Finite Group Databasebeta

← #13251646

Subgroups — #13251646

Order 1792.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2535C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic, complemented
3#11729371C211yesmaximal, characteristic, complemented
4#11729352C212yesmaximal, complemented
5#11725437C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C7×#397C412yes—
10C7×#392C412yes—
11C7×#386—22no—
12C7×#382C2211yescharacteristic
13C7×#380—21no—
14C2×#6407—21no—
15C2×C22⋊(C7⋊C8)—22no—
16D4×C7⋊C8—24no—
17#89697C2212yes—
18#89696C2212yes—
19D4⋊(C7⋊C8)—24no—
20#89325—21no—
21#9485—71nomaximal, Sylow
22C22×C7×D4—22no—
23C22×C7×D4C2×C411yescharacteristic
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7D412yes—
27C7×#121—42no—
28C7×#121—42no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4C2×C412yes—
40C2×C42×C7C2311yescharacteristic
41C2×#1136—22no—
42C22×C7⋊C8—24no—
43C7×#95—22no—
44C7×#95—21no—
45C7×#95C2×C412yes—
46#6407—42no—
47C22⋊(C7⋊C8)—24no—
48C22⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52C4×C7⋊C8D414yescomplemented
53#2972—71no—
54#2535D711yescharacteristic
55#1800—71no—
56#1486—72no—
57#1286—72no—
58C24×C7—21no—
59C24×C7C22⋊C412yes—
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—22no—
63C2×C7×D4—42no—
64C2×C7×D4—44no—
65C2×C7×D4—44no—
66C2×C7×D4—82no—
67C2×C7×D4D812yes—
68C2×C7×D4SD1612yes—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—21no—
73C22×C4×C7—42no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7—42no—
77C22×C4×C7—82no—
78C22×C4×C7C22⋊C412yes—
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×C4⋊C4—22no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—44no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—82no—
91C42×C7C2×D412yes—
92C42×C7#4112yes—
93#1136—42no—
94#1136—42no—
95C2×C7⋊C8—24no—
96C2×C7⋊C8—42no—
97C2×C7⋊C8—84no—
98C2×#121—21no—
99C2×C4×D4C2×D711yescharacteristic
100C2×C4×D4C2×D711yescharacteristic
101C8×D4—144no—
102#431—72no—
103#423—72no—
104C2×#100—141no—
105C2×C22⋊C8—142no—
106#397Dic712yes—
107#392Dic712yes—
108#386—22no—
109#382C2×D711yescharacteristic
110#380—21no—
111#343—141no—
112D4⋊C8—144no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—22no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7#10214yes—
125C23×C7C2×C22⋊C411yescharacteristic
126C7×D4—24no—
127C7×D4—24no—
128C7×D4—24no—
129C7×D4—42no—
130C7×D4—88no—
131C2×C4×C7—22no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—44no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—82no—
143C2×C4×C7—82no—
144C2×C4×C7—82no—
145C2×C4×C7C4×D412yes—
146C2×C4×C7#12011yescharacteristic
147C2×C4×C7#12112yes—
148C2×C4×C7#12311yescharacteristic
149C2×C4×C7C2×D811yescharacteristic
150C2×C4×C7C2×SD1611yescharacteristic
151C2×C4×C7#13512yes—
152C7⋊C8—44no—
153C7⋊C8—82no—
154C22×D4—22no—
155C22×D4C2×Dic711yescharacteristic
156C22×D4C22⋊D711yescharacteristic
157C23×C4—21no—
158C23×C4C22⋊D712yes—
159C2×M16—142no—
160C22×C8—144no—
161#121—42no—
162#121—42no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C2×Dic711yescharacteristic
169C2×C4⋊C4C22⋊D711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C2×Dic712yes—
174C2×C42C22×D711yescharacteristic
175C4⋊C8—142no—
176C4⋊C8—142no—
177C4⋊C8—142no—
178#100—282no—
179C22⋊C8—144no—
180C22⋊C8—142no—
181C4×C8—74no—
182#95—22no—
183#95—21no—
184#95C2×Dic712yes—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—24no—
188C22×C7—44no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—42no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—82no—
195C22×C7—81no—
196C22×C7—81no—
197C22×C7—81no—
198C22×C7—81no—
199C22×C7#38611yescharacteristic
200C22×C7C2×#10211yescharacteristic
201C22×C7#41611yescharacteristic
202C4×C7—22no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—42no—
206C4×C7—82no—
207C4×C7—81no—
208C4×C7#46512yes—
209C4×C7#46612yes—
210C24—21no—
211C24C22⋊Dic712yes—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—82no—
219C2×D4#114112yes—
220C2×D4#114212yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4C22⋊Dic712yes—
231C22×C4C22×Dic711yescharacteristic
232C22×C4C2×C22⋊D711yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—282no—
235M16—282no—
236C2×C8—144no—
237C2×C8—282no—
238C2×C8—564no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—24no—
244C22⋊C4—44no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—82no—
248C42D4×D712yes—
249C42#115912yes—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7—44no—
253C2×C7—82no—
254C2×C7#189111yescharacteristic
255C2×C7#189311yescharacteristic
256C2×C7#191111yescharacteristic
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#640614yes—
269C23C2×C22⋊Dic711yescharacteristic
270D4—24no—
271D4—24no—
272D4—24no—
273D4—42no—
274D4—88no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4C2×#114111yescharacteristic
290C2×C4C2×#114211yescharacteristic
291C2×C4D4×Dic712yes—
292C2×C4#649811yescharacteristic
293C2×C4#650011yescharacteristic
294C2×C4#650212yes—
295C2×C4#651312yes—
296C8—284no—
297C8—562no—
298C7#948511yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#640611yescharacteristic
314C22#8999211yescharacteristic
315C22#9000511yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4#8981012yes—
323C4#8981312yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#1172943511yescharacteristic
329C2#1172946811yescharacteristic
330C2#1172960711yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#2535C2
socleC22×C7#386
cosocleC2×C42×C7C23
hypercenter#2535D7
hypocenterC7#9485
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
2. #11739746C2semidirect
3. #11729371C2semidirect
4. #11729352C2semidirect
5. #11725437C2semidirect
52. C4×C7⋊C8D4semidirect
298. C7#9485semidirect

Maximal subgroups

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

subgroupindex
C7×#25352
#117397462
#117293712
#117293522
#117254372
#94857