Finite Group Databasebeta

← #13328696

Subgroups — #13328696

Order 1792.

906 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C7×#3502C211yesmaximal, characteristic
2#11740768C212yesmaximal
3#11739920C211yesmaximal, characteristic
4#11729652C218yesmaximal
5#11729589C212yesmaximal
6#11729378C211yesmaximal, characteristic
7C7×Q82—22no—
8C7×#557—22no—
9C7×#557—24no—
10C7×#544—21no—
11C7×#544C2214yes—
12C7×#533—22no—
13C7×#533C2212yes—
14C7×#519—21no—
15C2×C4×C7×Q8C2211yescharacteristic
16C2×#6507—21no—
17C2×#6432C2211yescharacteristic
18#89853—22no—
19#89851—22no—
20#89815C2218yes—
21#89792—24no—
22#89755C2214yes—
23#89746—21no—
24C4×C4⋊Dic7C2211yescharacteristic
25#89700—22no—
26#89697C2212yes—
27#89333C2218yes—
28#89323C2214yes—
29#18410—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×Dic14—21no—
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—22no—
34C7×C4⋊Q8—24no—
35C7×C4⋊Q8—28no—
36C7×C4⋊Q8D412yes—
37C7×C4⋊Q8D412yes—
38C7×#125—24no—
39C7×#125—24no—
40C7×#125—28no—
41C7×#125—22no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8C2314yes—
45C4×C7×Q8—24no—
46C4×C7×Q8—24no—
47C4×C7×Q8—24no—
48C4×C7×Q8—24no—
49C4×C7×Q8—22no—
50C4×C7×Q8—22no—
51C4×C7×Q8—22no—
52C4×C7×Q8D414yes—
53C7×#117—24no—
54C7×#117—22no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7D412yes—
58C2×C42×C7C2311yescharacteristic
59#6507—44no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7C2312yes—
63C2×Dic7⋊C4—24no—
64C2×C4×Dic7—21no—
65#6432—22no—
66#6432—22no—
67#6409—28no—
68C7⋊#95—24no—
69C7⋊#95C2312yes—
70C22⋊(C7⋊C8)C2314yes—
71#6382—28no—
72C4⋊(C7⋊C8)—22no—
73C4×C7⋊C8—22no—
74#3502D711yescharacteristic
75#3229—72no—
76#2546—71no—
77#1611—78no—
78#1503—72no—
79#1461—71no—
80C2×C7×Q8—24no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8SD1614yes—
85C2×C7×Q8C2×D414yes—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2411yescharacteristic
91C22×Dic7—22no—
92C22×Dic7—22no—
93C2×Dic14—42no—
94C2×Dic14—42no—
95C2×Dic14—42no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—24no—
99C7×C4⋊C4—28no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—24no—
107C7×C4⋊C4C2×D414yes—
108C7×C22⋊C4—24no—
109C42×C7—28no—
110C42×C7—22no—
111C42×C7—22no—
112C42×C7—24no—
113C42×C7C2×D412yes—
114C42×C7C2×D412yes—
115C4⋊Dic7—24no—
116C4⋊Dic7—24no—
117C4⋊Dic7—41no—
118C4⋊Dic7—41no—
119Dic7⋊C4—48no—
120C4×Dic7—42no—
121C2×C7⋊C8—24no—
122Q82—22no—
123#557—22no—
124#557—24no—
125#544—21no—
126#544C2×D714yes—
127#533—22no—
128#533C2×D712yes—
129C2×C4⋊Q8—71no—
130C2×C4⋊Q8—141no—
131#519—21no—
132C2×C4×Q8C2×D711yescharacteristic
133#498—142no—
134#487—142no—
135#483—78no—
136#475—144no—
137#431—72no—
138#398—74no—
139#391—141no—
140#389—142no—
141C4×C4⋊C4—71no—
142#328—78no—
143#324—74no—
144C23×C7C22×D411yescharacteristic
145C7×Q8—28no—
146C7×Q8—44no—
147C7×Q8—44no—
148C7×Q8—84no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—24no—
153C2×C4×C7—24no—
154C2×C4×C7—22no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7#12011yescharacteristic
158C2×C4×C7#12012yes—
159C2×C4×C7#12011yescharacteristic
160C2×C4×C7C2×SD1614yes—
161C2×C4×C7C2×SD1612yes—
162C2×C4×C7#13712yes—
163C2×C4×C7C22×D412yes—
164C2×Dic7—24no—
165C2×Dic7—42no—
166C2×Dic7—42no—
167C2×Dic7—42no—
168C2×Dic7—42no—
169C2×Dic7—42no—
170Dic14—84no—
171C7⋊C8—44no—
172C22×Q8—141no—
173C22×Q8C22⋊D711yescharacteristic
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—24no—
177C4⋊Q8—28no—
178C4⋊Q8—142no—
179C4⋊Q8—142no—
180C4⋊Q8—284no—
181C4⋊Q8C22⋊D712yes—
182C4⋊Q8C22⋊D712yes—
183#125—24no—
184#125—24no—
185#125—28no—
186#125—22no—
187C22⋊Q8—22no—
188C22⋊Q8—24no—
189C22⋊Q8C22×D714yes—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8C22⋊D714yes—
198#117—24no—
199#117—22no—
200C2×C4⋊C4—72no—
201C2×C4⋊C4—141no—
202C2×C4⋊C4—141no—
203C2×C4⋊C4—144no—
204C2×C4⋊C4C22⋊D711yescharacteristic
205C2×C4⋊C4C22×D712yes—
206C2×C42—141no—
207C2×C42C22⋊D712yes—
208C2×C42C22×D711yescharacteristic
209#106—148no—
210C4⋊C8—142no—
211#103—148no—
212C22⋊C8—74no—
213C4×C8—142no—
214#95—72no—
215#95—144no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7C2×#12011yescharacteristic
219C22×C7C22×SD1611yescharacteristic
220C22×C7C2×#13711yescharacteristic
221C4×C7—24no—
222C4×C7—21no—
223C4×C7—22no—
224C4×C7—44no—
225C4×C7—42no—
226C4×C7—44no—
227C4×C7#44814yes—
228C4×C7#45612yes—
229Dic7—44no—
230Dic7—82no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—282no—
236C2×Q8—282no—
237C2×Q8—282no—
238C2×Q8#114314yes—
239C2×Q8C2×C22⋊D714yes—
240C22×C4—142no—
241C22×C4—142no—
242C22×C4C2×C22⋊D712yes—
243C22×C4C2×C22⋊D711yescharacteristic
244C22×C4C2×C22⋊D712yes—
245C22×C4C2×C22⋊D711yescharacteristic
246C22×C4C23×D711yescharacteristic
247C2×C8—144no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—28no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—144no—
260C4⋊C4—144no—
261C4⋊C4—288no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4C2×C22⋊D714yes—
265C22⋊C4—24no—
266C42—28no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—282no—
271C42C2×C22⋊D712yes—
272C42C2×C22⋊D712yes—
273C2×C7—21no—
274C2×C7—21no—
275C2×C7C2×#44811yescharacteristic
276C2×C7C2×#45611yescharacteristic
277C2×C7#302811yescharacteristic
278C23C22×C22⋊D711yescharacteristic
279Q8—28no—
280Q8—44no—
281Q8—44no—
282Q8—84no—
283Q8—564no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—144no—
293C2×C4—282no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4C2×#114314yes—
299C2×C4C2×#114312yes—
300C2×C4#650512yes—
301C2×C4#651611yescharacteristic
302C2×C4#651612yes—
303C2×C4#651611yescharacteristic
304C2×C4C22×C22⋊D712yes—
305C8—284no—
306C7#1841011yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22C22×#114311yescharacteristic
310C22C2×#650511yescharacteristic
311C22C2×#651611yescharacteristic
312C4—24no—
313C4—21no—
314C4—22no—
315C4—44no—
316C4—42no—
317C4—44no—
318C4—284no—
319C4—562no—
320C4#8999914yes—
321C4#9001812yes—
322C2—21no—
323C2—21no—
324C2C2×#8999911yescharacteristic
325C2C2×#9001811yescharacteristic
326C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#3502D7
hypocenterC7#18410
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
306. C7#18410semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117407682
#117399202
#117296522
#117295892
#117293782
#184107