Finite Group Databasebeta

← #13328600

Subgroups — #13328600

Order 1792.

1010 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1010 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×#3501C211yesmaximal, characteristic
2#11740769C212yesmaximal, complemented
3#11739824C211yesmaximal, characteristic
4#11729645C218yesmaximal, complemented
5#11729589C212yesmaximal, complemented
6#11729355C211yesmaximal, characteristic
7C7×#554—22no—
8C7×#554—24no—
9C7×D4×Q8—22no—
10C7×#544—21no—
11C7×#543C2214yes—
12C7×#533C2212yes—
13C2×C7×C22⋊Q8—22no—
14C7×#518—21no—
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6487—22no—
17C2×#6432C2211yescharacteristic
18#89993—21no—
19#89853—22no—
20#89850—22no—
21#89816C2218yes—
22#89793—24no—
23#89748—21no—
24#89721C2214yes—
25C4×C4⋊Dic7C2211yescharacteristic
26#89697C2212yes—
27#89333C2218yes—
28#89320C2214yes—
29#18397—71nomaximal, Sylow
30C22×C7×Q8—22no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×Dic14—21no—
34C7×C4⋊Q8—24no—
35C7×C4⋊Q8D412yes—
36C7×C4⋊Q8D412yes—
37C7×#125—24no—
38C7×#125—22no—
39C7×C22⋊Q8—24no—
40C7×C22⋊Q8—24no—
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8C2314yes—
46C4×C7×Q8—22no—
47C4×C7×Q8D414yes—
48C4×C7×D4—24no—
49C4×C7×D4—24no—
50C4×C7×D4—22no—
51C4×C7×D4—22no—
52C7×#117—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4C2312yes—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—24no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7C2312yes—
63#6487—48no—
64C2×Dic7⋊C4—24no—
65C2×C4×Dic7—21no—
66#6432—22no—
67#6432—22no—
68#6409—28no—
69C7⋊#95C2312yes—
70C22⋊(C7⋊C8)C2314yes—
71#6381—28no—
72C4⋊(C7⋊C8)—22no—
73C4×C7⋊C8—22no—
74#3501D711yescharacteristic
75#3230—72no—
76#2542—71no—
77#1614—78no—
78#1503—72no—
79#1460—71no—
80C24×C7C2×D412yes—
81C2×C7×Q8—24no—
82C2×C7×Q8—22no—
83C2×C7×Q8—84no—
84C2×C7×Q8C2×D414yes—
85C2×C7×D4—22no—
86C2×C7×D4—22no—
87C2×C7×D4Q1614yes—
88C22×C4×C7—24no—
89C22×C4×C7—24no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—44no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D412yes—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2411yescharacteristic
97C22×Dic7—22no—
98C22×Dic7—22no—
99C2×Dic14—44no—
100C2×Dic14—41no—
101C2×Dic14—41no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—28no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4C2×D414yes—
111C7×C22⋊C4—28no—
112C7×C22⋊C4—24no—
113C7×C22⋊C4—24no—
114C42×C7—24no—
115C42×C7C2×D412yes—
116C42×C7C2×D412yes—
117C22⋊Dic7—48no—
118C4⋊Dic7—24no—
119C4⋊Dic7—24no—
120C4⋊Dic7—42no—
121Dic7⋊C4—48no—
122C4×Dic7—42no—
123C2×C7⋊C8—24no—
124#554—22no—
125#554—24no—
126D4×Q8—22no—
127#544—21no—
128#543C2×D714yes—
129#533C2×D712yes—
130C2×C4⋊Q8—71no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—142no—
133#518—21no—
134C2×C4×D4C2×D711yescharacteristic
135#500—142no—
136#487—142no—
137#484—78no—
138#477—144no—
139#431—72no—
140#399—74no—
141#390—141no—
142#384—141no—
143C4×C4⋊C4—71no—
144#328—78no—
145#323—74no—
146C23×C7—22no—
147C23×C7—22no—
148C23×C7—42no—
149C23×C7—42no—
150C23×C7C2×Q1614yes—
151C23×C7C22×D411yescharacteristic
152C7×Q8—44no—
153C7×Q8—44no—
154C7×Q8—84no—
155C7×D4—28no—
156C2×C4×C7—22no—
157C2×C4×C7—24no—
158C2×C4×C7—24no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—44no—
162C2×C4×C7—42no—
163C2×C4×C7—82no—
164C2×C4×C7—84no—
165C2×C4×C7—84no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12012yes—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7C2×Q1612yes—
170C2×C4×C7#13712yes—
171C2×C4×C7C22×D412yes—
172C2×Dic7—24no—
173C2×Dic7—42no—
174C2×Dic7—42no—
175C2×Dic7—44no—
176C2×Dic7—42no—
177Dic14—82no—
178Dic14—82no—
179C7⋊C8—44no—
180C22×Q8—22no—
181C22×Q8—141no—
182C22×D4C22⋊D711yescharacteristic
183C23×C4C22⋊D712yes—
184C4⋊Q8—24no—
185C4⋊Q8—142no—
186C4⋊Q8—142no—
187C4⋊Q8C22⋊D712yes—
188C4⋊Q8C22⋊D712yes—
189#125—24no—
190#125—22no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—28no—
194C22⋊Q8—22no—
195C22⋊Q8—24no—
196C22⋊Q8—28no—
197C22⋊Q8—288no—
198C22⋊Q8C22×D714yes—
199C4×Q8—22no—
200C4×Q8C22⋊D714yes—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—22no—
205#117—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—72no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—144no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4C22⋊D711yescharacteristic
213C2×C22⋊C4—24no—
214C2×C22⋊C4—144no—
215C2×C22⋊C4C22×D712yes—
216C2×C42—141no—
217C2×C42C22×D711yescharacteristic
218#107—148no—
219C4⋊C8—142no—
220#103—148no—
221C22⋊C8—74no—
222C4×C8—142no—
223#95—72no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—24no—
227C22×C7—44no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—82no—
231C22×C7C2×#12011yescharacteristic
232C22×C7C22×Q1611yescharacteristic
233C22×C7C2×#13711yescharacteristic
234C4×C7—21no—
235C4×C7—22no—
236C4×C7—42no—
237C4×C7—44no—
238C4×C7—44no—
239C4×C7#45114yes—
240C4×C7#45312yes—
241Dic7—44no—
242Dic7—82no—
243C24C2×C22⋊D712yes—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—84no—
247C2×Q8—284no—
248C2×Q8—281no—
249C2×Q8—281no—
250C2×Q8C2×C22⋊D714yes—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4#114414yes—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—142no—
260C22×C4—142no—
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C23×D711yescharacteristic
265C2×C8—144no—
266C4⋊C4—24no—
267C4⋊C4—28no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—144no—
275C4⋊C4—144no—
276C4⋊C4—288no—
277C4⋊C4—282no—
278C4⋊C4C2×C22⋊D714yes—
279C22⋊C4—28no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—288no—
283C42—24no—
284C42—282no—
285C42C2×C22⋊D712yes—
286C42C2×C22⋊D712yes—
287C2×C7—21no—
288C2×C7—21no—
289C2×C7—44no—
290C2×C7C2×#45111yescharacteristic
291C2×C7C2×#45311yescharacteristic
292C2×C7#302811yescharacteristic
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23C2×#114414yes—
298C23C22×C22⋊D711yescharacteristic
299Q8—44no—
300Q8—44no—
301Q8—84no—
302Q8—562no—
303Q8—562no—
304D4—28no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—144no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—284no—
319C2×C4—282no—
320C2×C4#650512yes—
321C2×C4C2×#114412yes—
322C2×C4#651611yescharacteristic
323C2×C4#651612yes—
324C2×C4#651611yescharacteristic
325C2×C4C22×C22⋊D712yes—
326C8—284no—
327C7#1839711yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—24no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22C2×#650511yescharacteristic
336C22C22×#114411yescharacteristic
337C22C2×#651611yescharacteristic
338C4—21no—
339C4—22no—
340C4—42no—
341C4—44no—
342C4—44no—
343C4—284no—
344C4—562no—
345C4#9000014yes—
346C4#9001312yes—
347C2—21no—
348C2—21no—
349C2—44no—
350C2C2×#9000011yescharacteristic
351C2C2×#9001311yescharacteristic
352C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18397
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11740769C2semidirect
4. #11729645C2semidirect
5. #11729589C2semidirect
327. C7#18397semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117407692
#117398242
#117296452
#117295892
#117293552
#183977