Finite Group Databasebeta

← #13251517

Subgroups — #13251517

Order 1792.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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#11740221C211yesmaximal, characteristic
2#11731499C211yesmaximal, characteristic
3C7×#1511C211yesmaximal, characteristic
4#11729443C211yesmaximal, characteristic
5#11727727C211yesmaximal, characteristic
6#11725488C211yesmaximal, characteristic
7#11725434C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C4⋊Dic7C2211yescharacteristic
10C7×#407C2211yescharacteristic
11C7×#381C2211yescharacteristic
12#90002C412yescomplemented
13#89983C2211yescharacteristic
14C4×C22⋊Dic7C2211yescharacteristic
15#89979C2211yescharacteristic
16#89758—22no—
17#89749C412yescomplemented
18#89715—21no—
19#89715—21no—
20#89714—21no—
21C22⋊C4×Dic7—21no—
22#89426—22no—
23#89425—22no—
24C7×#328—22no—
25#89349—22no—
26#89333—22no—
27#9460—71nomaximal, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C23×Dic7D411yescharacteristic
31C23×Dic7D411yescharacteristic
32C2×C7×M16—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C22⋊Dic7—21no—
42C2×C22⋊Dic7—21no—
43C2×C4⋊Dic7—21no—
44C2×C4⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×C4⋊Dic7—21no—
47C2×C4⋊Dic7C2×C412yes—
48C2×C4⋊Dic7D412yes—
49C2×Dic7⋊C4—22no—
50C2×C4×Dic7—22no—
51C2×C4×Dic7C2×C412yes—
52C2×#1136—21no—
53C7×C22⋊C8—21no—
54C7×C22⋊C8—21no—
55C7×#95—21no—
56C7×#95D412yes—
57C7⋊#95—22no—
58C7⋊#95—22no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—22no—
62C7⋊#95—22no—
63C7⋊#95—22no—
64C7⋊#95—22no—
65C7⋊#95D412yes—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68#2503—71no—
69#2313—71no—
70#1821—71no—
71#1511—71no—
72#1511D711yescharacteristic
73#1284—71no—
74#1284—71no—
75C24×C7C2×D411yescharacteristic
76C2×C7×Q8—21no—
77C2×C7×Q8—21no—
78C2×C7×Q8—42no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—22no—
82C22×C4×C7—42no—
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×Dic7—21no—
90C22×Dic7—21no—
91C22×Dic7—21no—
92C22×Dic7—21no—
93C22×Dic7—22no—
94C22×Dic7—22no—
95C22×Dic7—42no—
96C22×Dic7—42no—
97C22×Dic7C22⋊C412yes—
98C22×Dic7C22⋊C412yes—
99C7×M16—42no—
100C2×C7×C8—41no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C22⋊Dic7—42no—
109C22⋊Dic7—42no—
110C4⋊Dic7—22no—
111C4⋊Dic7—22no—
112C4⋊Dic7—22no—
113C4⋊Dic7—22no—
114Dic7⋊C4—82no—
115C4×Dic7—42no—
116C4×Dic7—82no—
117#1136—42no—
118C2×C7⋊C8—41no—
119C2×C22⋊Q8C2×D711yescharacteristic
120C22×C4⋊C4—71no—
121#407—71no—
122#407C2×D711yescharacteristic
123#400—142no—
124#391—72no—
125#387—142no—
126#387—142no—
127#382—72no—
128#381—71no—
129#381—141no—
130#381—141no—
131#381C2×D711yescharacteristic
132#380—141no—
133C4×C22⋊C4—71no—
134C4×C22⋊C4—141no—
135#343—142no—
136#328—22no—
137#328—142no—
138C23×C7—22no—
139C23×C7—22no—
140C23×C7—41no—
141C23×C7C2×C22⋊C411yescharacteristic
142C23×C7#12011yescharacteristic
143C23×C7#12311yescharacteristic
144C7×Q8—81no—
145C7×Q8—81no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7C4×D411yescharacteristic
160C2×C4×C7C4×D411yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×Dic7—22no—
164C2×Dic7—22no—
165C2×Dic7—42no—
166C2×Dic7—42no—
167C2×Dic7—42no—
168C2×Dic7—42no—
169C2×Dic7—42no—
170C2×Dic7—42no—
171C2×Dic7—42no—
172C2×Dic7—42no—
173C2×Dic7—42no—
174C2×Dic7—42no—
175C2×Dic7—81no—
176C2×Dic7—81no—
177C7×C8—81no—
178C7⋊C8—81no—
179C22×Q8C4×D711yescharacteristic
180C23×C4—71no—
181C23×C4—71no—
182C23×C4C22×D711yescharacteristic
183C2×M16—21no—
184C2×M16—141no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—72no—
191C2×C4⋊C4—72no—
192C2×C4⋊C4—141no—
193C2×C4⋊C4—141no—
194C2×C4⋊C4—141no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4—142no—
197C2×C4⋊C4C4×D711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—141no—
201C2×C22⋊C4—141no—
202C2×C42—72no—
203C2×C42—142no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—141no—
207C22⋊C8—141no—
208#95—21no—
209#95—72no—
210#95—142no—
211#95—142no—
212#95—142no—
213#95—142no—
214#95—142no—
215#95—142no—
216#95—141no—
217#95—141no—
218#95C22⋊D712yes—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—44no—
227C22×C7#38611yescharacteristic
228C22×C7#41012yes—
229C22×C7#45512yes—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—82no—
235C4×C7—81no—
236C4×C7—81no—
237Dic7—42no—
238Dic7—42no—
239Dic7—82no—
240Dic7—82no—
241C24D4×D711yescharacteristic
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—42no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—72no—
250C22×C4—72no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4—142no—
256C22×C4—142no—
257C22×C4—282no—
258C22×C4—282no—
259C22×C4C2×C4×D711yescharacteristic
260C22×C4#115711yescharacteristic
261C22×C4D4×D711yescharacteristic
262C22×C4D4×D711yescharacteristic
263C22×C4#115911yescharacteristic
264C22×C4C2×C22⋊D711yescharacteristic
265M16—42no—
266M16—282no—
267C2×C8—41no—
268C2×C8—281no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—142no—
277C4⋊C4—562no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—282no—
282C22⋊C4—282no—
283C42—282no—
284C42—562no—
285C2×C7—22no—
286C2×C7—22no—
287C2×C7—41no—
288C2×C7#188511yescharacteristic
289C2×C7#189911yescharacteristic
290C2×C7#191211yescharacteristic
291C23—22no—
292C23—22no—
293C23—41no—
294C23D7×C22⋊C411yescharacteristic
295C23#644611yescharacteristic
296C23#650011yescharacteristic
297Q8—81no—
298Q8—81no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—142no—
313C2×C4—142no—
314C2×C4—282no—
315C2×C4—282no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4—282no—
324C2×C4—561no—
325C2×C4—561no—
326C2×C4(C22⋊D7)⋊C411yescharacteristic
327C2×C4#644711yescharacteristic
328C2×C4C4×C22⋊D711yescharacteristic
329C2×C4#650111yescharacteristic
330C8—81no—
331C8—561no—
332C7#946011yescharacteristic, complemented, Sylow
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#8951212yes—
342C22#8972811yescharacteristic
343C22#8997412yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4—282no—
352C4—282no—
353C4—562no—
354C4—562no—
355C2—22no—
356C2—22no—
357C2—41no—
358C2#1172778311yescharacteristic
359C2#1172809711yescharacteristic
360C2#1172816511yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1511C2
socleC22×C7#386
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1511D7
hypocenterC7#9460
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
12. #90002C4semidirect
17. #89749C4semidirect
332. C7#9460semidirect

Maximal subgroups

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

subgroupindex
#117402212
#117314992
C7×#15112
#117294432
#117277272
#117254882
#117254342
#94607