Finite Group Databasebeta

← #13251645

Subgroups — #13251645

Order 1792.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2972C211yesmaximal, characteristic, complemented
2#11739861C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic, complemented
4#11728516C212yesmaximal, complemented
5#11725544C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6501—21no—
8C2×C4×D28C2211yescharacteristic
9C7×C8×D4—24no—
10C7×#431C2212yes—
11C2×C7×C22⋊C8—22no—
12C2×#6407—21no—
13#89985—22no—
14#89759—21no—
15#89732C412yes—
16#89725C412yes—
17#89708C2211yescharacteristic
18#89696C2212yes—
19#89325—21no—
20#89254—24no—
21#9485—71nomaximal, Sylow
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D412yes—
24C22×C22⋊D7—22no—
25C22×D28C2×C411yescharacteristic
26C22×C4×D7—21no—
27C22×C7×C8—24no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C22⋊C4C2×C412yes—
32C2×C42×C7C2311yescharacteristic
33C2×C22⋊Dic7—22no—
34#6501—42no—
35#6501—42no—
36C2×(C2×D7)⋊C4—21no—
37C2×(C2×D7)⋊C4—22no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7C2×C411yescharacteristic
40C2×#1136—22no—
41C4×D28—22no—
42C4×D28—22no—
43C7×C4⋊C8—22no—
44C7×C22⋊C8—24no—
45C7×C22⋊C8—22no—
46C4×C7×C8D414yescomplemented
47#6407—42no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C7⋊#95C2×C412yes—
51C4⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)—22no—
53#2972D711yescharacteristic, complemented
54#2535—71no—
55#1800—71no—
56#1486—72no—
57#1286—72no—
58C24×C7C22⋊C412yes—
59C23×D7—21no—
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4D812yes—
63C2×C7×D4SD1612yes—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—82no—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C2×C22⋊D7—44no—
72C2×C22⋊D7—44no—
73C2×C22⋊D7—82no—
74C22×Dic7—22no—
75C22×Dic7—21no—
76C2×D28—22no—
77C2×D28—42no—
78C2×C4×D7—42no—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C2×C4×D7—42no—
82C2×C7×C8—24no—
83C2×C7×C8—84no—
84C7×C4⋊C4—22no—
85C7×C22⋊C4—24no—
86C42×C7C2×D412yes—
87C42×C7#4112yes—
88C22⋊Dic7—44no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—82no—
92C4⋊Dic7—22no—
93C4⋊Dic7—41no—
94C4⋊Dic7—41no—
95#1136—42no—
96#1136—42no—
97C2×C7⋊C8—42no—
98C2×#121—141no—
99C2×C4×D4—71no—
100C2×C4×D4C2×D711yescharacteristic
101C8×D4—24no—
102#431C2×D712yes—
103#423—72no—
104C2×#100—141no—
105C2×C22⋊C8—22no—
106#397—72no—
107#392—72no—
108#386—142no—
109#382—71no—
110#380—141no—
111#343—141no—
112D4⋊C8—144no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—42no—
116C23×C7—42no—
117C23×C7#10214yes—
118C23×C7C2×C22⋊C411yescharacteristic
119C22×D7—22no—
120C22×D7—41no—
121C22×D7—41no—
122C22×D7—41no—
123C22×D7—41no—
124C22×D7—41no—
125C22×D7—41no—
126C7×D4—24no—
127C7×D4—24no—
128C2×C4×C7—22no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—44no—
132C2×C4×C7—42no—
133C2×C4×C7—82no—
134C2×C4×C7C4×D412yes—
135C2×C4×C7#12011yescharacteristic
136C2×C4×C7#12112yes—
137C2×C4×C7#12311yescharacteristic
138C2×C4×C7C2×D811yescharacteristic
139C2×C4×C7C2×SD1611yescharacteristic
140C2×C4×C7#13512yes—
141C22⋊D7—88no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—41no—
145C2×Dic7—42no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148D28—24no—
149D28—42no—
150C4×D7—82no—
151C4×D7—82no—
152C7×C8—44no—
153C7⋊C8—82no—
154C22×D4—71no—
155C22×D4—142no—
156C22×D4D2811yescharacteristic
157C23×C4—141no—
158C23×C4C22⋊D712yes—
159C2×M16—142no—
160C22×C8—24no—
161#121—282no—
162#121—282no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—142no—
166C4×D4—142no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4D2811yescharacteristic
170C2×C22⋊C4—142no—
171C2×C22⋊C4—141no—
172C2×C22⋊C4—142no—
173C2×C22⋊C4C4×D712yes—
174C2×C42C22×D711yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—142no—
177C4⋊C8—142no—
178#100—282no—
179C22⋊C8—24no—
180C22⋊C8—22no—
181C4×C8C22⋊D714yescomplemented
182#95—72no—
183#95—142no—
184#95—141no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—24no—
188C22×C7—44no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—82no—
192C22×C7#38611yescharacteristic
193C22×C7C2×#10211yescharacteristic
194C22×C7#41611yescharacteristic
195C2×D7—42no—
196C2×D7—42no—
197C2×D7—42no—
198C2×D7—81no—
199C2×D7—81no—
200C2×D7—81no—
201C2×D7—81no—
202C4×C7—22no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—42no—
206C4×C7#46512yes—
207C4×C7#46612yes—
208Dic7—82no—
209Dic7—81no—
210C24—141no—
211C24(C2×D7)⋊C412yes—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—142no—
215C2×D4—282no—
216C2×D4—284no—
217C2×D4—284no—
218C2×D4—562no—
219C2×D4#113212yes—
220C2×D4D5612yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—82no—
224C22×C4—142no—
225C22×C4—141no—
226C22×C4—281no—
227C22×C4—281no—
228C22×C4—282no—
229C22×C4—282no—
230C22×C4(C2×D7)⋊C412yes—
231C22×C4C2×C4×D711yescharacteristic
232C22×C4C2×D2811yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—282no—
235M16—282no—
236C2×C8—24no—
237C2×C8—84no—
238C2×C8—282no—
239C4⋊C4—22no—
240C4⋊C4—142no—
241C4⋊C4—281no—
242C4⋊C4—281no—
243C22⋊C4—24no—
244C22⋊C4—284no—
245C22⋊C4—281no—
246C22⋊C4—281no—
247C22⋊C4—562no—
248C42#115712yes—
249C42C2×C22⋊D712yes—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7—44no—
253C2×C7#189111yescharacteristic
254C2×C7#189311yescharacteristic
255C2×C7#191111yescharacteristic
256D7—82no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—142no—
262C23—281no—
263C23—281no—
264C23—281no—
265C23—281no—
266C23—281no—
267C23—281no—
268C23#639514yes—
269C23C2×(C2×D7)⋊C411yescharacteristic
270D4—24no—
271D4—24no—
272D4—144no—
273D4—282no—
274D4—568no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—282no—
282C2×C4—282no—
283C2×C4—281no—
284C2×C4—282no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4—562no—
288C2×C4—562no—
289C2×C4C2×#113211yescharacteristic
290C2×C4C2×D5611yescharacteristic
291C2×C4#646712yes—
292C2×C4C4×C22⋊D712yes—
293C2×C4#649211yescharacteristic
294C2×C4#649312yes—
295C2×C4#651611yescharacteristic
296C8—44no—
297C8—562no—
298C7#948511yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—282no—
307C22—282no—
308C22—282no—
309C22—561no—
310C22—561no—
311C22—561no—
312C22—561no—
313C22C2×#639511yescharacteristic
314C22#8988511yescharacteristic
315C22#8998511yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—562no—
321C4—561no—
322C4#8988612yes—
323C4#8988712yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—562no—
328C2#1172940611yescharacteristic
329C2#1172972711yescharacteristic
330C2#1172978311yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2972C2
socleC22×C7#386
prime radical#11739861C2
cosocleC2×C42×C7C23
hypercenter#2972D7
hypocenterC7#9485
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. C7×#2972C2semidirect
3. #11729371C2semidirect
4. #11728516C2semidirect
5. #11725544C2semidirect
46. C4×C7×C8D4semidirect
53. #2972D7semidirect
181. C4×C8C22⋊D7semidirect
298. C7#9485semidirect

Maximal subgroups

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

subgroupindex
C7×#29722
#117398612
#117293712
#117285162
#117255442
#94857