Finite Group Databasebeta

← #13251508

Subgroups — #13251508

Order 1792.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#11739921C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic, complemented
3C7×#1467C211yesmaximal, characteristic, complemented
4#11728572C212yesmaximal
5#11725555C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6510C412yes—
8C2×#6437C2211yescharacteristic
9C7×#432C2212yes—
10#89796—24no—
11#89763—21no—
12#89763—21no—
13#89760C412yes—
14#89711—21no—
15#89711—21no—
16C4×C4⋊Dic7C2211yescharacteristic
17#89698—21no—
18#89698C2212yes—
19#89697—21no—
20C7×#324—21no—
21C7×#324—21no—
22#89280—24no—
23#9458—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×D28—21no—
26C22×D28—21no—
27C22×D28C2×C412yes—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36#6510—44no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—22no—
40C2×(C2×D7)⋊C4—22no—
41C2×C4⋊Dic7—21no—
42C2×C4×Dic7C2×C412yes—
43#6437—22no—
44#6437—22no—
45C7×C4⋊C8—22no—
46C7×C22⋊C8—22no—
47C7×#97—22no—
48C7⋊#95—21no—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)D414yes—
53#6377—22no—
54#6377—22no—
55C4×C7⋊C8—22no—
56#2983—71no—
57#2548—71no—
58#1555—72no—
59#1467D711yescharacteristic, complemented
60#1285—72no—
61C23×D7—22no—
62C2×C7×Q8—22no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×Dic7—22no—
73C2×D28—24no—
74C2×D28—42no—
75C2×D28—42no—
76C2×D28—42no—
77C2×D28—42no—
78C2×D28—41no—
79C2×D28—42no—
80C2×D28—41no—
81C2×D28—82no—
82C2×C7×C8—42no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7C2×D412yes—
91C42×C7#4112yes—
92(C2×D7)⋊C4—44no—
93(C2×D7)⋊C4—44no—
94(C2×D7)⋊C4—82no—
95(C2×D7)⋊C4—82no—
96C4⋊Dic7—42no—
97C4×Dic7—42no—
98C2×C7⋊C8—24no—
99C2×#127—71no—
100C2×#124—72no—
101C2×C4×Q8C2×D711yescharacteristic
102#446—144no—
103#432—72no—
104#432—141no—
105#432C2×D712yes—
106#431—141no—
107#394—141no—
108#394—141no—
109#390—72no—
110#390—141no—
111#390—141no—
112C4×C4⋊C4—71no—
113#331—144no—
114#324—21no—
115#324—21no—
116C23×C7C2×C22⋊C411yescharacteristic
117C22×D7—42no—
118C22×D7—42no—
119C22×D7—42no—
120C22×D7—42no—
121C22×D7—42no—
122C22×D7—42no—
123C22×D7—42no—
124C7×Q8—44no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—22no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7#10012yes—
139C2×C4×C7C4×D412yes—
140C2×C4×C7#12011yescharacteristic
141C2×C4×C7#12112yes—
142C2×C4×C7#12311yescharacteristic
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146D28—84no—
147D28—82no—
148D28—84no—
149D28—82no—
150C7×C8—82no—
151C7⋊C8—44no—
152C22×Q8C4×D711yescharacteristic
153C22×D4—72no—
154C22×D4—141no—
155C22×D4—141no—
156#127—142no—
157#127—142no—
158#124—284no—
159C4×Q8—22no—
160C4×Q8—22no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4C4×D711yescharacteristic
163C2×C4⋊C4C22⋊D711yescharacteristic
164C2×C4⋊C4C22⋊D711yescharacteristic
165C2×C22⋊C4—142no—
166C2×C22⋊C4—142no—
167C2×C22⋊C4—142no—
168C2×C22⋊C4—142no—
169C2×C42—72no—
170C2×C42D2811yescharacteristic
171C2×C42D2811yescharacteristic
172C2×C42C22×D711yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—74no—
175C4⋊C8—142no—
176C4⋊C8—142no—
177C22⋊C8—22no—
178C22⋊C8—142no—
179#97—22no—
180#97—142no—
181#97—142no—
182C4×C8—142no—
183#95—141no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7#38611yescharacteristic
187C22×C7C2×#10011yescharacteristic
188C22×C7#41311yescharacteristic
189C2×D7—82no—
190C2×D7—82no—
191C2×D7—82no—
192C2×D7—82no—
193C2×D7—82no—
194C2×D7—82no—
195C2×D7—82no—
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—22no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7#45412yes—
204Dic7—82no—
205C24—142no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×D4—144no—
210C2×D4—282no—
211C2×D4—282no—
212C2×D4—282no—
213C2×D4—282no—
214C2×D4—281no—
215C2×D4—281no—
216C2×D4—282no—
217C2×D4—562no—
218C22×C4—142no—
219C22×C4(C2×D7)⋊C411yescharacteristic
220C22×C4(C2×D7)⋊C411yescharacteristic
221C22×C4(C2×D7)⋊C411yescharacteristic
222C22×C4(C2×D7)⋊C411yescharacteristic
223C22×C4C2×C4×D711yescharacteristic
224C22×C4C2×D2811yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C2×C8—42no—
227C2×C8—144no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—282no—
232C22⋊C4—284no—
233C22⋊C4—284no—
234C22⋊C4—562no—
235C22⋊C4—562no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—282no—
241C42D4×D712yes—
242C42#115912yes—
243C2×C7—21no—
244C2×C7—21no—
245C2×C7#188811yescharacteristic
246C2×C7#190011yescharacteristic
247C2×C7#191011yescharacteristic
248D7—162no—
249C23—282no—
250C23—282no—
251C23—282no—
252C23—282no—
253C23—282no—
254C23—282no—
255C23—282no—
256C23C2×(C2×D7)⋊C411yescharacteristic
257Q8—44no—
258D4—564no—
259D4—562no—
260D4—564no—
261D4—562no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—282no—
276C2×C4—282no—
277C2×C4—282no—
278C2×C4#639712yes—
279C2×C4(C22⋊D7)⋊C412yes—
280C2×C4#644511yescharacteristic
281C2×C4#644911yescharacteristic
282C2×C4#650212yes—
283C8—82no—
284C8—284no—
285C7#945811yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22—562no—
289C22—562no—
290C22—562no—
291C22—562no—
292C22—562no—
293C22—562no—
294C22—562no—
295C22#8973011yescharacteristic
296C22C2×#639711yescharacteristic
297C22#8990411yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—562no—
306C4#8952512yes—
307C2—21no—
308C2—21no—
309C2—1122no—
310C2#1172773911yescharacteristic
311C2#1172780111yescharacteristic
312C2#1172785811yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1467C2
socleC22×C7#386
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenter#1467D7
hypocenterC7#9458
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #11739881C2semidirect
3. C7×#1467C2semidirect
5. #11725555C2semidirect
59. #1467D7semidirect
285. C7#9458semidirect

Maximal subgroups

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

subgroupindex
#117399212
#117398812
C7×#14672
#117285722
#117255552
#94587