Finite Group Databasebeta

← #13328644

Subgroups — #13328644

Order 1792.

998 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 191 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, complemented
2C7×#2548C211yesmaximal, characteristic
3#11740614C212yesmaximal
4#11729378C211yesmaximal, characteristic, complemented
5#11727458C218yesmaximal, complemented
6#11727225C212yesmaximal, complemented
7C2×C7×#127C2211yescharacteristic
8C2×C7×#124—21no—
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×#394C2214yes—
19C7×#390—21no—
20C7×#390—22no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#89990C2218yes—
23#89848—22no—
24#89847—24no—
25#89842—22no—
26#89697C2212yes—
27#89323C2214yes—
28#89247C2218yes—
29#18403—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4C2312yes—
34C7×#127—22no—
35C7×#127—22no—
36C7×#124—44no—
37C4×C7×Q8—22no—
38C4×C7×Q8—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4C2312yes—
42C2×C7×C22⋊C4—24no—
43C2×C7×C22⋊C4—24no—
44C2×C42×C7—21no—
45C2×C42×C7D412yes—
46C2×C42×C7C2311yescharacteristic
47Q8×Dic7—24no—
48Q8×Dic7—24no—
49C2×#1142—28no—
50#6489—24no—
51#6489—22no—
52#6487—22no—
53#6487—24no—
54#6487C2314yes—
55#6453—24no—
56#6453—28no—
57#6451—24no—
58#6451—28no—
59#6433—24no—
60#6433—22no—
61#6432—22no—
62#6432—22no—
63#6432D412yes—
64#6432D412yes—
65C4×Dic14—24no—
66C4×Dic14—24no—
67C4×Dic14—22no—
68C4×Dic14D414yes—
69C7×#95C2312yes—
70#6406—28no—
71C22⋊(C7⋊C8)C2314yes—
72C4⋊(C7⋊C8)—22no—
73C4×C7⋊C8—22no—
74#3502—71no—
75#3208—72no—
76#2548D711yescharacteristic
77#1599—78no—
78#1503—72no—
79#1461—71no—
80C24×C7—22no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8SD1614yescomplemented
84C2×C7×D4—24no—
85C2×C7×D4—44no—
86C2×C7×D4—41no—
87C2×C7×D4—42no—
88C2×C7×D4—41no—
89C2×C7×D4—42no—
90C2×C7×D4—42no—
91C2×C7×D4—82no—
92C22×C4×C7—22no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D412yes—
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2411yescharacteristic
98C2×Dic14—22no—
99C2×Dic14—24no—
100C2×Dic14C2×D414yes—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—42no—
104C7×C22⋊C4—48no—
105C7×C22⋊C4—84no—
106C42×C7—22no—
107C42×C7—22no—
108C42×C7—42no—
109C42×C7C2×D412yes—
110C42×C7C2×D412yes—
111C22⋊Dic7—24no—
112#1142—88no—
113C4⋊Dic7—24no—
114C4⋊Dic7—24no—
115C4⋊Dic7—24no—
116C4⋊Dic7—22no—
117C4⋊Dic7—22no—
118C4⋊Dic7—22no—
119C4⋊Dic7C2×D414yes—
120Dic7⋊C4—28no—
121Dic7⋊C4—24no—
122Dic7⋊C4—24no—
123C4×Dic7—28no—
124C4×Dic7—24no—
125C2×C7⋊C8—24no—
126Q82—142no—
127#557—144no—
128#557—142no—
129#544—74no—
130#544—141no—
131#533—72no—
132#533—142no—
133C2×#127C2×D711yescharacteristic
134C2×#124—21no—
135#519—141no—
136C2×C4×Q8C2×D711yescharacteristic
137#492—142no—
138#486—142no—
139#460—144no—
140#450—78no—
141#431—72no—
142#394C2×D714yes—
143#390—21no—
144#390—22no—
145C4×C4⋊C4C2×D711yescharacteristic
146#328—78no—
147#324—74no—
148C23×C7—44no—
149C23×C7—44no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7—42no—
153C23×C7C22×D411yescharacteristic
154C7×Q8—28no—
155C7×D4—48no—
156C7×D4—82no—
157C7×D4—84no—
158C7×D4—82no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12012yes—
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7C2×SD1614yes—
172C2×C4×C7C2×SD1612yes—
173C2×C4×C7#13712yes—
174C2×C4×C7C22×D412yes—
175C2×Dic7—24no—
176C2×Dic7—24no—
177Dic14—44no—
178Dic14—44no—
179Dic14—84no—
180C7⋊C8—44no—
181C22×Q8C22⋊D711yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C22×D712yes—
185C2×SD16—148no—
186C4⋊Q8—72no—
187C4⋊Q8—72no—
188C4⋊Q8—142no—
189C4⋊Q8—142no—
190C4⋊Q8—148no—
191C4⋊Q8—144no—
192#127—22no—
193#127—22no—
194#125—144no—
195#125—148no—
196#125—144no—
197#125—142no—
198#124—44no—
199C22⋊Q8—74no—
200C22⋊Q8—142no—
201C22⋊Q8—144no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—74no—
205C4×Q8—144no—
206C4×Q8—144no—
207C4×Q8—144no—
208C4×Q8—144no—
209C4×Q8—142no—
210#117—142no—
211#117—144no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22⋊D711yescharacteristic
214C2×C4⋊C4C22×D712yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C42—21no—
218C2×C42C22⋊D712yes—
219C2×C42C22×D711yescharacteristic
220C4⋊C8—142no—
221#102—148no—
222C22⋊C8—74no—
223C4×C8—142no—
224#95C22×D712yes—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—84no—
228C22×C7—84no—
229C22×C7—82no—
230C22×C7—82no—
231C22×C7—82no—
232C22×C7C2×#12011yescharacteristic
233C22×C7C22×SD1611yescharacteristic
234C22×C7C2×#13711yescharacteristic
235C4×C7—24no—
236C4×C7—21no—
237C4×C7—22no—
238C4×C7—42no—
239C4×C7—82no—
240C4×C7#44814yes—
241C4×C7#45412yes—
242Dic7—44no—
243Dic7—44no—
244C24—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—74no—
248C2×Q8—144no—
249C2×Q8—142no—
250C2×Q8#114214yescomplemented
251C2×D4—24no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—82no—
259C22×C4—22no—
260C22×C4C2×C22⋊D712yes—
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C23×D711yescharacteristic
265SD16—568no—
266C2×C8—144no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—74no—
271C4⋊C4—148no—
272C4⋊C4—144no—
273C4⋊C4—144no—
274C4⋊C4—144no—
275C4⋊C4—144no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—144no—
280C22⋊C4—48no—
281C22⋊C4—84no—
282C22⋊C4—144no—
283C42—22no—
284C42—22no—
285C42—42no—
286C42—148no—
287C42—144no—
288C42D4×D712yes—
289C42D4×D712yes—
290C2×C7—21no—
291C2×C7—21no—
292C2×C7—162no—
293C2×C7C2×#44811yescharacteristic
294C2×C7C2×#45411yescharacteristic
295C2×C7#302711yescharacteristic
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23C22×C22⋊D711yescharacteristic
302Q8—28no—
303Q8—284no—
304Q8—284no—
305Q8—564no—
306D4—48no—
307D4—82no—
308D4—84no—
309D4—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—144no—
320C2×C4—144no—
321C2×C4C2×#114214yes—
322C2×C4C2×#114212yes—
323C2×C4#650011yescharacteristic
324C2×C4#650012yes—
325C2×C4#650011yescharacteristic
326C2×C4#651412yes—
327C2×C4C2×D4×D712yes—
328C8—284no—
329C7#1840311yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22—84no—
333C22—84no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22C22×#114211yescharacteristic
338C22C2×#650011yescharacteristic
339C22C2×#651411yescharacteristic
340C4—22no—
341C4—24no—
342C4—21no—
343C4—42no—
344C4—82no—
345C4—284no—
346C4—284no—
347C4#8981214yes—
348C4#8983412yes—
349C2—21no—
350C2—21no—
351C2—162no—
352C2C2×#8981211yescharacteristic
353C2C2×#8983411yescharacteristic
354C2#1174084111yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2548C2
socleC22×C7C2×#120
prime radicalC2×C7×#127C22
hypercenter#2548D7
hypocenterC7#18403
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #11742871C2semidirect
4. #11729378C2semidirect
5. #11727458C2semidirect
6. #11727225C2semidirect
83. C2×C7×Q8SD16semidirect
250. C2×Q8#1142semidirect
329. C7#18403semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#25482
#117406142
#117293782
#117274582
#117272252
#184037