Finite Group Databasebeta

← #13326929

Subgroups — #13326929

Order 1792.

890 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 203 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#11740813C212yesmaximal
3#11739926C211yesmaximal, characteristic
4#11729658C214yesmaximal
5#11729653C214yesmaximal
6#11729592C212yesmaximal
7#11729378C211yesmaximal, characteristic
8C7×Q82—22no—
9C7×#557—22no—
10C7×#557—24no—
11C7×#544—21no—
12C7×#544C2214yes—
13C7×#533—22no—
14C7×#533C2212yes—
15C7×#519—21no—
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×Dic14C2211yescharacteristic
18#90002—21no—
19#89863—22no—
20#89862—22no—
21#89822C2214yes—
22#89816C2214yes—
23C4×#1144—24no—
24#89758C2212yes—
25#89751C2212yes—
26#89745—21no—
27#89708C2211yescharacteristic
28#89705—22no—
29#89697C2212yes—
30#89333C2218yes—
31#89323C2214yes—
32#18221—71nomaximal, Sylow
33C22×C7×Q8D411yescharacteristic
34C22×Dic14C2311yescharacteristic
35C7×C4⋊Q8—22no—
36C7×C4⋊Q8—22no—
37C7×C4⋊Q8—24no—
38C7×C4⋊Q8—28no—
39C7×C4⋊Q8D412yes—
40C7×C4⋊Q8D412yes—
41C7×#125—24no—
42C7×#125—24no—
43C7×#125—28no—
44C7×#125—22no—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—24no—
47C7×C22⋊Q8C2314yes—
48C4×C7×Q8—24no—
49C4×C7×Q8—24no—
50C4×C7×Q8—24no—
51C4×C7×Q8—24no—
52C4×C7×Q8—22no—
53C4×C7×Q8—22no—
54C4×C7×Q8—22no—
55C4×C7×Q8D414yes—
56C7×#117—24no—
57C7×#117—22no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4C2312yes—
60C2×C42×C7D412yes—
61C2×C42×C7C2311yescharacteristic
62C2×#1144—24no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7C2311yescharacteristic
65C2×Dic7⋊C4—21no—
66C2×Dic7⋊C4—22no—
67C2×C4×Dic7—21no—
68C4×Dic14—22no—
69C4×Dic14—22no—
70#6409—24no—
71C7⋊#95—21no—
72C7⋊#95—22no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75C7⋊#95C2312yes—
76C22⋊(C7⋊C8)C2314yes—
77#6384—24no—
78#6381—24no—
79C4⋊(C7⋊C8)—22no—
80C4×C7⋊C8—22no—
81#3502D711yescharacteristic
82#3104—72no—
83#2581—71no—
84#1617—74no—
85#1605—74no—
86#1518—72no—
87#1461—71no—
88C2×C7×Q8—24no—
89C2×C7×Q8—22no—
90C2×C7×Q8—22no—
91C2×C7×Q8—22no—
92C2×C7×Q8Q1614yescomplemented
93C2×C7×Q8C2×D414yes—
94C22×C4×C7C2×D412yes—
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2411yescharacteristic
99C22×Dic7—21no—
100C22×Dic7—22no—
101C22×Dic7—21no—
102C2×Dic14—22no—
103C2×Dic14—42no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—24no—
107C7×C4⋊C4—28no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—24no—
113C7×C4⋊C4—24no—
114C7×C4⋊C4—24no—
115C7×C4⋊C4C2×D414yes—
116C7×C22⋊C4—24no—
117C42×C7—28no—
118C42×C7—22no—
119C42×C7—22no—
120C42×C7—24no—
121C42×C7C2×D412yes—
122C42×C7C2×D412yes—
123#1144—48no—
124C4⋊Dic7—22no—
125C4⋊Dic7—42no—
126Dic7⋊C4—44no—
127Dic7⋊C4—41no—
128Dic7⋊C4—41no—
129C4×Dic7—42no—
130C2×C7⋊C8—24no—
131Q82—22no—
132#557—22no—
133#557—24no—
134#544—21no—
135#544C2×D714yes—
136#533—22no—
137#533C2×D712yes—
138#519—21no—
139C2×C4×Q8—71no—
140C2×C4×Q8C2×D711yescharacteristic
141#484—74no—
142#477—142no—
143#462—142no—
144#451—74no—
145C4×Q16—144no—
146#431—72no—
147#400—72no—
148#395—72no—
149#391—141no—
150#389—141no—
151#387—142no—
152#382—71no—
153#328—78no—
154#324—74no—
155C23×C7C22×D411yescharacteristic
156C7×Q8—28no—
157C7×Q8—44no—
158C7×Q8—44no—
159C7×Q8—84no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—24no—
163C2×C4×C7—24no—
164C2×C4×C7—22no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12012yes—
171C2×C4×C7C2×Q1614yes—
172C2×C4×C7C2×Q1612yes—
173C2×C4×C7#13712yes—
174C2×C4×C7C22×D412yes—
175C2×Dic7—22no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—42no—
179C2×Dic7—42no—
180C2×Dic7—42no—
181C2×Dic7—41no—
182C2×Dic7—42no—
183Dic14—24no—
184Dic14—42no—
185C7⋊C8—44no—
186C22×Q8—71no—
187C22×Q8C22⋊D711yescharacteristic
188C2×Q16—144no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—24no—
192C4⋊Q8—28no—
193C4⋊Q8C22⋊D712yes—
194C4⋊Q8C22⋊D712yes—
195#125—24no—
196#125—24no—
197#125—28no—
198#125—22no—
199C22⋊Q8—22no—
200C22⋊Q8—24no—
201C22⋊Q8C22×D714yes—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—142no—
210C4×Q8—142no—
211C4×Q8C22⋊D714yes—
212#117—24no—
213#117—22no—
214C2×C4⋊C4—71no—
215C2×C4⋊C4—141no—
216C2×C4⋊C4—142no—
217C2×C4⋊C4—141no—
218C2×C4⋊C4C22⋊D711yescharacteristic
219C2×C4⋊C4C22×D712yes—
220C2×C42—141no—
221C2×C42C22⋊D712yes—
222C2×C42C22×D711yescharacteristic
223#107—144no—
224C4⋊C8—142no—
225#103—144no—
226#103—144no—
227C22⋊C8—74no—
228C4×C8—142no—
229#95—72no—
230#95—141no—
231#95—142no—
232#95—142no—
233#95—142no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7C2×#12011yescharacteristic
237C22×C7C22×Q1611yescharacteristic
238C22×C7C2×#13711yescharacteristic
239C4×C7—24no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—22no—
243C4×C7—44no—
244C4×C7—42no—
245C4×C7—44no—
246C4×C7#45114yes—
247Dic7—42no—
248Dic7—82no—
249Dic7—81no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—142no—
255C2×Q8—282no—
256C2×Q8#114414yescomplemented
257C2×Q8C2×C22⋊D714yes—
258C22×C4—141no—
259C22×C4—142no—
260C22×C4—141no—
261C22×C4C2×C22⋊D712yes—
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C2×C22⋊D711yescharacteristic
265C22×C4C23×D711yescharacteristic
266Q16—288no—
267C2×C8—144no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—142no—
280C4⋊C4—284no—
281C4⋊C4—282no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C4⋊C4C2×C22⋊D714yes—
285C22⋊C4—24no—
286C42—28no—
287C42—22no—
288C42—22no—
289C42—24no—
290C42—282no—
291C42C2×C22⋊D712yes—
292C42C2×C22⋊D712yes—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7C2×#45111yescharacteristic
296C2×C7#302011yescharacteristic
297C2×C7#302511yescharacteristic
298C23C22×C22⋊D711yescharacteristic
299Q8—28no—
300Q8—44no—
301Q8—44no—
302Q8—84no—
303Q8—144no—
304Q8—282no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—142no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4—281no—
321C2×C4#650512yes—
322C2×C4C2×#114414yes—
323C2×C4C2×#114412yes—
324C2×C4#651611yescharacteristic
325C2×C4#651611yescharacteristic
326C2×C4#651612yes—
327C2×C4C22×C22⋊D712yes—
328C8—284no—
329C7#1822111yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22C2×#650511yescharacteristic
333C22C22×#114411yescharacteristic
334C22C2×#651611yescharacteristic
335C4—24no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—44no—
340C4—42no—
341C4—44no—
342C4—282no—
343C4—562no—
344C4—561no—
345C4#9000014yes—
346C2—21no—
347C2—21no—
348C2C2×#9000011yescharacteristic
349C2#1174156111yescharacteristic
350C2#1174157011yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#3502D7
hypocenterC7#18221
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
92. C2×C7×Q8Q16semidirect
256. C2×Q8#1144semidirect
329. C7#18221semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117408132
#117399262
#117296582
#117296532
#117295922
#117293782
#182217