Finite Group Databasebeta

← #13242517

Subgroups — #13242517

Order 1792.

604 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2683C211yesmaximal, characteristic, complemented
2#11727412C211yesmaximal, characteristic
3#11727381C212yesmaximal, complemented
4#11725445C211yesmaximal, characteristic, complemented
5#11725444C212yesmaximal, complemented
6C2×C7×C22⋊Q8—22no—
7C22×#1141—21no—
8C2×#6437C2211yescharacteristic
9C7×#396—22no—
10C7×#395—21no—
11C7×#390C2211yescharacteristic
12C7×#389C2212yes—
13C7×#386—21no—
14C2×#6383—22no—
15C2×#6383—22no—
16C2×#6383—21no—
17#89319C2212yes—
18#89319C2211yescharacteristic
19#7738—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C22×D28—21no—
24C22×D28—22no—
25C7×C22⋊Q8—44no—
26C7×C22⋊Q8—44no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—21no—
35C2×C42×C7C2311yescharacteristic
36C2×#1141—42no—
37C2×#1141—42no—
38C2×#1141—42no—
39C22×C7⋊C8—21no—
40C22×C7⋊C8—22no—
41#6437—42no—
42C7×#95—21no—
43C7×#95—22no—
44#6383—42no—
45#6383—44no—
46#6383—44no—
47#2683D711yescharacteristic, complemented
48#2041—72no—
49#2011—71no—
50#1271—72no—
51#1270—71no—
52C24×C7—21no—
53C23×D7—41no—
54C2×C7×Q8—44no—
55C2×C7×Q8—81no—
56C2×C7×D4—21no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4SD1612yes—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—21no—
63C22×C4×C7—44no—
64C22×C4×C7—81no—
65C22×C4×C7C2×D412yes—
66C22×C4×C7C2×D411yescharacteristic
67C2×D28—22no—
68C2×D28—24no—
69C2×D28—41no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—41no—
73C2×D28—42no—
74C2×D28—41no—
75C2×D28—42no—
76C2×D28—41no—
77C2×D28—42no—
78C2×D28—41no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—44no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4D814yescomplemented
86C7×C22⋊C4—44no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—81no—
90C42×C7—41no—
91#1141—48no—
92C2×C7⋊C8—22no—
93C2×C7⋊C8—41no—
94C2×C7⋊C8—41no—
95C2×C7⋊C8—42no—
96C2×C7⋊C8—42no—
97C2×C7⋊C8—42no—
98C22×D8—141no—
99C2×#127—71no—
100C2×C22⋊Q8—22no—
101C2×#102—141no—
102C2×#102—142no—
103C2×#102—142no—
104#396—22no—
105#395—21no—
106#390C2×D711yescharacteristic
107#389C2×D712yes—
108#386—21no—
109#336—71no—
110#336—72no—
111C23×C7—22no—
112C23×C7—42no—
113C23×C7—41no—
114C23×C7—42no—
115C23×C7—41no—
116C23×C7#12011yescharacteristic
117C22×D7—42no—
118C22×D7—82no—
119C22×D7—81no—
120C22×D7—81no—
121C22×D7—82no—
122C7×Q8—84no—
123C7×D4—42no—
124C7×D4—81no—
125C2×C4×C7—21no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—24no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—84no—
138C2×C4×C7C2×D812yes—
139C2×C4×C7C2×SD1611yescharacteristic
140C2×C4×C7AGL(1,ℤ/8)12yes—
141C2×C4×C7#13711yescharacteristic
142D28—42no—
143D28—42no—
144D28—44no—
145D28—44no—
146D28—81no—
147D28—81no—
148D28—82no—
149D28—82no—
150C7⋊C8—42no—
151C7⋊C8—82no—
152C22×Q8—21no—
153C22×D4—141no—
154C22×D4—142no—
155C22×D4C22⋊D711yescharacteristic
156C23×C4—21no—
157C2×D8—282no—
158C2×D8—282no—
159C2×D8—282no—
160C22×C8—141no—
161C22×C8—142no—
162#127—282no—
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊D712yes—
168C2×C4⋊C4C22⋊D712yes—
169C2×C4⋊C4C22⋊D711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C42C22×D711yescharacteristic
174#102—282no—
175#102—284no—
176#102—284no—
177#95—21no—
178#95—22no—
179C22×C7—42no—
180C22×C7—44no—
181C22×C7—82no—
182C22×C7—81no—
183C22×C7—81no—
184C22×C7#44712yes—
185C22×C7#44811yescharacteristic
186C22×C7#45311yescharacteristic
187C22×C7#45412yes—
188C22×C7B(4,𝔽2)11yescharacteristic
189C2×D7—82no—
190C2×D7—84no—
191C2×D7—161no—
192C2×D7—162no—
193C2×D7—161no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—22no—
197C4×C7—22no—
198C4×C7—44no—
199C4×C7—82no—
200C4×C7—81no—
201C24—21no—
202C24—281no—
203C2×Q8—44no—
204C2×Q8—81no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—144no—
209C2×D4—142no—
210C2×D4—281no—
211C2×D4—281no—
212C2×D4—282no—
213C2×D4—282no—
214C2×D4—282no—
215C2×D4—281no—
216C2×D4—282no—
217C2×D4—282no—
218C2×D4—281no—
219C2×D4—281no—
220C2×D4#114312yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—44no—
225C22×C4—81no—
226C22×C4C2×C22⋊D712yes—
227C22×C4C2×C22⋊D711yescharacteristic
228D8—288no—
229C2×C8—142no—
230C2×C8—281no—
231C2×C8—281no—
232C2×C8—282no—
233C2×C8—282no—
234C2×C8—282no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4#114114yescomplemented
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C42—41no—
247C2×C7—82no—
248C2×C7#159611yescharacteristic
249C2×C7#159812yes—
250C2×C7#161911yescharacteristic
251C2×C7#162712yes—
252C2×C7#165711yescharacteristic
253D7—162no—
254C23—22no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—282no—
260C23—562no—
261C23—561no—
262C23—561no—
263C23—562no—
264C23#651611yescharacteristic
265Q8—84no—
266D4—42no—
267D4—81no—
268D4—282no—
269D4—284no—
270D4—284no—
271D4—282no—
272D4—562no—
273D4—562no—
274D4—561no—
275D4—561no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—84no—
289C2×C4C2×#114112yes—
290C2×C4#649512yes—
291C2×C4C2×#114311yescharacteristic
292C2×C4#650511yescharacteristic
293C8—282no—
294C8—562no—
295C7#773811yescharacteristic, complemented, Sylow
296C22—44no—
297C22—42no—
298C22—81no—
299C22—82no—
300C22—81no—
301C22—564no—
302C22—562no—
303C22—1122no—
304C22—1121no—
305C22—1121no—
306C22#8998912yes—
307C22#8999911yescharacteristic
308C22#9001311yescharacteristic
309C22#9001611yescharacteristic
310C22#9001712yes—
311C4—22no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—44no—
316C4—82no—
317C4—81no—
318C2—82no—
319C2—1122no—
320C2#1172963212yes—
321C2#1172964611yescharacteristic
322C2#1172966011yescharacteristic
323C2#1172966812yes—
324C2#1172970911yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2683C2
socleC23×C7#120
prime radical#11727412C2
hypercenter#2683D7
hypocenterC7#7738
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. C7×#2683C2semidirect
3. #11727381C2semidirect
4. #11725445C2semidirect
5. #11725444C2semidirect
47. #2683D7semidirect
85. C7×C4⋊C4D8semidirect
241. C4⋊C4#1141semidirect
295. C7#7738semidirect

Maximal subgroups

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

subgroupindex
C7×#26832
#117274122
#117273812
#117254452
#117254442
#77387