Finite Group Databasebeta

← #13328650

Subgroups — #13328650

Order 1792.

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

normal subgroups only

This group has 910 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
2C7×#2567C211yesmaximal, characteristic
3#11740667C212yesmaximal
4#11729378C211yesmaximal, characteristic
5#11727461C218yesmaximal
6#11727225C212yesmaximal
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C4×C7×Q8C2211yescharacteristic
9#90503—21no—
10#90261—22no—
11#90260—24no—
12Q8×Dic14—22no—
13#90191—21no—
14#90191C2214yes—
15#90161—22no—
16#90161C2212yes—
17C7×#395C2214yes—
18C7×#391—21no—
19C7×#389—22no—
20C7×#384—21no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#90000C2218yes—
23#89864—22no—
24#89862—24no—
25#89853—22no—
26#89697C2212yes—
27#89323C2214yes—
28#89247C2218yes—
29#18404—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×C7×Q8C2312yes—
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—22no—
34C4×C7×Q8—22no—
35C4×C7×Q8—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—24no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4C2312yes—
42C2×C42×C7—21no—
43C2×C42×C7D412yes—
44C2×C42×C7C2311yescharacteristic
45Q8×Dic7—24no—
46Q8×Dic7—24no—
47C2×#1144—28no—
48#6489—24no—
49#6489—22no—
50#6487—22no—
51#6487—24no—
52#6487C2314yes—
53#6453—24no—
54#6453—28no—
55#6451—24no—
56#6451—28no—
57#6433—24no—
58#6433—22no—
59#6432—22no—
60#6432—22no—
61#6432D412yes—
62#6432D412yes—
63C4×Dic14—24no—
64C4×Dic14—24no—
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#3502—71no—
74#3209—72no—
75#2567D711yescharacteristic
76#1605—78no—
77#1503—72no—
78#1461—71no—
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—24no—
82C2×C7×Q8—82no—
83C2×C7×Q8Q1614yescomplemented
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2411yescharacteristic
91C2×Dic14—22no—
92C2×Dic14—24no—
93C2×Dic14C2×D414yes—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—84no—
101C42×C7—22no—
102C42×C7—22no—
103C42×C7—42no—
104C42×C7C2×D412yes—
105C42×C7C2×D412yes—
106C22⋊Dic7—24no—
107#1144—88no—
108C4⋊Dic7—24no—
109C4⋊Dic7—24no—
110C4⋊Dic7—24no—
111C4⋊Dic7—22no—
112C4⋊Dic7—22no—
113C4⋊Dic7—22no—
114C4⋊Dic7C2×D414yes—
115Dic7⋊C4—28no—
116Dic7⋊C4—24no—
117Dic7⋊C4—24no—
118C4×Dic7—28no—
119C4×Dic7—24no—
120C2×C7⋊C8—24no—
121Q82—142no—
122#557—144no—
123#557—142no—
124#544—74no—
125#544—141no—
126#533—72no—
127#533—142no—
128C2×C4⋊Q8C2×D711yescharacteristic
129#519—141no—
130C2×C4×Q8C2×D711yescharacteristic
131#494—142no—
132#487—142no—
133#462—144no—
134#451—78no—
135#431—72no—
136#395C2×D714yes—
137#391—21no—
138#389—22no—
139#384—21no—
140C4×C4⋊C4C2×D711yescharacteristic
141#328—78no—
142#324—74no—
143C23×C7C22×D411yescharacteristic
144C7×Q8—28no—
145C7×Q8—48no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—24no—
150C2×C4×C7—22no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—44no—
156C2×C4×C7—42no—
157C2×C4×C7#12011yescharacteristic
158C2×C4×C7#12012yes—
159C2×C4×C7#12011yescharacteristic
160C2×C4×C7C2×Q1614yes—
161C2×C4×C7C2×Q1612yes—
162C2×C4×C7#13712yes—
163C2×C4×C7C22×D412yes—
164C2×Dic7—24no—
165C2×Dic7—24no—
166Dic14—44no—
167Dic14—44no—
168Dic14—84no—
169C7⋊C8—44no—
170C22×Q8C22⋊D711yescharacteristic
171C22×Q8C22×D712yes—
172C2×Q16—148no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—72no—
176C4⋊Q8—72no—
177C4⋊Q8—142no—
178C4⋊Q8—142no—
179C4⋊Q8—148no—
180C4⋊Q8—144no—
181#125—144no—
182#125—148no—
183#125—144no—
184#125—142no—
185C22⋊Q8—74no—
186C22⋊Q8—142no—
187C22⋊Q8—144no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—74no—
191C4×Q8—144no—
192C4×Q8—144no—
193C4×Q8—144no—
194C4×Q8—144no—
195C4×Q8—142no—
196#117—142no—
197#117—144no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C22⋊D711yescharacteristic
203C2×C4⋊C4C22×D712yes—
204C2×C42—21no—
205C2×C42C22⋊D712yes—
206C2×C42C22×D711yescharacteristic
207C4⋊C8—142no—
208#103—148no—
209C22⋊C8—74no—
210C4×C8—142no—
211#95—24no—
212#95C22×D712yes—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7C2×#12011yescharacteristic
216C22×C7C22×Q1611yescharacteristic
217C22×C7C2×#13711yescharacteristic
218C4×C7—24no—
219C4×C7—21no—
220C4×C7—22no—
221C4×C7—44no—
222C4×C7—42no—
223C4×C7—82no—
224C4×C7#45114yes—
225C4×C7#45612yes—
226Dic7—44no—
227Dic7—44no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—74no—
232C2×Q8—82no—
233C2×Q8—144no—
234C2×Q8—142no—
235C2×Q8#114414yescomplemented
236C22×C4—22no—
237C22×C4—22no—
238C22×C4C2×C22⋊D711yescharacteristic
239C22×C4C2×C22⋊D712yes—
240C22×C4C2×C22⋊D712yes—
241C22×C4C2×C22⋊D711yescharacteristic
242C22×C4C23×D711yescharacteristic
243Q16—568no—
244C2×C8—144no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—74no—
252C4⋊C4—84no—
253C4⋊C4—148no—
254C4⋊C4—144no—
255C4⋊C4—144no—
256C4⋊C4—144no—
257C4⋊C4—144no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—144no—
262C22⋊C4—144no—
263C42—22no—
264C42—22no—
265C42—42no—
266C42—148no—
267C42—144no—
268C42D4×D712yes—
269C42D4×D712yes—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7C2×#45111yescharacteristic
273C2×C7C2×#45611yescharacteristic
274C2×C7#302711yescharacteristic
275C23C22×C22⋊D711yescharacteristic
276Q8—28no—
277Q8—48no—
278Q8—284no—
279Q8—284no—
280Q8—564no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—144no—
293C2×C4—144no—
294C2×C4#650011yescharacteristic
295C2×C4#650011yescharacteristic
296C2×C4#650012yes—
297C2×C4C2×#114414yes—
298C2×C4C2×#114412yes—
299C2×C4#651412yes—
300C2×C4C2×D4×D712yes—
301C8—284no—
302C7#1840411yescharacteristic, complemented, Sylow
303C22—21no—
304C22—21no—
305C22C2×#650011yescharacteristic
306C22C22×#114411yescharacteristic
307C22C2×#651411yescharacteristic
308C4—24no—
309C4—22no—
310C4—21no—
311C4—44no—
312C4—42no—
313C4—82no—
314C4—284no—
315C4—284no—
316C4#8982214yes—
317C4#8986712yes—
318C2—21no—
319C2—21no—
320C2C2×#8982211yescharacteristic
321C2C2×#8986711yescharacteristic
322C2#1174084111yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2567C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#2567D7
hypocenterC7#18404
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
83. C2×C7×Q8Q16semidirect
235. C2×Q8#1144semidirect
302. C7#18404semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#25672
#117406672
#117293782
#117274612
#117272252
#184047