Finite Group Databasebeta

← #13251601

Subgroups — #13251601

Order 1792.

818 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2406C211yesmaximal, characteristic
2#11739589C211yesmaximal, characteristic, complemented
3#11729892C212yesmaximal
4#11729306C211yesmaximal, characteristic, complemented
5#11725434C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×#6489—21no—
10C7×#395C412yescomplemented
11C7×#394C412yescomplemented
12C7×#386—21no—
13C7×#385—22no—
14C7×#381C2211yescharacteristic
15C2×#6411—24no—
16C4×C22⋊Dic7—21no—
17C4×C22⋊Dic7C2212yes—
18#89979C2212yes—
19C4⋊C4×Dic7—24no—
20#89323—22no—
21#9476—71nomaximal, Sylow
22C2×C7×#41—22no—
23C2×C7×#41—22no—
24C2×C7×#41—44no—
25C22×C7×Q8C2×C411yescharacteristic
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C7×#123—44no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4D412yes—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C22⋊Dic7—22no—
42#6489—28no—
43C2×C4⋊Dic7—22no—
44C2×C4×Dic7—22no—
45C2×C4×Dic7—22no—
46C2×C4×Dic7D414yes—
47C2×#1136—22no—
48C7×#95—22no—
49C7×#95—21no—
50C7×#95C2×C412yes—
51#6411—48no—
52C7⋊#95—22no—
53C7⋊#95—22no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56#2406D711yescharacteristic
57#2274—71no—
58#1789—72no—
59#1463—71no—
60#1284—72no—
61C24×C7—21no—
62C24×C7C2×D411yescharacteristic
63C7×#41—28no—
64C7×#41—44no—
65C7×#41—44no—
66C7×#41—44no—
67C2×C7×Q8—22no—
68C2×C7×Q8—42no—
69C2×C7×D4—24no—
70C2×C7×D4—22no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—24no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—22no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7C22⋊C412yes—
88C22×C4×C7C22⋊C412yes—
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D412yes—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7#4112yes—
93C22×Dic7—24no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—44no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—44no—
100C22⋊Dic7—28no—
101C4⋊Dic7—28no—
102C4×Dic7—28no—
103C4×Dic7—44no—
104C4×Dic7—82no—
105#1136—44no—
106C2×C7⋊C8—42no—
107C22×#41C2×D711yescharacteristic
108C2×#123—21no—
109C2×#117—141no—
110C22×C4⋊C4C2×D711yescharacteristic
111C2×#104—144no—
112#407—72no—
113#395Dic712yescomplemented
114#394Dic712yescomplemented
115#386—21no—
116#385—22no—
117#381C2×D711yescharacteristic
118C4×C4⋊C4—144no—
119C4×C22⋊C4—72no—
120C4×C22⋊C4—141no—
121#324—142no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—42no—
128C23×C7—41no—
129C23×C7—42no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#12011yescharacteristic
132C23×C7#12311yescharacteristic
133C7×Q8—24no—
134C7×Q8—42no—
135C7×D4—24no—
136C7×D4—42no—
137C7×D4—44no—
138C7×D4—44no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—24no—
142C2×C4×C7—22no—
143C2×C4×C7—22no—
144C2×C4×C7—24no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—48no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—81no—
159C2×C4×C7#10414yescomplemented
160C2×C4×C7C4×D412yes—
161C2×C4×C7#12112yes—
162C2×Dic7—28no—
163C2×Dic7—44no—
164C2×Dic7—44no—
165C7⋊C8—82no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—44no—
169C22×Q8C2×Dic711yescharacteristic
170C22×D4—21no—
171C22×D4C2×Dic711yescharacteristic
172C23×C4—21no—
173C23×C4C22⋊D711yescharacteristic
174C23×C4C22⋊D711yescharacteristic
175C23×C4C22×D711yescharacteristic
176C2×M16—142no—
177#123—44no—
178#117—148no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—142no—
183C2×C4⋊C4C2×Dic712yes—
184C2×C4⋊C4C22⋊D712yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—142no—
189C2×C42—74no—
190C2×C42—142no—
191C2×C42—142no—
192#104—288no—
193C22⋊C8—142no—
194C22⋊C8—142no—
195#95—22no—
196#95—21no—
197#95—142no—
198#95—142no—
199#95C2×Dic712yes—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—44no—
208C22×C7—44no—
209C22×C7—42no—
210C22×C7—82no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7#38611yescharacteristic
214C22×C7C2×#10412yes—
215C22×C7B(4,𝔽2)12yes—
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—24no—
219C4×C7—42no—
220C4×C7—42no—
221C4×C7—82no—
222Dic7—48no—
223C24—21no—
224C24C2×C22⋊D711yescharacteristic
225#41—28no—
226#41—44no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—144no—
250C22×C4C22⋊Dic712yes—
251C22×C4C22⋊Dic712yes—
252C22×C4D4×D712yes—
253C22×C4#115912yes—
254C22×C4C22×Dic711yescharacteristic
255C22×C4C2×C22⋊D711yescharacteristic
256M16—284no—
257C2×C8—282no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—148no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—148no—
266C42—148no—
267C42—284no—
268C42—562no—
269C2×C7—22no—
270C2×C7—22no—
271C2×C7—41no—
272C2×C7—82no—
273C2×C7#188911yescharacteristic
274C2×C7#189611yescharacteristic
275C2×C7#191211yescharacteristic
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—41no—
283C23—42no—
284C23#649811yescharacteristic
285C23#650011yescharacteristic
286C23C2×C22⋊Dic711yescharacteristic
287Q8—24no—
288Q8—42no—
289D4—24no—
290D4—42no—
291D4—44no—
292D4—44no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—48no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—148no—
314C2×C4—284no—
315C2×C4—284no—
316C2×C4#641114yescomplemented
317C2×C4D4×Dic712yes—
318C2×C4#650212yes—
319C8—562no—
320C7#947611yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—82no—
332C22—81no—
333C22—81no—
334C22#8980412yes—
335C22#8999211yescharacteristic
336C22C2×#641112yes—
337C4—22no—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—288no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—82no—
348C2#1172942311yescharacteristic
349C2#1172946511yescharacteristic
350C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2406C2
socleC22×C7#386
prime radicalC22×C7×#41C22
cosocleC23×C4×C7C23
hypercenter#2406D7
hypocenterC7#9476
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. #11739589C2semidirect
4. #11729306C2semidirect
5. #11725434C2semidirect
10. C7×#395C4semidirect
11. C7×#394C4semidirect
113. #395Dic7semidirect
114. #394Dic7semidirect
159. C2×C4×C7#104semidirect
316. C2×C4#6411semidirect
320. C7#9476semidirect

Maximal subgroups

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

subgroupindex
C7×#24062
#117395892
#117298922
#117293062
#117254342
#94767