Finite Group Databasebeta

← #13267730

Subgroups — #13267730

Order 1792.

302 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 67 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×#3403C211yesmaximal, characteristic
2#11729073C211yesmaximal, characteristic, complemented
3#11728718C211yesmaximal, characteristic, complemented
4#11728641C211yesmaximal, characteristic, complemented
5#11725602C211yesmaximal, characteristic, complemented
6#11725575C211yesmaximal, characteristic, complemented
7#11725568C211yesmaximal, characteristic, complemented
8C7×#559—21no—
9C7×#496—21no—
10C7×#495C2211yescharacteristic
11C7×#464—21no—
12C7×#463—21no—
13C7×#459—21no—
14C7×#446C2211yescharacteristic
15C7×#440—21no—
16C4×C7×D8C2211yescharacteristic
17#89803—21no—
18#89454C2211yescharacteristic
19#89359—21no—
20#89357—21no—
21#89357—21no—
22#89285—21no—
23#89281—21no—
24C4⋊(C7⋊C16)C2211yescharacteristic
25#89256C2211yescharacteristic
26#89252C2211yescharacteristic
27#12390—71nomaximal, Sylow
28C2×C7×Q16D411yescharacteristic
29C2×C7×SD16—21no—
30C2×C7×SD16—21no—
31C2×C7×D8—21no—
32C2×C7×D8D411yescharacteristic
33C2×C7×D8D411yescharacteristic
34C7×#127—21no—
35C7×#124—21no—
36C7×#124—21no—
37C7×#124—41no—
38C7×#123—41no—
39C7×#123—41no—
40C4×C7×Q8—21no—
41C4×C7×Q8D411yescharacteristic
42C4×C7×D4—21no—
43C4×C7×D4—21no—
44C4×C7×D4—41no—
45C2×#1144—41no—
46#6432—21no—
47C7×#107D411yescharacteristic
48C7×C4⋊C8—21no—
49C7×C4⋊C8D411yescharacteristic
50C7×#103—21no—
51C7×#102—21no—
52C7×#102—21no—
53C7×#97—21no—
54C4×C7×C8C2311yescharacteristic
55#6392—21no—
56#6392—21no—
57C2×C7⋊C16—21no—
58C2×C7⋊C16—21no—
59#6384—41no—
60C4⋊(C7⋊C8)—21no—
61#3403D711yescharacteristic
62#2243—71no—
63#2226—71no—
64#1694—71no—
65#1360—71no—
66#1338—71no—
67#1333—71no—
68C2×C7×Q8—21no—
69C2×C7×Q8D811yescharacteristic
70C2×C7×D4—21no—
71C2×C7×D4—21no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C2×Dic14—41no—
77C7×Q16—41no—
78C7×SD16—41no—
79C7×SD16—41no—
80C7×SD16—81no—
81C7×D8—21no—
82C7×D8—21no—
83C7×D8—41no—
84C7×D8—81no—
85C2×C7×C8—21no—
86C2×C7×C8C2×D411yescharacteristic
87C2×C7×C8C2×D411yescharacteristic
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4D811yescharacteristic
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—41no—
98C42×C7—21no—
99C42×C7—41no—
100C42×C7C2×D411yescharacteristic
101#1144—82no—
102C4⋊Dic7—41no—
103C4⋊Dic7—41no—
104C2×C7⋊C8—41no—
105C7⋊C16—41no—
106C7⋊C16—41no—
107#559—21no—
108#500—71no—
109#496—21no—
110#495C2×D711yescharacteristic
111#464—21no—
112#463—21no—
113#462—141no—
114#459—21no—
115#446C2×D711yescharacteristic
116#440—21no—
117C4×D8C2×D711yescharacteristic
118#367—141no—
119C4⋊C16—71no—
120#358—141no—
121#357—141no—
122#357—141no—
123#346—71no—
124#334—71no—
125#332—141no—
126C23×C7—41no—
127C23×C7—41no—
128C7×Q8—41no—
129C7×Q8—41no—
130C7×Q8—41no—
131C7×D4—41no—
132C7×D4—41no—
133C7×D4—41no—
134C7×D4—41no—
135C7×D4—82no—
136C7×D4—161no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—161no—
142C2×C4×C7—161no—
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7C2×D811yescharacteristic
145C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
146C2×Dic7—81no—
147Dic14—81no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—41no—
151C7×C8#13511yescharacteristic
152C7×C8#13711yescharacteristic
153C7⋊C8—81no—
154C2×Q16—281no—
155C2×Q16C22⋊D711yescharacteristic
156C2×SD16—21no—
157C2×SD16—21no—
158C2×D8—21no—
159C2×D8C22⋊D711yescharacteristic
160C2×D8C22⋊D711yescharacteristic
161C4⋊Q8—141no—
162#127—21no—
163#124—21no—
164#124—21no—
165#124—41no—
166#123—41no—
167#123—41no—
168C4×Q8—21no—
169C4×Q8C22⋊D711yescharacteristic
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—41no—
173C2×C16—141no—
174C2×C16—141no—
175#107—141no—
176#107—141no—
177#107C22⋊D711yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—141no—
180C4⋊C8C22⋊D711yescharacteristic
181#103—21no—
182#103—281no—
183#102—21no—
184#102—21no—
185#97—21no—
186C4×C8C22×D711yescharacteristic
187C22×C7—81no—
188C22×C7—81no—
189C22×C7—81no—
190C22×C7—81no—
191C22×C7—81no—
192C22×C7—81no—
193C22×C7#44711yescharacteristic
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—81no—
199C4×C7—81no—
200C4×C7#45211yescharacteristic
201C4×C7#45511yescharacteristic
202Dic7—161no—
203C2×Q8—21no—
204C2×Q8—281no—
205C2×Q8#114111yescharacteristic
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—41no—
211C22×C4—41no—
212Q16—41no—
213Q16—562no—
214SD16—41no—
215SD16—41no—
216SD16—81no—
217D8—21no—
218D8—21no—
219D8—41no—
220D8—81no—
221C2×C8—21no—
222C2×C8—281no—
223C2×C8C2×C22⋊D711yescharacteristic
224C2×C8C2×C22⋊D711yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—281no—
229C4⋊C4—281no—
230C4⋊C4#114111yescharacteristic
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C42—21no—
238C42—41no—
239C42C2×C22⋊D711yescharacteristic
240C16—281no—
241C16—281no—
242C2×C7—161no—
243C2×C7—161no—
244C2×C7#164111yescharacteristic
245C2×C7#219011yescharacteristic
246C2×C7#219411yescharacteristic
247C23—41no—
248C23—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—561no—
253Q8—561no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—82no—
259D4—161no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—161no—
265C2×C4—161no—
266C2×C4—561no—
267C2×C4C2×#114111yescharacteristic
268C2×C4#649511yescharacteristic
269C2×C4#651611yescharacteristic
270C8—21no—
271C8—41no—
272C8—561no—
273C8#651311yescharacteristic
274C8#651411yescharacteristic
275C7#1239011yescharacteristic, complemented, Sylow
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#8998911yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—1121no—
290C4#9001111yescharacteristic
291C4#9001411yescharacteristic
292C2—161no—
293C2—161no—
294C2#1172967411yescharacteristic
295C2#1172986611yescharacteristic
296C2#1172987711yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
Fitting, prime radicalC7×#3403C2
socleC22×C7#447
hypercenter#3403D7
hypocenterC7#12390
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3403

Complements

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

NH ≅ G/Ntype
2. #11729073C2semidirect
3. #11728718C2semidirect
4. #11728641C2semidirect
5. #11725602C2semidirect
6. #11725575C2semidirect
7. #11725568C2semidirect
275. C7#12390semidirect

Maximal subgroups

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

subgroupindex
C7×#34032
#117290732
#117287182
#117286412
#117256022
#117255752
#117255682
#123907