Finite Group Databasebeta

← #13264115

Subgroups — #13264115

Order 1792.

432 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2628C211yesmaximal, characteristic
2#11729341C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725454C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C7×#117C2211yescharacteristic
7C22×#1136—21no—
8C7×#400—21no—
9C7×#399—21no—
10C7×#395C2212yes—
11C7×#393C412yes—
12C7×#389—21no—
13C7×#382C411yescharacteristic
14C7×#380—21no—
15#89697—21no—
16#89325—21no—
17#89325C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11724—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C23×C4×C7D411yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C7×#117—41no—
26C7×#117—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4C2311yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4C2×C411yescharacteristic
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C42×C7C2×C411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×#1136—24no—
39C2×#1136—22no—
40C2×#1136—41no—
41C2×#1136—42no—
42C22×C7⋊C8—41no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95—22no—
46C7×#95C2×C412yes—
47C22⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—41no—
49C4×C7⋊C8—41no—
50#2628D711yescharacteristic
51#1742—71no—
52#1458—71no—
53#1354—74no—
54C24×C7C22⋊C411yescharacteristic
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C22×C4×C7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7SD1612yes—
66C22×C4×C7Q1612yes—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—82no—
77C7×C22⋊C4—41no—
78C7×C22⋊C4—41no—
79C7×C22⋊C4—82no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7—21no—
84#1136—24no—
85#1136—44no—
86#1136—42no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—44no—
89C2×C7⋊C8—42no—
90C22×M16—141no—
91C2×C4⋊Q8Dic711yescharacteristic
92C2×#117C2×D711yescharacteristic
93#431—141no—
94#400—21no—
95#399—21no—
96#395C2×D712yes—
97#393Dic712yes—
98#389—21no—
99#382Dic711yescharacteristic
100#380—21no—
101#343—72no—
102#343—141no—
103#324—72no—
104#323—141no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—41no—
108C23×C7—41no—
109C23×C7#10312yes—
110C23×C7C2×C22⋊C411yescharacteristic
111C7×Q8—84no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7#9911yescharacteristic
133C2×C4×C7#9911yescharacteristic
134C2×C4×C7#10312yes—
135C2×C4×C7#10412yes—
136C2×C4×C7C2×SD1611yescharacteristic
137C2×C4×C7C2×Q1611yescharacteristic
138C7⋊C8—44no—
139C7⋊C8—82no—
140C22×Q8C2×Dic712yes—
141C23×C4C22⋊D711yescharacteristic
142C2×M16—144no—
143C2×M16—142no—
144C2×M16—281no—
145C2×M16—282no—
146C22×C8—281no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D711yescharacteristic
156C2×C4⋊C4C22×D711yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4C2×Dic711yescharacteristic
159C2×C22⋊C4C22⋊D711yescharacteristic
160C2×C42C2×Dic711yescharacteristic
161C2×C42C22⋊D711yescharacteristic
162C4⋊C8—281no—
163C22⋊C8—142no—
164C4×C8—281no—
165#95—21no—
166#95—22no—
167#95—22no—
168#95C2×Dic712yes—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—22no—
172C22×C7—42no—
173C22×C7—41no—
174C22×C7—41no—
175C22×C7—81no—
176C22×C7C2×#9911yescharacteristic
177C22×C7C2×#10311yescharacteristic
178C22×C7C2×#10411yescharacteristic
179C4×C7—21no—
180C4×C7—21no—
181C4×C7—41no—
182C4×C7—41no—
183C4×C7—41no—
184C4×C7—41no—
185C4×C7—82no—
186C4×C7—82no—
187C4×C7#32812yes—
188C4×C7#32812yes—
189C24C22⋊Dic711yescharacteristic
190C2×Q8—42no—
191C2×Q8—82no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4#114312yes—
198C22×C4#114412yes—
199C22×C4C22⋊Dic711yescharacteristic
200C22×C4C22⋊Dic711yescharacteristic
201C22×C4C22⋊Dic711yescharacteristic
202C22×C4C22×Dic711yescharacteristic
203C22×C4C2×C22⋊D711yescharacteristic
204C22×C4C2×C22⋊D711yescharacteristic
205M16—144no—
206M16—284no—
207M16—282no—
208C2×C8—282no—
209C2×C8—284no—
210C2×C8—282no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—82no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C2×C7—21no—
226C2×C7—21no—
227C2×C7—42no—
228C2×C7C2×#32811yescharacteristic
229C2×C7#212211yescharacteristic
230C2×C7#213511yescharacteristic
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23#640912yes—
236C23C2×C22⋊Dic711yescharacteristic
237Q8—84no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4#640811yescharacteristic
259C2×C4#640811yescharacteristic
260C2×C4#640912yes—
261C2×C4#641112yes—
262C2×C4C2×#114311yescharacteristic
263C2×C4C2×#114411yescharacteristic
264C8—284no—
265C8—562no—
266C7#1172411yescharacteristic, complemented, Sylow
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#640811yescharacteristic
275C22C2×#640911yescharacteristic
276C22C2×#641111yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#8933312yes—
286C4#8933312yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#8933311yescharacteristic
291C2#1172961311yescharacteristic
292C2#1172962711yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2628C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2628D7
hypocenterC7#11724
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
4. #11725454C2semidirect
266. C7#11724semidirect

Maximal subgroups

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

subgroupindex
C7×#26282
#117293412
#117293252
#117254542
#117247