Finite Group Databasebeta

← #13250728

Subgroups — #13250728

Order 1792.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3113C211yesmaximal, characteristic
2#11739899C211yesmaximal, characteristic
3#11729595C211yesmaximal, characteristic
4#11729590C211yesmaximal, characteristic
5#11729377C211yesmaximal, characteristic
6#11725556C211yesmaximal, characteristic
7#11725554C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C7×#483—21no—
11C7×#475C412yescomplemented
12C7×#462C412yescomplemented
13C7×#451—21no—
14C7×#441—22no—
15C7×#432C2211yescharacteristic
16#90002—21no—
17#89757—21no—
18#89753—22no—
19#89709—21no—
20#89708—21no—
21C4×C4⋊Dic7C2211yescharacteristic
22#89704C2211yescharacteristic
23#89698C2211yescharacteristic
24C4×#1136C2211yescharacteristic
25#89333—21no—
26#89333—21no—
27Q8⋊(C7⋊C8)—22no—
28#89327—22no—
29#9328—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C2×C7×Q16—22no—
32C7×C4⋊Q8C2×C411yescharacteristic
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×#125—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C4×C7×Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8C2×C412yes—
42C4×C7×Q8D412yes—
43C7×#117—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C42×C7—21no—
47C2×C42×C7C2311yescharacteristic
48C2×C4⋊Dic7—21no—
49C2×C4⋊Dic7—22no—
50C2×C4×Dic7—21no—
51C2×C4×Dic7D412yes—
52C2×#1136—21no—
53C7×#106—22no—
54C7×C4⋊C8C2×C412yes—
55C7×#103—22no—
56C7×#103—22no—
57C7×C22⋊C8—21no—
58C7×#97—21no—
59C7⋊#95—22no—
60C7⋊#95—22no—
61C7⋊#95—22no—
62C7⋊#95—21no—
63C7⋊#95—21no—
64C7⋊#95D411yescharacteristic
65C7⋊#95D411yescharacteristic
66C22⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68#6377—21no—
69#6377—21no—
70C4×C7⋊C8—21no—
71#3113D711yescharacteristic
72#2320—71no—
73#1986—71no—
74#1506—71no—
75#1491—71no—
76#1300—71no—
77#1285—71no—
78C2×C7×Q8—21no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—42no—
82C2×C7×Q8C22⋊C412yes—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×Dic7—21no—
89C22×Dic7—22no—
90C22×Dic7—21no—
91C7×Q16—44no—
92C2×C7×C8—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4C22⋊C412yes—
103C7×C22⋊C4—41no—
104C42×C7—22no—
105C42×C7—22no—
106C42×C7—41no—
107C42×C7C22×C411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7#4111yescharacteristic
110C42×C7#4111yescharacteristic
111C4⋊Dic7—42no—
112C4⋊Dic7—82no—
113C4×Dic7—42no—
114C4×Dic7—42no—
115#1136—22no—
116#1136—41no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119#533C2×D711yescharacteristic
120C2×C4×Q8C2×D711yescharacteristic
121#483—21no—
122#475Dic712yescomplemented
123#462Dic712yescomplemented
124#451—21no—
125#441—22no—
126#432—71no—
127#432C2×D711yescharacteristic
128C4×M16—71no—
129#391—141no—
130#389—141no—
131#387—142no—
132#383—141no—
133#382—141no—
134C4×C4⋊C4—71no—
135#376—71no—
136#339—142no—
137#328—141no—
138#328—141no—
139Q8⋊C8—142no—
140C23×C7#12011yescharacteristic
141C7×Q8—22no—
142C7×Q8—22no—
143C7×Q8—42no—
144C7×Q8—42no—
145C7×Q8—81no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7C2×C22⋊C411yescharacteristic
158C2×C4×C7C4×D411yescharacteristic
159C2×C4×C7C4×D411yescharacteristic
160C2×C4×C7#12111yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C2×C4×C7#12311yescharacteristic
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—42no—
166C2×Dic7—41no—
167C2×Dic7—42no—
168C2×Dic7—41no—
169C2×Dic7—42no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C7×C8—42no—
173C7⋊C8—42no—
174C7⋊C8—81no—
175C22×Q8C22⋊D711yescharacteristic
176C2×Q16—22no—
177C2×M16—141no—
178C4⋊Q8C2×Dic711yescharacteristic
179C4⋊Q8C2×Dic711yescharacteristic
180#125—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8C2×Dic712yes—
188C4×Q8C22⋊D712yes—
189#117—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—141no—
192C2×C4⋊C4—142no—
193C2×C4⋊C4C22⋊D711yescharacteristic
194C2×C42—21no—
195C2×C42—72no—
196C2×C42—141no—
197C2×C42C22×D711yescharacteristic
198#106—22no—
199C4⋊C8—141no—
200C4⋊C8C2×Dic712yes—
201#103—22no—
202#103—22no—
203C22⋊C8—21no—
204C22⋊C8—141no—
205#97—21no—
206#97—141no—
207#97—141no—
208C4×C8—141no—
209#95—71no—
210#95—71no—
211#95—142no—
212#95—142no—
213#95—142no—
214#95—141no—
215#95—141no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7#38611yescharacteristic
219C22×C7#45511yescharacteristic
220C22×C7#45611yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—42no—
227C4×C7—42no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7#42112yes—
231Dic7—81no—
232Dic7—82no—
233Dic7—81no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—42no—
238C2×Q8C22⋊Dic712yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—142no—
244C22×C4D4×D711yescharacteristic
245C22×C4D4×D711yescharacteristic
246C22×C4C2×C22⋊D711yescharacteristic
247Q16—44no—
248M16—142no—
249M16—281no—
250C2×C8—22no—
251C2×C8—281no—
252C2×C8—281no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—282no—
263C4⋊C4—562no—
264C4⋊C4C22⋊Dic712yes—
265C22⋊C4—41no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—282no—
270C42—282no—
271C42#115911yescharacteristic
272C42#115911yescharacteristic
273C42C22×Dic711yescharacteristic
274C42C2×C22⋊D711yescharacteristic
275C2×C7—21no—
276C2×C7—21no—
277C2×C7#188211yescharacteristic
278C2×C7#189711yescharacteristic
279C2×C7#190511yescharacteristic
280C23#650011yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—42no—
284Q8—42no—
285Q8—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—282no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4D4×Dic711yescharacteristic
307C2×C4D4×Dic711yescharacteristic
308C2×C4#649811yescharacteristic
309C2×C4#650211yescharacteristic
310C2×C4#650211yescharacteristic
311C2×C4C2×C22⋊Dic711yescharacteristic
312C8—42no—
313C8—282no—
314C8—561no—
315C7#932811yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22#8997411yescharacteristic
319C22#8997711yescharacteristic
320C22#8999211yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—561no—
331C4—562no—
332C4—561no—
333C4#9000812yes—
334C2—21no—
335C2—21no—
336C2#1172946611yescharacteristic
337C2#1172985411yescharacteristic
338C2#1172985511yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3113C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#3113D7
hypocenterC7#9328
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
11. C7×#475C4semidirect
12. C7×#462C4semidirect
122. #475Dic7semidirect
123. #462Dic7semidirect
315. C7#9328semidirect

Maximal subgroups

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

subgroupindex
C7×#31132
#117398992
#117295952
#117295902
#117293772
#117255562
#117255542
#93287