Finite Group Databasebeta

← #13326928

Subgroups — #13326928

Order 1792.

890 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 187 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#11742871C211yesmaximal, characteristic
2C7×#2581C211yesmaximal, characteristic
3#11739892C212yesmaximal
4#11729378C211yesmaximal, characteristic
5#11727473C214yesmaximal
6#11727461C214yesmaximal
7#11727300C212yesmaximal
8C2×C4×C7×Q8C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10#90503—21no—
11#90261—22no—
12#90260—24no—
13Q8×Dic14—22no—
14#90191—21no—
15#90191C2214yes—
16#90161—22no—
17#90161C2212yes—
18C7×#400C2212yes—
19C7×#395C2212yes—
20C7×#391—21no—
21C7×#389—21no—
22C7×#387—22no—
23C7×#382C2211yescharacteristic
24#90000C2214yes—
25#89803—22no—
26C4×#1144—24no—
27#89793—22no—
28#89742C2214yes—
29#89697C2212yes—
30#89323C2214yes—
31#89247C2218yes—
32#18221—71nomaximal, Sylow
33C22×C7×Q8D411yescharacteristic
34C22×C7×Q8C2311yescharacteristic
35C4×C7×Q8—22no—
36C4×C7×Q8—22no—
37C4×C7×Q8—22no—
38C4×C7×Q8—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—22no—
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4C2311yescharacteristic
44C2×C7×C4⋊C4C2312yes—
45C2×C42×C7—21no—
46C2×C42×C7D412yes—
47C2×C42×C7C2311yescharacteristic
48Q8×Dic7—24no—
49Q8×Dic7—24no—
50C2×#1144—24no—
51#6489—24no—
52#6489—22no—
53#6487—22no—
54#6487—24no—
55#6487C2314yes—
56#6453—24no—
57#6453—28no—
58#6451—24no—
59#6451—28no—
60#6433—24no—
61#6433—22no—
62#6432—22no—
63#6432—22no—
64#6432D412yes—
65#6432D412yes—
66C4×Dic14—24no—
67C4×Dic14—24no—
68C4×Dic14—22no—
69C4×Dic14D414yes—
70C7×#95—21no—
71C7×#95—22no—
72C7×#95—22no—
73C7×#95—22no—
74C7×#95C2312yes—
75#6409—24no—
76C22⋊(C7⋊C8)C2314yes—
77#6384—24no—
78#6381—24no—
79C4⋊(C7⋊C8)—22no—
80C4×C7⋊C8—22no—
81#3502—71no—
82#3104—72no—
83#2581D711yescharacteristic
84#1617—74no—
85#1605—74no—
86#1518—72no—
87#1461—71no—
88C2×C7×Q8—22no—
89C2×C7×Q8—22no—
90C2×C7×Q8—22no—
91C2×C7×Q8—42no—
92C2×C7×Q8Q1614yescomplemented
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—21no—
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D412yes—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2411yescharacteristic
101C2×Dic14—22no—
102C2×Dic14—24no—
103C2×Dic14C2×D414yes—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—44no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—42no—
111C42×C7—22no—
112C42×C7—22no—
113C42×C7—42no—
114C42×C7C2×D412yes—
115C42×C7C2×D412yes—
116C22⋊Dic7—24no—
117#1144—48no—
118C4⋊Dic7—24no—
119C4⋊Dic7—24no—
120C4⋊Dic7—24no—
121C4⋊Dic7—22no—
122C4⋊Dic7—22no—
123C4⋊Dic7—22no—
124C4⋊Dic7C2×D414yes—
125Dic7⋊C4—28no—
126Dic7⋊C4—24no—
127Dic7⋊C4—24no—
128C4×Dic7—28no—
129C4×Dic7—24no—
130C2×C7⋊C8—24no—
131Q82—142no—
132#557—144no—
133#557—142no—
134#544—74no—
135#544—141no—
136#533—72no—
137#533—142no—
138#519—141no—
139C2×C4×Q8C2×D711yescharacteristic
140C2×C4×Q8C2×D711yescharacteristic
141#484—74no—
142#477—142no—
143#462—142no—
144#451—74no—
145C4×Q16—144no—
146#431—72no—
147#400C2×D712yes—
148#395C2×D712yes—
149#391—21no—
150#389—21no—
151#387—22no—
152#382C2×D711yescharacteristic
153#328—78no—
154#324—74no—
155C23×C7C22×D411yescharacteristic
156C7×Q8—28no—
157C7×Q8—24no—
158C7×Q8—42no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—41no—
172C2×C4×C7—42no—
173C2×C4×C7#12011yescharacteristic
174C2×C4×C7#12011yescharacteristic
175C2×C4×C7#12012yes—
176C2×C4×C7C2×Q1614yes—
177C2×C4×C7C2×Q1612yes—
178C2×C4×C7#13712yes—
179C2×C4×C7C22×D412yes—
180C2×Dic7—24no—
181C2×Dic7—24no—
182Dic14—44no—
183Dic14—44no—
184Dic14—84no—
185C7⋊C8—44no—
186C22×Q8C22⋊D711yescharacteristic
187C22×Q8C22×D711yescharacteristic
188C2×Q16—144no—
189C4⋊Q8—72no—
190C4⋊Q8—72no—
191C4⋊Q8—142no—
192C4⋊Q8—142no—
193C4⋊Q8—148no—
194C4⋊Q8—144no—
195#125—144no—
196#125—148no—
197#125—144no—
198#125—142no—
199C22⋊Q8—74no—
200C22⋊Q8—142no—
201C22⋊Q8—144no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—74no—
207C4×Q8—144no—
208C4×Q8—144no—
209C4×Q8—144no—
210C4×Q8—144no—
211C4×Q8—142no—
212#117—142no—
213#117—144no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊D711yescharacteristic
218C2×C4⋊C4C22×D711yescharacteristic
219C2×C4⋊C4C22×D712yes—
220C2×C42—21no—
221C2×C42C22⋊D712yes—
222C2×C42C22×D711yescharacteristic
223#107—144no—
224C4⋊C8—142no—
225#103—144no—
226#103—144no—
227C22⋊C8—74no—
228C4×C8—142no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95C22×D712yes—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7C2×#12011yescharacteristic
237C22×C7C22×Q1611yescharacteristic
238C22×C7C2×#13711yescharacteristic
239C4×C7—24no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—22no—
243C4×C7—42no—
244C4×C7—42no—
245C4×C7—82no—
246C4×C7—81no—
247C4×C7#45114yes—
248Dic7—44no—
249Dic7—44no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—74no—
255C2×Q8—144no—
256C2×Q8—142no—
257C2×Q8#114414yescomplemented
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4C2×C22⋊D712yes—
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C2×C22⋊D711yescharacteristic
265C22×C4C23×D711yescharacteristic
266Q16—288no—
267C2×C8—144no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—74no—
276C4⋊C4—148no—
277C4⋊C4—144no—
278C4⋊C4—144no—
279C4⋊C4—144no—
280C4⋊C4—144no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C4⋊C4—144no—
285C22⋊C4—144no—
286C42—22no—
287C42—22no—
288C42—42no—
289C42—148no—
290C42—144no—
291C42D4×D712yes—
292C42D4×D712yes—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7C2×#45111yescharacteristic
296C2×C7#302011yescharacteristic
297C2×C7#302511yescharacteristic
298C23C22×C22⋊D711yescharacteristic
299Q8—28no—
300Q8—24no—
301Q8—42no—
302Q8—284no—
303Q8—284no—
304Q8—564no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—144no—
320C2×C4—144no—
321C2×C4#650011yescharacteristic
322C2×C4#650011yescharacteristic
323C2×C4#650012yes—
324C2×C4C2×#114414yes—
325C2×C4C2×#114412yes—
326C2×C4#651412yes—
327C2×C4C2×D4×D712yes—
328C8—284no—
329C7#1822111yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22C2×#650011yescharacteristic
333C22C22×#114411yescharacteristic
334C22C2×#651411yescharacteristic
335C4—24no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—81no—
343C4—284no—
344C4—284no—
345C4#8982214yes—
346C2—21no—
347C2—21no—
348C2C2×#8982211yescharacteristic
349C2#1174038111yescharacteristic
350C2#1174086011yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2581C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#2581D7
hypocenterC7#18221
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
92. C2×C7×Q8Q16semidirect
257. C2×Q8#1144semidirect
329. C7#18221semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#25812
#117398922
#117293782
#117274732
#117274612
#117273002
#182217