Finite Group Databasebeta

← #13328701

Subgroups — #13328701

Order 1792.

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

normal subgroups only

This group has 870 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×#2568C211yesmaximal, characteristic
3#11740669C212yesmaximal
4#11729378C211yesmaximal, characteristic
5#11727473C218yesmaximal
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×#400C2214yes—
18C7×#391—21no—
19C7×#391—22no—
20C7×#382—21no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#89865—22no—
23#89863—24no—
24#89850—22no—
25#89742C2218yes—
26#89697C2212yes—
27#89323C2214yes—
28#89247C2218yes—
29#18411—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D411yescharacteristic
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⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2312yes—
40C2×C7×C4⋊C4C2312yes—
41C2×C42×C7—21no—
42C2×C42×C7D412yes—
43C2×C42×C7C2311yescharacteristic
44Q8×Dic7—24no—
45Q8×Dic7—24no—
46#6489—24no—
47#6489—22no—
48#6487—22no—
49#6487—24no—
50#6487C2314yes—
51#6453—24no—
52#6453—28no—
53#6451—24no—
54#6451—28no—
55#6433—24no—
56#6433—22no—
57#6432—22no—
58#6432—22no—
59#6432D412yes—
60#6432D412yes—
61C4×Dic14—24no—
62C4×Dic14—24no—
63C4×Dic14—22no—
64C4×Dic14D414yes—
65C7×#95—24no—
66C7×#95—24no—
67C7×#95C2312yes—
68C22⋊(C7⋊C8)C2314yes—
69#6384—28no—
70#6381—28no—
71C4⋊(C7⋊C8)—22no—
72C4×C7⋊C8—22no—
73#3502—71no—
74#3230—72no—
75#2568D711yescharacteristic
76#1617—78no—
77#1503—72no—
78#1461—71no—
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—44no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×Q8Q1614yes—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2411yescharacteristic
92C2×Dic14—22no—
93C2×Dic14—24no—
94C2×Dic14C2×D414yes—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4—42no—
100C42×C7—22no—
101C42×C7—22no—
102C42×C7—42no—
103C42×C7C2×D412yes—
104C42×C7C2×D412yes—
105C22⋊Dic7—24no—
106C4⋊Dic7—24no—
107C4⋊Dic7—24no—
108C4⋊Dic7—24no—
109C4⋊Dic7—22no—
110C4⋊Dic7—22no—
111C4⋊Dic7—22no—
112C4⋊Dic7C2×D414yes—
113Dic7⋊C4—28no—
114Dic7⋊C4—24no—
115Dic7⋊C4—24no—
116C4×Dic7—28no—
117C4×Dic7—24no—
118C2×C7⋊C8—24no—
119Q82—142no—
120#557—144no—
121#557—142no—
122#544—74no—
123#544—141no—
124#533—72no—
125#533—142no—
126C2×C4⋊Q8C2×D711yescharacteristic
127#519—141no—
128C2×C4×Q8C2×D711yescharacteristic
129#500—142no—
130#487—142no—
131#484—78no—
132#477—144no—
133#431—72no—
134#400C2×D714yes—
135#391—21no—
136#391—22no—
137#382—21no—
138C4×C4⋊C4C2×D711yescharacteristic
139#328—78no—
140#324—74no—
141C23×C7C22×D411yescharacteristic
142C7×Q8—28no—
143C7×Q8—82no—
144C7×Q8—82no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—44no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7#12011yescharacteristic
157C2×C4×C7#12012yes—
158C2×C4×C7#12011yescharacteristic
159C2×C4×C7C2×Q1614yes—
160C2×C4×C7C2×Q1612yes—
161C2×C4×C7#13712yes—
162C2×C4×C7C22×D412yes—
163C2×Dic7—24no—
164C2×Dic7—24no—
165Dic14—44no—
166Dic14—44no—
167Dic14—84no—
168C7⋊C8—44no—
169C22×Q8—21no—
170C22×Q8C22⋊D711yescharacteristic
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—72no—
174C4⋊Q8—72no—
175C4⋊Q8—142no—
176C4⋊Q8—142no—
177C4⋊Q8—148no—
178C4⋊Q8—144no—
179#125—144no—
180#125—148no—
181#125—144no—
182#125—142no—
183C22⋊Q8—74no—
184C22⋊Q8—142no—
185C22⋊Q8—144no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—74no—
189C4×Q8—144no—
190C4×Q8—144no—
191C4×Q8—144no—
192C4×Q8—144no—
193C4×Q8—142no—
194#117—142no—
195#117—144no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D711yescharacteristic
199C2×C4⋊C4C22×D712yes—
200C2×C4⋊C4C22×D712yes—
201C2×C42—21no—
202C2×C42C22⋊D712yes—
203C2×C42C22×D711yescharacteristic
204#107—148no—
205C4⋊C8—142no—
206#103—148no—
207C22⋊C8—74no—
208C4×C8—142no—
209#95—24no—
210#95—24no—
211#95C22×D712yes—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7C2×#12011yescharacteristic
215C22×C7C22×Q1611yescharacteristic
216C22×C7C2×#13711yescharacteristic
217C4×C7—24no—
218C4×C7—21no—
219C4×C7—22no—
220C4×C7—42no—
221C4×C7—82no—
222C4×C7—82no—
223C4×C7#45114yes—
224C4×C7#45412yes—
225Dic7—44no—
226Dic7—44no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—44no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—74no—
233C2×Q8—144no—
234C2×Q8—142no—
235C2×Q8#114414yes—
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
243C2×C8—144no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—74no—
250C4⋊C4—148no—
251C4⋊C4—144no—
252C4⋊C4—144no—
253C4⋊C4—144no—
254C4⋊C4—144no—
255C4⋊C4—142no—
256C4⋊C4—142no—
257C4⋊C4—142no—
258C4⋊C4—144no—
259C22⋊C4—144no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42—148no—
264C42—144no—
265C42D4×D712yes—
266C42D4×D712yes—
267C2×C7—21no—
268C2×C7—21no—
269C2×C7C2×#45111yescharacteristic
270C2×C7C2×#45411yescharacteristic
271C2×C7#302811yescharacteristic
272C23C22×C22⋊D711yescharacteristic
273Q8—28no—
274Q8—82no—
275Q8—82no—
276Q8—284no—
277Q8—284no—
278Q8—564no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—144no—
291C2×C4—144no—
292C2×C4#650011yescharacteristic
293C2×C4#650012yes—
294C2×C4#650011yescharacteristic
295C2×C4C2×#114414yes—
296C2×C4C2×#114412yes—
297C2×C4#651412yes—
298C2×C4C2×D4×D712yes—
299C8—284no—
300C7#1841111yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22C2×#650011yescharacteristic
304C22C22×#114411yescharacteristic
305C22C2×#651411yescharacteristic
306C4—24no—
307C4—22no—
308C4—21no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—284no—
313C4—284no—
314C4#8982214yes—
315C4#8983412yes—
316C2—21no—
317C2—21no—
318C2C2×#8982211yescharacteristic
319C2C2×#8983411yescharacteristic
320C2#1174086211yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2568C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#2568D7
hypocenterC7#18411
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
300. C7#18411semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#25682
#117406692
#117293782
#117274732
#117272252
#184117