Finite Group Databasebeta

← #13267183

Subgroups — #13267183

Order 1792.

280 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3256C211yesmaximal, characteristic
2#11728844C211yesmaximal, characteristic, complemented
3#11728842C211yesmaximal, characteristic
4#11728280C211yesmaximal, characteristic, complemented
5#11725586C211yesmaximal, characteristic, complemented
6#11725584C211yesmaximal, characteristic
7#11725523C211yesmaximal, characteristic, complemented
8C7×#529—21no—
9C7×#479—21no—
10C7×#478—21no—
11C7×#470C2211yescharacteristic
12C7×#469C2211yescharacteristic
13C7×#468—21no—
14C7×#464—21no—
15C7×#463—21no—
16C7×#423C2211yescharacteristic
17#89897C2211yescharacteristic
18#89360—21no—
19#89358—21no—
20#89354—21no—
21#89352C2211yescharacteristic
22#89348C2211yescharacteristic
23#89348C2211yescharacteristic
24#89318—21no—
25#89282—21no—
26#89282—21no—
27#12309—71nomaximal, Sylow
28C2×C7×Q16D411yescharacteristic
29C2×C7×SD16—21no—
30C2×C7×D8D411yescharacteristic
31C2×C7×M16D411yescharacteristic
32C2×C7×M16C2311yescharacteristic
33C7×#126—41no—
34C7×#125—21no—
35C7×#124—21no—
36C7×#123—41no—
37C7×C22⋊Q8—21no—
38C7×#121—21no—
39C4×C7×Q8—21no—
40C4×C7×D4—21no—
41C4×C7×D4—41no—
42C7×#117—41no—
43C2×C42×C7D411yescharacteristic
44#6489—21no—
45C2×#1136—21no—
46C7×#107—21no—
47C7×#106D411yescharacteristic
48C7×#106D411yescharacteristic
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×#103—21no—
52C7×#103—21no—
53C7×#102—21no—
54C7×#102—21no—
55#6393—21no—
56#6393—21no—
57#6386—21no—
58#6386—21no—
59#3256D711yescharacteristic
60#2239—71no—
61#2238—71no—
62#2102—71no—
63#1340—71no—
64#1339—71no—
65#1327—71no—
66C2×C7×Q8—21no—
67C2×C7×D4—21no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
70C22×C4×C7—41no—
71C22×C4×C7C2×D411yescharacteristic
72C7×Q16—41no—
73C7×SD16—81no—
74C7×D8—41no—
75C7×M16—41no—
76C7×M16—41no—
77C2×C7×C8—21no—
78C2×C7×C8C2×D411yescharacteristic
79C2×C7×C8C2×D411yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C42×C7—21no—
92C42×C7—41no—
93C42×C7SD1611yescharacteristic
94C42×C7SD1611yescharacteristic
95C22⋊Dic7—41no—
96C4⋊Dic7—41no—
97C4×Dic7—41no—
98#1136—41no—
99#1136—41no—
100C2×C7⋊C8—41no—
101C7⋊C16—41no—
102C7⋊C16—41no—
103#529—21no—
104#479—21no—
105#478—21no—
106#470C2×D711yescharacteristic
107#469C2×D711yescharacteristic
108#468—21no—
109#464—21no—
110#463—21no—
111#430—71no—
112#423C2×D711yescharacteristic
113#365—141no—
114#365—141no—
115#362—141no—
116#361—141no—
117#349—71no—
118#349—71no—
119#344—141no—
120#339—141no—
121H(3,ℤ/4)—71no—
122C23×C7—41no—
123C23×C7AGL(1,ℤ/8)11yescharacteristic
124C7×Q8—41no—
125C7×Q8—41no—
126C7×D4—41no—
127C7×D4—41no—
128C7×D4—82no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—81no—
137C2×C4×C7—82no—
138C2×C4×C7#12011yescharacteristic
139C2×C4×C7C2×SD1611yescharacteristic
140C2×Dic7—81no—
141C7×C8—21no—
142C7×C8—21no—
143C7×C8—41no—
144C7⋊C8—81no—
145C2×Q16C22⋊D711yescharacteristic
146C2×SD16—21no—
147C2×D8C22⋊D711yescharacteristic
148C2×M16—141no—
149C2×M16C22⋊D711yescharacteristic
150C2×M16C22×D711yescharacteristic
151#126—41no—
152#125—21no—
153#124—21no—
154#123—41no—
155C22⋊Q8—21no—
156#121—21no—
157C4×Q8—21no—
158C4×D4—21no—
159C4×D4—41no—
160#117—41no—
161#117—141no—
162C2×C42C22⋊D711yescharacteristic
163M32—141no—
164M32—141no—
165#108—141no—
166#108—141no—
167#107—21no—
168#106C22⋊D711yescharacteristic
169#106C22⋊D711yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172#103—21no—
173#103—21no—
174#102—21no—
175#102—21no—
176C22×C7—21no—
177C22×C7—41no—
178C22×C7—81no—
179C22×C7—81no—
180C22×C7—81no—
181C22×C7#45011yescharacteristic
182C4×C7—21no—
183C4×C7—41no—
184C4×C7—41no—
185C4×C7—81no—
186C4×C7—81no—
187C4×C7—81no—
188C4×C7#45411yescharacteristic
189C4×C7#45811yescharacteristic
190Dic7—161no—
191C2×Q8—21no—
192C2×D4—21no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4C2×C22⋊D711yescharacteristic
197Q16—41no—
198SD16—81no—
199D8—41no—
200M16—41no—
201M16—41no—
202M16—281no—
203M16—281no—
204C2×C8—21no—
205C2×C8—281no—
206C2×C8C2×C22⋊D711yescharacteristic
207C2×C8C2×C22⋊D711yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—281no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—281no—
221C42—21no—
222C42—41no—
223C42—281no—
224C42#114211yescharacteristic
225C42#114211yescharacteristic
226C16—281no—
227C16—281no—
228C2×C7—21no—
229C2×C7—41no—
230C2×C7—161no—
231C2×C7#161611yescharacteristic
232C23—41no—
233C23#649511yescharacteristic
234Q8—41no—
235Q8—41no—
236D4—41no—
237D4—41no—
238D4—82no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—561no—
249C2×C4C2×#114211yescharacteristic
250C2×C4#651611yescharacteristic
251C8—21no—
252C8—21no—
253C8—41no—
254C8—561no—
255C7#1230911yescharacteristic, complemented, Sylow
256C22—21no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#8999011yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—1121no—
269C4#9001511yescharacteristic
270C4#9001711yescharacteristic
271C2—21no—
272C2—41no—
273C2—161no—
274C2#1172962911yescharacteristic
subgroupG/H
centerC2#11729629
commutator, cosocleC2×C7×M16C23
FrattiniC2×M16C22×D7
FittingC7×#3256C2
socleC2×C7#1616
prime radicalC7×#469C22
hypercenter#3256D7
hypocenterC7#12309
LCSG ▷ C2×C7×M16 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3256

Complements

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

NH ≅ G/Ntype
2. #11728844C2semidirect
4. #11728280C2semidirect
5. #11725586C2semidirect
7. #11725523C2semidirect
255. C7#12309semidirect

Maximal subgroups

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

subgroupindex
C7×#32562
#117288442
#117288422
#117282802
#117255862
#117255842
#117255232
#123097