Finite Group Databasebeta

← #13242296

Subgroups — #13242296

Order 1792.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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#11733730C211yesmaximal, characteristic
2D7×#336C211yesmaximal, characteristic
3#11726354C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic, complemented
5#11725444C211yesmaximal, characteristic, complemented
6#11725284C211yesmaximal, characteristic, complemented
7#11725283C211yesmaximal, characteristic, complemented
8C22×C8×D7—21no—
9C2×#6453C411yescharacteristic
10C2×C42×D7C2211yescharacteristic
11C7×#389C2211yescharacteristic
12C2×(C2×D7)⋊C8—21no—
13C2×#6389—21no—
14#89761—21no—
15#89759C411yescharacteristic
16#89759C411yescharacteristic
17#89753—21no—
18#89749C411yescharacteristic
19#89700C2211yescharacteristic
20#89440—21no—
21#89424—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7691—71nomaximal, Sylow
28C22×C4×D7C2×C411yescharacteristic
29C22×C4×D7D411yescharacteristic
30C22×C4×D7D411yescharacteristic
31C22×C7×C8—21no—
32C22×C7×C8—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C42×C7C2311yescharacteristic
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7C2×C411yescharacteristic
45C2×C4×Dic7D411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C22×C7⋊C8—21no—
48C22×C7⋊C8—21no—
49C2×C8×D7—24no—
50C2×C8×D7—42no—
51C2×C8×D7—42no—
52#6453—42no—
53C42×D7—42no—
54C7×#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60(C2×D7)⋊C8—24no—
61#6389—42no—
62#2851—71no—
63#1749—71no—
64C2×#336—71no—
65#1271—71no—
66#1271—71no—
67#1271—71no—
68#1271D711yescharacteristic, complemented
69C23×D7C22⋊C411yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7C22⋊C411yescharacteristic
78C22×Dic7C22⋊C411yescharacteristic
79C22×Dic7C22⋊C411yescharacteristic
80C2×C4×D7—22no—
81C2×C4×D7—21no—
82C2×C4×D7—21no—
83C2×C4×D7—22no—
84C2×C4×D7—22no—
85C2×C4×D7—22no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—41no—
91C2×C4×D7—41no—
92C2×C7×C8—22no—
93C2×C7×C8—22no—
94C2×C7×C8—41no—
95C2×C7×C8—41no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C42×C7—41no—
104(C2×D7)⋊C4—42no—
105(C2×D7)⋊C4—42no—
106C4⋊Dic7—21no—
107C4⋊Dic7—21no—
108C4⋊Dic7—42no—
109C4⋊Dic7—41no—
110Dic7⋊C4—41no—
111Dic7⋊C4—41no—
112Dic7⋊C4—42no—
113C4×Dic7—22no—
114C4×Dic7—22no—
115C4×Dic7—41no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—22no—
118C2×C7⋊C8—41no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121C2×C7⋊C8—41no—
122C8×D7—48no—
123C23×C8—141no—
124C2×#125—71no—
125C22×C42—71no—
126C2×C22⋊C8—141no—
127C2×#97—141no—
128#389—71no—
129#389C2×D711yescharacteristic
130#387—141no—
131#387—141no—
132#385—141no—
133#385—141no—
134#382—71no—
135#380—71no—
136#380—71no—
137#336—71no—
138#336—71no—
139#336—141no—
140#336C2×D711yescharacteristic
141#336C2×D711yescharacteristic
142C23×C7C2×C22⋊C411yescharacteristic
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7#10112yes—
148C22×D7#10412yes—
149C22×D7#10412yes—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7#13111yescharacteristic
163C2×C4×C7#13111yescharacteristic
164C2×C4×C7#13511yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C7#13711yescharacteristic
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7#9912yes—
183C4×D7—24no—
184C4×D7—24no—
185C4×D7—42no—
186C4×D7—42no—
187C4×D7—42no—
188C4×D7—42no—
189C7×C8—42no—
190C7×C8—42no—
191C7⋊C8—42no—
192C7⋊C8—42no—
193C23×C4—71no—
194C23×C4—71no—
195C23×C4—71no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—144no—
199C22×C8—141no—
200C22×C8—141no—
201C22×C8—282no—
202C22×C8—282no—
203#125—282no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—141no—
208C2×C4⋊C4—141no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4C4×D711yescharacteristic
211C2×C4⋊C4C4×D711yescharacteristic
212C2×C22⋊C4—141no—
213C2×C22⋊C4—141no—
214C2×C42—71no—
215C2×C42—71no—
216C2×C42—71no—
217C2×C42—282no—
218C2×C42C22×D711yescharacteristic
219C22⋊C8—144no—
220#97—282no—
221#95—21no—
222#95—141no—
223#95—141no—
224#95—141no—
225#95—141no—
226#95—141no—
227C22×C7#40811yescharacteristic
228C22×C7C2×#9911yescharacteristic
229C22×C7C2×#10111yescharacteristic
230C22×C7#41611yescharacteristic
231C22×C7#41711yescharacteristic
232C22×C7C2×#10411yescharacteristic
233C22×C7C2×#10411yescharacteristic
234C2×D7—22no—
235C2×D7—22no—
236C2×D7—22no—
237C2×D7—22no—
238C2×D7—22no—
239C2×D7—22no—
240C2×D7#33014yes—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—81no—
248C4×C7—81no—
249Dic7—22no—
250Dic7—41no—
251Dic7—41no—
252Dic7—81no—
253Dic7—81no—
254C24—71no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—142no—
261C22×C4—142no—
262C22×C4—142no—
263C22×C4—142no—
264C22×C4—141no—
265C22×C4—141no—
266C22×C4—141no—
267C22×C4—141no—
268C22×C4—141no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—281no—
273C22×C4—281no—
274C22×C4C2×C4×D711yescharacteristic
275C22×C4D4×D711yescharacteristic
276C22×C4D4×D711yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—142no—
284C2×C8—142no—
285C2×C8—288no—
286C2×C8—281no—
287C2×C8—281no—
288C2×C8—281no—
289C2×C8—281no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—141no—
296C4⋊C4—141no—
297C4⋊C4—281no—
298C4⋊C4—282no—
299C4⋊C4—282no—
300C4⋊C4—281no—
301C4⋊C4—281no—
302C22⋊C4—282no—
303C22⋊C4—282no—
304C42—41no—
305C42—142no—
306C42—142no—
307C42—281no—
308C2×C7#147211yescharacteristic
309C2×C7#148311yescharacteristic
310C2×C7#149411yescharacteristic
311C2×C7#150111yescharacteristic
312C2×C7#150411yescharacteristic
313C2×C7C2×#33011yescharacteristic
314C2×C7#155211yescharacteristic
315D7—24no—
316C23—72no—
317C23—72no—
318C23—72no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23D7×C22⋊C411yescharacteristic
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—72no—
337C2×C4—144no—
338C2×C4—144no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—282no—
348C2×C4—282no—
349C2×C4—282no—
350C2×C4—282no—
351C2×C4—281no—
352C2×C4—281no—
353C2×C4—281no—
354C2×C4—281no—
355C2×C4—281no—
356C2×C4—281no—
357C2×C4#646411yescharacteristic
358C2×C4#647011yescharacteristic
359C2×C4#647411yescharacteristic
360C2×C4#647811yescharacteristic
361C2×C4#647911yescharacteristic
362C2×C4#648211yescharacteristic
363C8—42no—
364C8—42no—
365C8—282no—
366C8—282no—
367C7#769111yescharacteristic, complemented, Sylow
368C22—74no—
369C22—142no—
370C22—142no—
371C22—142no—
372C22—142no—
373C22—142no—
374C22—142no—
375C22#8949911yescharacteristic
376C22D7×#9911yescharacteristic
377C22D7×#10111yescharacteristic
378C22#8954311yescharacteristic
379C22#8957611yescharacteristic
380C22D7×#10411yescharacteristic
381C22D7×#10411yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—142no—
391C4—281no—
392C4—281no—
393C4—561no—
394C4—561no—
395C2—144no—
396C2#1172590111yescharacteristic
397C2#1172592411yescharacteristic
398C2#1172599511yescharacteristic
399C2#1172605611yescharacteristic
400C2#1172608711yescharacteristic
401C2D7×#33011yescharacteristic
402C2#1172617911yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1271D7
hypocenterC7#7691
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. C7×#1271C2semidirect
5. #11725444C2semidirect
6. #11725284C2semidirect
7. #11725283C2semidirect
68. #1271D7semidirect
367. C7#7691semidirect

Maximal subgroups

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

subgroupindex
#117337302
D7×#3362
#117263542
C7×#12712
#117254442
#117252842
#117252832
#76917