
161C2×C42C22⋊D711yescharacteristic
162C2×C42C22⋊D711yescharacteristic
163C4⋊C8—281no—
164C4⋊C8—281no—
165C22⋊C8—142no—
166C22⋊C8—142no—
167#97—281no—
168C4×C8—281no—
169#95—21no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—82no—
173C22×C7—82no—
174C22×C7—82no—
175C22×C7—82no—
176C22×C7—82no—
177C22×C7—82no—
178C22×C7—82no—
179C22×C7#40811yescharacteristic
180C22×C7C2×#9911yescharacteristic
181C22×C7C2×#10011yescharacteristic
182C4×C7—21no—
183C4×C7—22no—
184C4×C7—22no—
185C4×C7—41no—
186C4×C7—41no—
187C4×C7—41no—
188C4×C7—41no—
189C4×C7—82no—
190C4×C7#35512yes—
191C24—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×D4—24no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—82no—
204C22×C4—22no—
205C22×C4C22⋊Dic711yescharacteristic
206C22×C4C22⋊Dic711yescharacteristic
207C22×C4C22⋊Dic711yescharacteristic
208C22×C4C22⋊Dic711yescharacteristic
209C22×C4C22×Dic711yescharacteristic
210C22×C4C2×C22⋊D711yescharacteristic
211C22×C4C2×C22⋊D711yescharacteristic
212C2×C8—282no—
213C2×C8—282no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—42no—
229C2×C7—21no—
230C2×C7—21no—
231C2×C7—162no—
232C2×C7#147111yescharacteristic
233C2×C7#212811yescharacteristic
234C2×C7C2×#35511yescharacteristic
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23C2×C22⋊Dic711yescharacteristic
243Q8—44no—
244D4—84no—
245D4—82no—
246D4—84no—
247D4—82no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4#640712yes—
266C2×C4#640811yescharacteristic
267C2×C4#640811yescharacteristic
268C2×C4#651112yes—
269C8—562no—
270C8—562no—
271C7#1156711yescharacteristic, complemented, Sylow
272C22—21no—
273C22—21no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22C2×#640711yescharacteristic
282C22C2×#640811yescharacteristic
283C22#9000611yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4#8933712yes—
293C2—21no—
294C2—21no—
295C2—162no—
296C2#1172943011yescharacteristic
297C2C2×#8933711yescharacteristic
298C2#1172961511yescharacteristic
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#11567
LCSG ▷ C22×C4×C7 ▷ C22×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. #11729379C2semidirect
3. #11729378C2semidirect
4. #11725446C2semidirect
271. C7#11567semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
#117293792
#117293782
#117254462
#115677