Finite Group Databasebeta

← #13242286

Subgroups — #13242286

Order 1792.

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

normal subgroups only

This group has 410 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#11733637C211yesmaximal, characteristic
2#11726357C211yesmaximal, characteristic
3#11726356C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic
5#11725444C211yesmaximal, characteristic
6#11725286C211yesmaximal, characteristic
7#11725286C211yesmaximal, characteristic
8C7×#389C2211yescharacteristic
9C2×#6389—21no—
10C2×Dic7⋊C8—21no—
11C2×Dic7⋊C8—21no—
12C2×C8×Dic7—21no—
13#89757C411yescharacteristic
14#89753—21no—
15#89750—21no—
16#89746C411yescharacteristic
17#89746C411yescharacteristic
18#89745C411yescharacteristic
19C42×Dic7C2211yescharacteristic
20#89701C2211yescharacteristic
21#89426—21no—
22#89423—21no—
23C7×#336C2211yescharacteristic
24C7×#336C2211yescharacteristic
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7689—71nomaximal, Sylow
28C22×Dic14C2×C411yescharacteristic
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—21no—
38C2×C4×Dic7C2×C411yescharacteristic
39C2×C4×Dic7C2×C411yescharacteristic
40C2×C4×Dic7D411yescharacteristic
41C2×C4×Dic7D411yescharacteristic
42C2×C4×Dic7D411yescharacteristic
43C2×C4×Dic7D411yescharacteristic
44C22×C7⋊C8—21no—
45C22×C7⋊C8—21no—
46C7×#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55#6389—42no—
56Dic7⋊C8—24no—
57Dic7⋊C8—24no—
58C8×Dic7—42no—
59#2863—71no—
60#1767—71no—
61C4×C4⋊C8—71no—
62#1272—71no—
63#1272—71no—
64#1271—71no—
65#1271D711yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×Dic7—21no—
72C22×Dic7—21no—
73C22×Dic7C22⋊C411yescharacteristic
74C22×Dic7C22⋊C411yescharacteristic
75C22×Dic7C22⋊C411yescharacteristic
76C22×Dic7C22⋊C411yescharacteristic
77C2×Dic14—21no—
78C2×Dic14—21no—
79C2×Dic14—41no—
80C2×Dic14—41no—
81C2×C7×C8—22no—
82C2×C7×C8—22no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C2×C7×C8—41no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C42×C7—41no—
93C4⋊Dic7—42no—
94C4⋊Dic7—41no—
95Dic7⋊C4—42no—
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—22no—
101C4×Dic7—22no—
102C4×Dic7—41no—
103C4×Dic7SD1612yes—
104C4×Dic7Q1612yes—
105C2×C7⋊C8—22no—
106C2×C7⋊C8—22no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—41no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111C2×C4⋊C8—141no—
112C2×C4⋊C8—141no—
113C2×#97—141no—
114C2×C4×C8—141no—
115#391—71no—
116#391—71no—
117#391—71no—
118#389—71no—
119#389—71no—
120#389C2×D711yescharacteristic
121#387—141no—
122#387—141no—
123#387—141no—
124#383—141no—
125C43—71no—
126#336—71no—
127#336—71no—
128#336C2×D711yescharacteristic
129#336C2×D711yescharacteristic
130C23×C7C2×C22⋊C411yescharacteristic
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7#13111yescharacteristic
144C2×C4×C7#13111yescharacteristic
145C2×C4×C7C2×SD1611yescharacteristic
146C2×C4×C7C2×Q1611yescharacteristic
147C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C7#13711yescharacteristic
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7#10312yes—
164C2×Dic7#10312yes—
165C2×Dic7#10412yes—
166C2×Dic7#10412yes—
167Dic14—81no—
168Dic14—81no—
169C7×C8—42no—
170C7×C8—42no—
171C7⋊C8—42no—
172C7⋊C8—42no—
173C22×Q8—71no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—141no—
177C22×C8—141no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—141no—
181C2×C4⋊C4—141no—
182C2×C4⋊C4C4×D711yescharacteristic
183C2×C4⋊C4C4×D711yescharacteristic
184C2×C42—71no—
185C2×C42—71no—
186C2×C42—71no—
187C2×C42—71no—
188C2×C42—71no—
189C2×C42—71no—
190C2×C42C22×D711yescharacteristic
191C4⋊C8—144no—
192C4⋊C8—144no—
193#97—282no—
194C4×C8—282no—
195#95—21no—
196#95—141no—
197#95—141no—
198#95—141no—
199#95—141no—
200#95—141no—
201#95—141no—
202#95—141no—
203#95—141no—
204C22×C7#40811yescharacteristic
205C22×C7#41011yescharacteristic
206C22×C7#41311yescharacteristic
207C22×C7C2×#10311yescharacteristic
208C22×C7#41711yescharacteristic
209C22×C7C2×#10411yescharacteristic
210C22×C7C2×#10411yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—81no—
218C4×C7—81no—
219Dic7—22no—
220Dic7—22no—
221Dic7—22no—
222Dic7—22no—
223Dic7—81no—
224Dic7—81no—
225C2×Q8—141no—
226C2×Q8—141no—
227C2×Q8—281no—
228C2×Q8—281no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4C2×C4×D711yescharacteristic
238C22×C4D4×D711yescharacteristic
239C22×C4D4×D711yescharacteristic
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—142no—
247C2×C8—142no—
248C2×C8—281no—
249C2×C8—281no—
250C2×C8—281no—
251C2×C8—281no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—281no—
258C4⋊C4—282no—
259C4⋊C4—282no—
260C42—41no—
261C42—72no—
262C42—72no—
263C42—142no—
264C42—142no—
265C42—141no—
266C42—141no—
267C42—141no—
268C42—141no—
269C42—281no—
270C2×C7#147311yescharacteristic
271C2×C7#148311yescharacteristic
272C2×C7#149311yescharacteristic
273C2×C7#150311yescharacteristic
274C2×C7#150911yescharacteristic
275C2×C7#152411yescharacteristic
276C2×C7#155011yescharacteristic
277C23D7×C22⋊C411yescharacteristic
278Q8—561no—
279Q8—561no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—72no—
293C2×C4—72no—
294C2×C4—72no—
295C2×C4—72no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4—281no—
307C2×C4—281no—
308C2×C4—281no—
309C2×C4—281no—
310C2×C4#646411yescharacteristic
311C2×C4#647011yescharacteristic
312C2×C4#647411yescharacteristic
313C2×C4D7×SD1611yescharacteristic
314C2×C4#647811yescharacteristic
315C2×C4D7×Q1611yescharacteristic
316C8—42no—
317C8—42no—
318C8—282no—
319C8—282no—
320C7#768911yescharacteristic, complemented, Sylow
321C22#8949911yescharacteristic
322C22#8951211yescharacteristic
323C22#8952311yescharacteristic
324C22#8954311yescharacteristic
325C22D7×#10311yescharacteristic
326C22D7×#10411yescharacteristic
327C22D7×#10411yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—81no—
335C4—81no—
336C4—142no—
337C4—142no—
338C4—142no—
339C4—142no—
340C4—561no—
341C4—561no—
342C2#1172588311yescharacteristic
343C2#1172592411yescharacteristic
344C2#1172599411yescharacteristic
345C2#1172603311yescharacteristic
346C2#1172606311yescharacteristic
347C2#1172611511yescharacteristic
348C2#1172617711yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socle, prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1271D7
hypocenterC7#7689
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
320. C7#7689semidirect

Maximal subgroups

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

subgroupindex
#117336372
#117263572
#117263562
C7×#12712
#117254442
#117252862
#76897