Finite Group Databasebeta

← #13328548

Subgroups — #13328548

Order 1792.

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

normal subgroups only

This group has 978 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×#2563C211yesmaximal, characteristic
3#11740667C212yesmaximal, complemented
4#11729355C211yesmaximal, characteristic
5#11727455C218yesmaximal, complemented
6#11727225C212yesmaximal, complemented
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9#90489—21no—
10C2×#6487—22no—
11#90234—22no—
12#90230—24no—
13D4×Dic14—22no—
14#90191—21no—
15#90174C2214yes—
16#90161C2212yes—
17C7×#393C2214yes—
18C7×#390—21no—
19C7×#382—21no—
20C7×#380—22no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#90000C2218yes—
23#89864—22no—
24#89862—24no—
25#89853—22no—
26#89697C2212yes—
27#89320C2214yes—
28#89247C2218yes—
29#18390—71nomaximal, Sylow
30C22×C7×Q8C2312yes—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×Dic14—22no—
34C7×C4⋊Q8—22no—
35C7×C4⋊Q8—22no—
36C4×C7×D4—22no—
37C4×C7×D4—22no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—24no—
43C2×C7×C22⋊C4C2312yes—
44C2×C42×C7—21no—
45C2×C42×C7C2311yescharacteristic
46C2×C22⋊Dic7—24no—
47C2×#1144—28no—
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—
69#6409—28no—
70C22⋊(C7⋊C8)C2314yes—
71C4⋊(C7⋊C8)—22no—
72C4×C7⋊C8—22no—
73#3501—71no—
74#3209—72no—
75#2563D711yescharacteristic
76#1602—78no—
77#1503—72no—
78#1460—71no—
79C24×C7C2×D412yes—
80C2×C7×Q8—24no—
81C2×C7×Q8—82no—
82C2×C7×D4—22no—
83C2×C7×D4—22no—
84C2×C7×D4Q1614yescomplemented
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—44no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2411yescharacteristic
94C22×Dic7—24no—
95C22×Dic7—24no—
96C2×Dic14—22no—
97C2×Dic14—24no—
98C2×Dic14—84no—
99C2×Dic14C2×D414yes—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—42no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—41no—
105C7×C22⋊C4—24no—
106C7×C22⋊C4—84no—
107C42×C7—42no—
108C42×C7C2×D412yes—
109C42×C7C2×D412yes—
110C22⋊Dic7—28no—
111C22⋊Dic7—24no—
112#1144—88no—
113C4⋊Dic7—24no—
114C4⋊Dic7—22no—
115C4⋊Dic7—22no—
116C4⋊Dic7—22no—
117C4⋊Dic7C2×D414yes—
118Dic7⋊C4—28no—
119Dic7⋊C4—24no—
120Dic7⋊C4—24no—
121C4×Dic7—24no—
122C2×C7⋊C8—24no—
123#554—144no—
124#554—142no—
125D4×Q8—142no—
126#544—141no—
127#543—74no—
128#533—72no—
129C2×C4⋊Q8C2×D711yescharacteristic
130C2×C22⋊Q8—142no—
131#518—141no—
132C2×C4×D4C2×D711yescharacteristic
133#494—142no—
134#487—142no—
135#462—144no—
136#451—78no—
137#431—72no—
138#393C2×D714yes—
139#390—21no—
140#382—21no—
141#380—22no—
142C4×C4⋊C4C2×D711yescharacteristic
143#328—78no—
144#323—74no—
145C23×C7—22no—
146C23×C7—22no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7C2×Q1614yes—
150C23×C7C22×D411yescharacteristic
151C7×Q8—48no—
152C7×D4—28no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—41no—
156C2×C4×C7—44no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—44no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—82no—
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12012yes—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7C2×Q1612yes—
168C2×C4×C7#13712yes—
169C2×C4×C7C22×D412yes—
170C2×Dic7—24no—
171C2×Dic7—24no—
172C2×Dic7—84no—
173C2×Dic7—84no—
174Dic14—44no—
175Dic14—44no—
176Dic14—84no—
177C7⋊C8—44no—
178C22×Q8—142no—
179C22×Q8C22×D712yes—
180C22×D4C22⋊D711yescharacteristic
181C23×C4C22⋊D712yes—
182C2×Q16—148no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—72no—
186C4⋊Q8—72no—
187C4⋊Q8—144no—
188#125—144no—
189#125—142no—
190C22⋊Q8—74no—
191C22⋊Q8—148no—
192C22⋊Q8—148no—
193C22⋊Q8—144no—
194C22⋊Q8—144no—
195C22⋊Q8—142no—
196C22⋊Q8—144no—
197C4×Q8—74no—
198C4×Q8—142no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—144no—
202C4×D4—144no—
203#117—142no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—144no—
208C2×C4⋊C4—142no—
209C2×C4⋊C4C22⋊D711yescharacteristic
210C2×C22⋊C4—24no—
211C2×C22⋊C4—144no—
212C2×C22⋊C4C22×D712yes—
213C2×C42—21no—
214C2×C42C22×D711yescharacteristic
215C4⋊C8—142no—
216#103—148no—
217C22⋊C8—74no—
218C4×C8—142no—
219#95—24no—
220#95C22×D712yes—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—24no—
224C22×C7—44no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—82no—
228C22×C7C2×#12011yescharacteristic
229C22×C7C22×Q1611yescharacteristic
230C22×C7C2×#13711yescharacteristic
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—42no—
234C4×C7—82no—
235C4×C7—82no—
236C4×C7#45114yes—
237C4×C7#45512yes—
238Dic7—44no—
239Dic7—44no—
240C24C2×C22⋊D712yes—
241C2×Q8—24no—
242C2×Q8—74no—
243C2×Q8—82no—
244C2×Q8—144no—
245C2×Q8—142no—
246C2×Q8—564no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#114414yescomplemented
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—144no—
256C22×C4—144no—
257C22×C4C2×C22⋊D711yescharacteristic
258C22×C4C2×C22⋊D712yes—
259C22×C4C2×C22⋊D711yescharacteristic
260C22×C4C23×D711yescharacteristic
261Q16—568no—
262C2×C8—144no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—74no—
269C4⋊C4—148no—
270C4⋊C4—144no—
271C4⋊C4—144no—
272C4⋊C4—142no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—144no—
276C22⋊C4—24no—
277C22⋊C4—84no—
278C22⋊C4—148no—
279C22⋊C4—144no—
280C42—42no—
281C42—144no—
282C42D4×D712yes—
283C42D4×D712yes—
284C2×C7—21no—
285C2×C7—21no—
286C2×C7—44no—
287C2×C7C2×#45111yescharacteristic
288C2×C7C2×#45511yescharacteristic
289C2×C7#302711yescharacteristic
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23C2×#114414yes—
295C23C22×C22⋊D711yescharacteristic
296Q8—48no—
297Q8—284no—
298Q8—284no—
299Q8—564no—
300D4—28no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—144no—
313C2×C4—144no—
314C2×C4—564no—
315C2×C4—564no—
316C2×C4#650011yescharacteristic
317C2×C4#650012yes—
318C2×C4#650011yescharacteristic
319C2×C4C2×#114412yes—
320C2×C4#651412yes—
321C2×C4C2×D4×D712yes—
322C8—284no—
323C7#1839011yescharacteristic, complemented, Sylow
324C22—21no—
325C22—21no—
326C22—24no—
327C22—44no—
328C22—42no—
329C22—42no—
330C22—82no—
331C22C2×#650011yescharacteristic
332C22C22×#114411yescharacteristic
333C22C2×#651411yescharacteristic
334C4—22no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—82no—
339C4—284no—
340C4—284no—
341C4#8982214yes—
342C4#8983312yes—
343C2—21no—
344C2—21no—
345C2—44no—
346C2C2×#8982211yescharacteristic
347C2C2×#8983311yescharacteristic
348C2#1174084111yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2563C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#2563D7
hypocenterC7#18390
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #11740667C2semidirect
5. #11727455C2semidirect
6. #11727225C2semidirect
84. C2×C7×D4Q16semidirect
249. C2×D4#1144semidirect
323. C7#18390semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#25632
#117406672
#117293552
#117274552
#117272252
#183907