Finite Group Databasebeta

← #13249811

Subgroups — #13249811

Order 1792.

458 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 105 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#11740707C211yesmaximal, characteristic
2#11736484C211yesmaximal, characteristic
3C7×#1549C211yesmaximal, characteristic, complemented
4#11729529C211yesmaximal, characteristic, complemented
5#11728536C211yesmaximal, characteristic
6#11725544C211yesmaximal, characteristic, complemented
7#11725519C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×#6435C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12#90003—21no—
13#90001—21no—
14#89898—21no—
15M16×Dic7—21no—
16#89764C412yes—
17#89759—21no—
18#89754C412yes—
19#89745—21no—
20#89708C2211yescharacteristic
21#89697C2211yescharacteristic
22#89696C2211yescharacteristic
23#89612—22no—
24#89612—22no—
25C7×#332—22no—
26#89342—22no—
27#89264—21no—
28#89263—21no—
29#9194—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C4×D7D411yescharacteristic
32C2×C7×M16—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C412yes—
39C2×C42×C7C2311yescharacteristic
40C2×(C2×D7)⋊C4—22no—
41C2×(C2×D7)⋊C4C2×C411yescharacteristic
42C2×(C2×D7)⋊C4D411yescharacteristic
43C2×C4⋊Dic7—21no—
44C2×Dic7⋊C4—22no—
45C2×Dic7⋊C4C2×C411yescharacteristic
46C2×Dic7⋊C4D411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C2×#1136—21no—
49#6435—21no—
50#6435—21no—
51#6435—21no—
52#6435—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8D412yes—
56C7×C22⋊C8—21no—
57C4×C7×C8—21no—
58C7⋊#95—21no—
59C7⋊#95—22no—
60C7⋊#95—22no—
61C7⋊#95C2×C412yes—
62C22⋊(C7⋊C8)—21no—
63(C2×D7)⋊C8—24no—
64#6389—22no—
65C8×Dic7—22no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)D412yes—
69C4×C7⋊C8—21no—
70#2921—71no—
71#2628—71no—
72#1549—71no—
73#1549D711yescharacteristic, complemented
74#1458—71no—
75#1286—71no—
76#1286—71no—
77C23×D7C22⋊C411yescharacteristic
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C2×C7×Q8—82no—
81C22×C4×C7—22no—
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×Dic7—22no—
86C22×Dic7C22⋊C411yescharacteristic
87C22×Dic7C22⋊C411yescharacteristic
88C22×Dic7C22⋊C411yescharacteristic
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×C4×D7—21no—
92C2×C4×D7—21no—
93C2×C4×D7—21no—
94C2×C4×D7—21no—
95C7×M16—41no—
96C7×M16—41no—
97C2×C7×C8—22no—
98C2×C7×C8—41no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C42×C7C2×D411yescharacteristic
104C42×C7C2×D411yescharacteristic
105C42×C7#4111yescharacteristic
106C42×C7#4111yescharacteristic
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110(C2×D7)⋊C4—21no—
111(C2×D7)⋊C4—82no—
112C4⋊Dic7—41no—
113C4⋊Dic7—41no—
114Dic7⋊C4—21no—
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117Dic7⋊C4—21no—
118Dic7⋊C4—82no—
119C4×Dic7—21no—
120C4×Dic7—21no—
121#1136—41no—
122#1136—41no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—41no—
125C2×C4⋊Q8C2×D711yescharacteristic
126C2×#117—71no—
127#433—142no—
128#433—142no—
129#431—71no—
130#431C2×D711yescharacteristic
131#423—71no—
132#423C2×D711yescharacteristic
133#407—141no—
134C4×M16—141no—
135#400—72no—
136#399—72no—
137#395—141no—
138#393—141no—
139#389—141no—
140#382—71no—
141#380—141no—
142#332—22no—
143#332—142no—
144#324—141no—
145#323—141no—
146C23×C7C2×C22⋊C411yescharacteristic
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—41no—
151C22×D7—41no—
152C7×Q8—82no—
153C7×Q8—82no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7C4×D411yescharacteristic
160C2×C4×C7C4×D411yescharacteristic
161C2×C4×C7#12011yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12311yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×C4×C7#13511yescharacteristic
167C2×C4×C7#13711yescharacteristic
168C2×C4×C7#13711yescharacteristic
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—42no—
177C2×Dic7—42no—
178C2×Dic7—42no—
179C4×D7—41no—
180C4×D7—41no—
181C4×D7—41no—
182C4×D7—41no—
183C7×C8—42no—
184C7×C8—81no—
185C7⋊C8—42no—
186C7⋊C8—81no—
187C22×Q8—21no—
188C23×C4—71no—
189C2×M16—21no—
190C2×M16—141no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195#117—141no—
196#117—141no—
197#117—141no—
198#117—141no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—71no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—141no—
203C2×C4⋊C4—142no—
204C2×C4⋊C4C4×D712yes—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4—142no—
208C2×C42—71no—
209C2×C42C22×D711yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—72no—
213C4⋊C8—141no—
214C4⋊C8—141no—
215C4⋊C8C22⋊D712yes—
216C22⋊C8—21no—
217C22⋊C8—144no—
218C22⋊C8—141no—
219#97—142no—
220C4×C8—21no—
221C4×C8—142no—
222C4×C8—141no—
223#95—72no—
224#95—142no—
225#95—142no—
226#95—141no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7#38611yescharacteristic
230C22×C7#41611yescharacteristic
231C22×C7#41911yescharacteristic
232C2×D7—42no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—81no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—82no—
242C4×C7#45211yescharacteristic
243C4×C7#45211yescharacteristic
244C4×C7#48511yescharacteristic
245C4×C7#48511yescharacteristic
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—82no—
250C24—71no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—82no—
254C22×C4—22no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4—141no—
264C22×C4—142no—
265C22×C4C2×C4×D711yescharacteristic
266C22×C4D4×D711yescharacteristic
267C22×C4D4×D711yescharacteristic
268M16—41no—
269M16—41no—
270M16—281no—
271M16—281no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—142no—
275C2×C8—281no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—141no—
281C4⋊C4—141no—
282C4⋊C4—141no—
283C4⋊C4—141no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—562no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—562no—
292C42—141no—
293C42—141no—
294C42#115711yescharacteristic
295C42D4×D711yescharacteristic
296C42#115911yescharacteristic
297C42C2×C22⋊D711yescharacteristic
298C2×C7—21no—
299C2×C7—21no—
300C2×C7#187311yescharacteristic
301C2×C7#191011yescharacteristic
302C2×C7#191511yescharacteristic
303D7—81no—
304C23—141no—
305C23—141no—
306C23—141no—
307C23—281no—
308C23—281no—
309C23D7×C22⋊C411yescharacteristic
310Q8—82no—
311Q8—82no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—281no—
323C2×C4—281no—
324C2×C4—281no—
325C2×C4—281no—
326C2×C4—281no—
327C2×C4—281no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—282no—
331C2×C4(C22⋊D7)⋊C411yescharacteristic
332C2×C4#644611yescharacteristic
333C2×C4#644711yescharacteristic
334C2×C4#647811yescharacteristic
335C2×C4#647911yescharacteristic
336C2×C4#648111yescharacteristic
337C2×C4#648211yescharacteristic
338C2×C4C4×C22⋊D711yescharacteristic
339C2×C4#650011yescharacteristic
340C2×C4#650111yescharacteristic
341C8—42no—
342C8—81no—
343C8—282no—
344C8—561no—
345C7#919411yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—282no—
349C22—281no—
350C22—281no—
351C22—281no—
352C22—561no—
353C22#8957411yescharacteristic
354C22#8957611yescharacteristic
355C22#8972811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—281no—
362C4—281no—
363C4—281no—
364C4—562no—
365C4#8958611yescharacteristic
366C4#8958711yescharacteristic
367C4#8994511yescharacteristic
368C4#8995911yescharacteristic
369C2—21no—
370C2—21no—
371C2—561no—
372C2#1172787811yescharacteristic
373C2#1172802311yescharacteristic
374C2#1172808411yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1549C2
socleC22×C7#386
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1549D7
hypocenterC7#9194
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C7×#1549C2semidirect
4. #11729529C2semidirect
6. #11725544C2semidirect
7. #11725519C2semidirect
73. #1549D7semidirect
345. C7#9194semidirect

Maximal subgroups

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

subgroupindex
#117407072
#117364842
C7×#15492
#117295292
#117285362
#117255442
#117255192
#91947