Finite Group Databasebeta

← #13272119

Subgroups — #13272119

Order 1792.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C2×#89337C214yesmaximal, complemented
3#11729391C211yesmaximal, characteristic, complemented
4C4×#6410C211yesmaximal, characteristic, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#124C2212yes—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#394—21no—
9C7×#394—21no—
10C7×#390—21no—
11C7×#390—21no—
12C7×#390C412yes—
13C4×C7×C4⋊C4C411yescharacteristic
14C2×#6410C2212yes—
15C2×#6410C2212yes—
16#89696—21no—
17C4×#1136—21no—
18#89337—28no—
19#89325—21no—
20#89325—21no—
21#13226—71nomaximal, Sylow
22C22×C7×Q8C2311yescharacteristic
23C22×C7×D4—21no—
24C22×C7×D4—21no—
25C22×C7×D4C2×C412yes—
26C7×#127—41no—
27C7×#127—41no—
28C7×#124—24no—
29C7×#124—24no—
30C4×C7×Q8—41no—
31C4×C7×Q8—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C42×C7C2×C412yes—
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×#1136—22no—
45C2×#1136—22no—
46C7×#95—21no—
47#6410—22no—
48#6410—24no—
49#6410D414yescomplemented
50C4⋊(C7⋊C8)—41no—
51C4⋊(C7⋊C8)—41no—
52#6377—41no—
53C4×C7⋊C8—41no—
54#2548D711yescharacteristic
55C2×#355—74no—
56#1801—71no—
57C4×#101—71no—
58C24×C7—22no—
59C2×C7×Q8—41no—
60C2×C7×Q8—41no—
61C2×C7×Q8C2×D412yes—
62C2×C7×Q8#4112yes—
63C2×C7×D4—24no—
64C2×C7×D4—42no—
65C2×C7×D4—42no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—82no—
72C22×C4×C7—22no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—42no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—82no—
88C7×C22⋊C4—82no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—24no—
96#1136—24no—
97#1136—42no—
98#1136—42no—
99#1136—42no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102C2×#127Dic711yescharacteristic
103C2×#124C2×D712yes—
104C2×C4×Q8C2×D711yescharacteristic
105#423—141no—
106C2×#101—72no—
107C2×#101—72no—
108C4×M16—141no—
109#394—21no—
110#394—21no—
111#390—21no—
112#390—21no—
113#390Dic712yes—
114C4×C4⋊C4Dic711yescharacteristic
115#355—148no—
116#343—141no—
117#343—141no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7C2×C22⋊C411yescharacteristic
126C7×Q8—42no—
127C7×Q8—42no—
128C7×D4—84no—
129C7×D4—82no—
130C7×D4—84no—
131C7×D4—82no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—22no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7#9911yescharacteristic
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7C4×D412yes—
151C2×C4×C7#12112yes—
152C2×C4×C7#12411yescharacteristic
153C2×C4×C7#12711yescharacteristic
154C7⋊C8—44no—
155C7⋊C8—82no—
156C22×Q8C22×D711yescharacteristic
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×Dic712yes—
160C2×M16—142no—
161C2×M16—142no—
162#127—41no—
163#127—41no—
164#124—24no—
165#124—24no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×Dic711yescharacteristic
170C2×C4⋊C4C22⋊D711yescharacteristic
171C2×C4⋊C4C22⋊D711yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42C2×Dic712yes—
177C2×C42C2×Dic711yescharacteristic
178C2×C42C22⋊D711yescharacteristic
179C2×C42C22⋊D711yescharacteristic
180C4⋊C8—281no—
181C4⋊C8—281no—
182#101—74no—
183#101—142no—
184#101—144no—
185#97—281no—
186C4×C8—281no—
187#95—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—82no—
191C22×C7—82no—
192C22×C7—82no—
193C22×C7—82no—
194C22×C7—82no—
195C22×C7—82no—
196C22×C7—82no—
197C22×C7#39011yescharacteristic
198C22×C7C2×#9911yescharacteristic
199C22×C7#41011yescharacteristic
200C4×C7—21no—
201C4×C7—22no—
202C4×C7—21no—
203C4×C7—22no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—82no—
209C4×C7#35512yescomplemented
210C24—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8D4×D712yes—
214C2×Q8#115912yes—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—82no—
224C22×C4—22no—
225C22×C4C22⋊Dic711yescharacteristic
226C22×C4C22⋊Dic711yescharacteristic
227C22×C4C22⋊Dic711yescharacteristic
228C22×C4C22⋊Dic711yescharacteristic
229C22×C4C22×Dic711yescharacteristic
230C22×C4C2×C22⋊D711yescharacteristic
231C22×C4C2×C22⋊D711yescharacteristic
232M16—144no—
233M16—282no—
234M16—282no—
235M16—282no—
236C2×C8—282no—
237C2×C8—282no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—24no—
254C2×C7—21no—
255C2×C7—21no—
256C2×C7—162no—
257C2×C7#196411yescharacteristic
258C2×C7C2×#35511yescharacteristic
259C2×C7#213411yescharacteristic
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23C2×C22⋊Dic711yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—84no—
271D4—82no—
272D4—84no—
273D4—82no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4#640811yescharacteristic
291C2×C4#640811yescharacteristic
292C2×C4D4×Dic712yes—
293C2×C4#649911yescharacteristic
294C2×C4#650112yes—
295C2×C4#650311yescharacteristic
296C8—284no—
297C8—562no—
298C7#1322611yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×#640811yescharacteristic
309C22#8999311yescharacteristic
310C22#9001211yescharacteristic
311C4—22no—
312C4—21no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#8933712yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2#1172944011yescharacteristic
325C2C2×#8933711yescharacteristic
326C2#1172961611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC7×#2548C2
socleC22×C7C2×#99
prime radicalC2×C7×#127C4
cosocleC22×C7×Q8C23
hypercenter#2548D7
hypocenterC7#13226
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. C2×#89337C2semidirect
3. #11729391C2semidirect
4. C4×#6410C2semidirect
49. #6410D4semidirect
209. C4×C7#355semidirect
298. C7#13226semidirect
320. C4#89337semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
C2×#893372
#117293912
C4×#64102
#132267