Finite Group Databasebeta

← #13243134

Subgroups — #13243134

Order 1792.

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2806C211yesmaximal, characteristic
2#11727438C211yesmaximal, characteristic
3#11727438C211yesmaximal, characteristic
4#11727366C211yesmaximal, characteristic
5#11725448C211yesmaximal, characteristic
6#11725448C211yesmaximal, characteristic
7#11725444C211yesmaximal, characteristic
8C7×#398—21no—
9C7×#398—21no—
10C7×#395—21no—
11C7×#391—21no—
12C7×#391C2211yescharacteristic
13C7×#391C2211yescharacteristic
14C7×#389—21no—
15C7×#389C2211yescharacteristic
16C7×#387—21no—
17C2×#6409—21no—
18C2×#6409—21no—
19C2×#6381—21no—
20C2×#6381—21no—
21#89707C2211yescharacteristic
22#89324—21no—
23#89324—21no—
24#89319C2211yescharacteristic
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7836—71nomaximal, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×Q8D411yescharacteristic
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7C2311yescharacteristic
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—21no—
43C2×C4⋊Dic7—21no—
44C22×C7⋊C8—21no—
45C22×C7⋊C8—21no—
46C22×C7⋊C8—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#6409—42no—
56#6409—42no—
57#6381—42no—
58#6381—42no—
59#2806D711yescharacteristic
60#2100—71no—
61#2095—71no—
62#2095—71no—
63#1272—71no—
64#1272—71no—
65#1271—71no—
66C2×C7×Q8—21no—
67C2×C7×Q8—21no—
68C2×C7×Q8—21no—
69C2×C7×Q8—41no—
70C2×C7×Q8—41no—
71C2×C7×Q8—41no—
72C2×C7×Q8—41no—
73C2×C7×Q8Q1612yes—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×Dic7—41no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—81no—
91C7×C4⋊C4—81no—
92C7×C4⋊C4—81no—
93C7×C4⋊C4—81no—
94C7×C4⋊C4Q1612yes—
95C42×C7—41no—
96C42×C7—81no—
97C4⋊Dic7—41no—
98C4⋊Dic7—41no—
99C4⋊Dic7—42no—
100C4⋊Dic7—41no—
101C4⋊Dic7—41no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8—22no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—41no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111C2×#107—141no—
112C2×#107—141no—
113C2×#103—141no—
114C2×#103—141no—
115#398—21no—
116#398—21no—
117#395—21no—
118#391—21no—
119#391C2×D711yescharacteristic
120#391C2×D711yescharacteristic
121#389—21no—
122#389C2×D711yescharacteristic
123#387—21no—
124#384—71no—
125#340—141no—
126#340—141no—
127#336—71no—
128#336—71no—
129#336—71no—
130C23×C7#12011yescharacteristic
131C7×Q8—42no—
132C7×Q8—42no—
133C7×Q8—81no—
134C7×Q8—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7C2×Q1611yescharacteristic
160C2×C4×C7C2×Q1611yescharacteristic
161C2×C4×C7#13511yescharacteristic
162C2×C4×C7#13711yescharacteristic
163C2×C4×C7#13711yescharacteristic
164C2×C4×C7#13711yescharacteristic
165C2×Dic7—81no—
166C2×Dic7—81no—
167C2×Dic7—81no—
168C7⋊C8—42no—
169C7⋊C8—42no—
170C7⋊C8—81no—
171C22×Q8C22⋊D711yescharacteristic
172C22×Q8C22⋊D711yescharacteristic
173C22×C8—141no—
174C22×C8—141no—
175C22×C8—141no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—141no—
182C2×C4⋊C4—141no—
183C2×C4⋊C4—141no—
184C2×C4⋊C4C22⋊D711yescharacteristic
185C2×C4⋊C4C22⋊D711yescharacteristic
186C2×C4⋊C4C22⋊D711yescharacteristic
187C2×C4⋊C4C22⋊D711yescharacteristic
188C2×C42—21no—
189C2×C42C22×D711yescharacteristic
190#107—282no—
191#107—282no—
192#103—282no—
193#103—282no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202C22×C7#45111yescharacteristic
203C22×C7#45111yescharacteristic
204C22×C7#45211yescharacteristic
205C22×C7#45311yescharacteristic
206C22×C7#45311yescharacteristic
207C22×C7#45411yescharacteristic
208C22×C7#45811yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—42no—
216C4×C7—42no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—81no—
221Dic7—161no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8#114412yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—281no—
237C22×C4C2×C22⋊D711yescharacteristic
238C22×C4C2×C22⋊D711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C2×C8—142no—
241C2×C8—142no—
242C2×C8—281no—
243C2×C8—281no—
244C2×C8—281no—
245C2×C8—281no—
246C2×C8—281no—
247C2×C8—281no—
248C2×C8—281no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C4⋊C4—281no—
262C4⋊C4—281no—
263C4⋊C4—282no—
264C4⋊C4#114412yes—
265C42—41no—
266C42—81no—
267C2×C7#161411yescharacteristic
268C2×C7#161711yescharacteristic
269C2×C7#161811yescharacteristic
270C2×C7#163211yescharacteristic
271C2×C7#163311yescharacteristic
272C2×C7#163611yescharacteristic
273C2×C7#165611yescharacteristic
274C23#651611yescharacteristic
275Q8—42no—
276Q8—42no—
277Q8—81no—
278Q8—81no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—561no—
304C2×C4—561no—
305C2×C4—561no—
306C2×C4#650511yescharacteristic
307C2×C4#650511yescharacteristic
308C2×C4C2×#114411yescharacteristic
309C2×C4C2×#114411yescharacteristic
310C2×C4#651311yescharacteristic
311C2×C4#651411yescharacteristic
312C8—282no—
313C8—282no—
314C8—561no—
315C7#783611yescharacteristic, complemented, Sylow
316C22#9000011yescharacteristic
317C22#9000011yescharacteristic
318C22#9001111yescharacteristic
319C22#9001311yescharacteristic
320C22#9001311yescharacteristic
321C22#9001511yescharacteristic
322C22#9001711yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—1121no—
336C2#1172963711yescharacteristic
337C2#1172964511yescharacteristic
338C2#1172965311yescharacteristic
339C2#1172967811yescharacteristic
340C2#1172969011yescharacteristic
341C2#1172969411yescharacteristic
342C2#1172970611yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2806C2
socle, prime radicalC23×C7#120
hypercenter#2806D7
hypocenterC7#7836
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2806

Complements

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

NH ≅ G/Ntype
315. C7#7836semidirect

Maximal subgroups

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

subgroupindex
C7×#28062
#117274382
#117273662
#117254482
#117254442
#78367