Finite Group Databasebeta

← #13328593

Subgroups — #13328593

Order 1792.

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

normal subgroups only

This group has 974 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#11742834C211yesmaximal, characteristic
2C7×#2547C211yesmaximal, characteristic
3#11740668C212yesmaximal, complemented
4#11729355C211yesmaximal, characteristic
5#11727464C218yesmaximal, complemented
6#11727225C212yesmaximal, complemented
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C7×#124—21no—
9C2×C4×C7×D4C2211yescharacteristic
10#90489—21no—
11C2×#6487—22no—
12#90234—22no—
13#90230—24no—
14D4×Dic14—22no—
15#90191—21no—
16#90174C2214yes—
17#90161C2212yes—
18C7×#397C2214yes—
19C7×#391—21no—
20C7×#380—22no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#89866—24no—
23#89865—22no—
24#89851—22no—
25#89743C2218yes—
26#89697C2212yes—
27#89320C2214yes—
28#89247C2218yes—
29#18396—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×Dic14—22no—
34C7×C4⋊Q8—22no—
35C7×C4⋊Q8—22no—
36C7×#124—44no—
37C4×C7×D4—22no—
38C4×C7×D4—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4C2312yes—
43C2×C7×C22⋊C4—24no—
44C2×C7×C22⋊C4C2312yes—
45C2×C42×C7—21no—
46C2×C42×C7C2311yescharacteristic
47C2×C22⋊Dic7—24no—
48D4×Dic7—24no—
49D4×Dic7—24no—
50#6489—22no—
51C2×C4⋊Dic7—22no—
52#6487—24no—
53#6487—24no—
54#6487—22no—
55#6487—24no—
56#6487C2314yes—
57C2×Dic7⋊C4—24no—
58#6453—24no—
59#6451—24no—
60#6441—28no—
61#6441—28no—
62#6433—22no—
63#6432D412yes—
64#6432D412yes—
65C4×Dic14—22no—
66C4×Dic14D414yes—
67C7×#95—24no—
68C7×#95C2312yes—
69C22⋊(C7⋊C8)C2314yes—
70#6384—28no—
71#6382—28no—
72C4⋊(C7⋊C8)—22no—
73C4×C7⋊C8—22no—
74#3501—71no—
75#3229—72no—
76#2547D711yescharacteristic
77#1608—78no—
78#1503—72no—
79#1460—71no—
80C24×C7C2×D412yes—
81C2×C7×Q8—42no—
82C2×C7×Q8—42no—
83C2×C7×Q8—42no—
84C2×C7×D4—22no—
85C2×C7×D4—22no—
86C2×C7×D4SD1614yes—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—44no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D412yes—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2411yescharacteristic
96C22×Dic7—24no—
97C22×Dic7—24no—
98C2×Dic14—22no—
99C2×Dic14—24no—
100C2×Dic14—84no—
101C2×Dic14C2×D414yes—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—24no—
104C7×C4⋊C4—42no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C22⋊C4—24no—
108C7×C22⋊C4—48no—
109C42×C7—42no—
110C42×C7C2×D412yes—
111C42×C7C2×D412yes—
112C22⋊Dic7—28no—
113C22⋊Dic7—24no—
114C4⋊Dic7—24no—
115C4⋊Dic7—22no—
116C4⋊Dic7—22no—
117C4⋊Dic7—22no—
118C4⋊Dic7C2×D414yes—
119Dic7⋊C4—28no—
120Dic7⋊C4—24no—
121Dic7⋊C4—24no—
122C4×Dic7—24no—
123C2×C7⋊C8—24no—
124#554—144no—
125#554—142no—
126D4×Q8—142no—
127#544—141no—
128#543—74no—
129#533—72no—
130C2×C4⋊Q8C2×D711yescharacteristic
131C2×#124—21no—
132C2×C22⋊Q8—142no—
133#518—141no—
134C2×C4×D4C2×D711yescharacteristic
135#498—142no—
136#487—142no—
137#483—78no—
138#475—144no—
139#431—72no—
140#397C2×D714yes—
141#391—21no—
142#380—22no—
143C4×C4⋊C4C2×D711yescharacteristic
144#328—78no—
145#323—74no—
146C23×C7—22no—
147C23×C7—22no—
148C23×C7—42no—
149C23×C7—42no—
150C23×C7C2×SD1614yes—
151C23×C7C22×D411yescharacteristic
152C7×Q8—84no—
153C7×D4—28no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—41no—
157C2×C4×C7—44no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—82no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12012yes—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7C2×SD1612yes—
170C2×C4×C7#13712yes—
171C2×C4×C7C22×D412yes—
172C2×Dic7—24no—
173C2×Dic7—24no—
174C2×Dic7—84no—
175C2×Dic7—84no—
176Dic14—44no—
177Dic14—44no—
178Dic14—84no—
179C7⋊C8—44no—
180C22×Q8—21no—
181C22×Q8—142no—
182C22×D4C22⋊D711yescharacteristic
183C23×C4C22⋊D712yes—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—72no—
187C4⋊Q8—72no—
188C4⋊Q8—144no—
189#125—144no—
190#125—142no—
191#124—44no—
192C22⋊Q8—74no—
193C22⋊Q8—148no—
194C22⋊Q8—148no—
195C22⋊Q8—144no—
196C22⋊Q8—144no—
197C22⋊Q8—142no—
198C22⋊Q8—144no—
199C4×Q8—74no—
200C4×Q8—142no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—144no—
204C4×D4—144no—
205#117—142no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—144no—
209C2×C4⋊C4—142no—
210C2×C4⋊C4C22⋊D711yescharacteristic
211C2×C4⋊C4C22×D712yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—144no—
214C2×C22⋊C4C22×D712yes—
215C2×C42—21no—
216C2×C42C22×D711yescharacteristic
217#106—148no—
218C4⋊C8—142no—
219#103—148no—
220C22⋊C8—74no—
221C4×C8—142no—
222#95—24no—
223#95C22×D712yes—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—24no—
227C22×C7—44no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—82no—
231C22×C7C2×#12011yescharacteristic
232C22×C7C22×SD1611yescharacteristic
233C22×C7C2×#13711yescharacteristic
234C4×C7—21no—
235C4×C7—22no—
236C4×C7—42no—
237C4×C7—82no—
238C4×C7—82no—
239C4×C7#44814yes—
240C4×C7#45512yes—
241Dic7—44no—
242Dic7—44no—
243C24C2×C22⋊D712yes—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—74no—
248C2×Q8—144no—
249C2×Q8—142no—
250C2×Q8—564no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4#114214yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—144no—
260C22×C4—144no—
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C23×D711yescharacteristic
265C2×C8—144no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—74no—
272C4⋊C4—148no—
273C4⋊C4—144no—
274C4⋊C4—144no—
275C4⋊C4—142no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—144no—
279C22⋊C4—24no—
280C22⋊C4—48no—
281C22⋊C4—148no—
282C22⋊C4—144no—
283C42—42no—
284C42—144no—
285C42D4×D712yes—
286C42D4×D712yes—
287C2×C7—21no—
288C2×C7—21no—
289C2×C7—44no—
290C2×C7C2×#44811yescharacteristic
291C2×C7C2×#45511yescharacteristic
292C2×C7#302811yescharacteristic
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23C2×#114214yes—
298C23C22×C22⋊D711yescharacteristic
299Q8—84no—
300Q8—284no—
301Q8—284no—
302Q8—564no—
303D4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—144no—
317C2×C4—144no—
318C2×C4—564no—
319C2×C4—564no—
320C2×C4C2×#114212yes—
321C2×C4#650011yescharacteristic
322C2×C4#650012yes—
323C2×C4#650011yescharacteristic
324C2×C4#651412yes—
325C2×C4C2×D4×D712yes—
326C8—284no—
327C7#1839611yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—24no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22C22×#114211yescharacteristic
336C22C2×#650011yescharacteristic
337C22C2×#651411yescharacteristic
338C4—22no—
339C4—21no—
340C4—42no—
341C4—82no—
342C4—82no—
343C4—284no—
344C4—284no—
345C4#8981214yes—
346C4#8983312yes—
347C2—21no—
348C2—21no—
349C2—44no—
350C2C2×#8981211yescharacteristic
351C2C2×#8983311yescharacteristic
352C2#1174086211yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2547C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#2547D7
hypocenterC7#18396
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #11740668C2semidirect
5. #11727464C2semidirect
6. #11727225C2semidirect
327. C7#18396semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#25472
#117406682
#117293552
#117274642
#117272252
#183967