Finite Group Databasebeta

← #13267519

Subgroups — #13267519

Order 1792.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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#11737375C211yesmaximal, characteristic
2C7×#1670C211yesmaximal, characteristic, complemented
3#11729069C211yesmaximal, characteristic
4#11729064C211yesmaximal, characteristic, complemented
5Q8⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
6#11725282C211yesmaximal, characteristic, complemented
7#11725277C211yesmaximal, characteristic
8#90270—21no—
9C7×#494C2211yescharacteristic
10C7×#475—21no—
11C7×C8×Q8C2211yescharacteristic
12C2×#6402—21no—
13#89619—22no—
14#89614—21no—
15C4×D56—21no—
16C4×D56C2211yescharacteristic
17#89455—21no—
18#89454C2211yescharacteristic
19C7×#335C2211yescharacteristic
20C7×#332—21no—
21#89359—22no—
22#89359—21no—
23C4⋊(C7⋊C16)C2211yescharacteristic
24C4×C7⋊C16C2211yescharacteristic
25#12359—71nomaximal, Sylow
26C2×C7×Q16—21no—
27C2×C7×Q16—21no—
28C7×C4⋊Q8—21no—
29C4×C7×Q8D411yescharacteristic
30#6509—41no—
31C2×D56D411yescharacteristic
32#6459—42no—
33D7×C4⋊C4—41no—
34C4×D28—22no—
35C4×D28—21no—
36#6433—22no—
37#6432—21no—
38C7×#106—41no—
39C7×C4⋊C8—21no—
40C7×C4⋊C8—21no—
41C7×C4⋊C8D411yescharacteristic
42C7×#103—41no—
43C4×C7×C8—21no—
44C4×C7×C8C2311yescharacteristic
45#6402—42no—
46#6395—22no—
47#6395—21no—
48#6392—21no—
49#6392D412yes—
50#6392D411yescharacteristic
51#6391—22no—
52C2×C7⋊C16—21no—
53C2×C7⋊C16—21no—
54#3380—71no—
55#2241—71no—
56#2208—71no—
57#1670D711yescharacteristic, complemented
58#1365—71no—
59Q8⋊C16—71no—
60#1331—71no—
61C2×C7×Q8—41no—
62C2×C7×Q8D811yescharacteristic
63C2×D28—21no—
64C2×C4×D7—41no—
65C2×C4×D7—42no—
66C2×Dic14—41no—
67C7×Q16—42no—
68C7×Q16—41no—
69C7×Q16—41no—
70C2×C7×C8—21no—
71C2×C7×C8—41no—
72C2×C7×C8C2×D411yescharacteristic
73C2×C7×C8C2×D411yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4D811yescharacteristic
78C42×C7—21no—
79C42×C7C2×D411yescharacteristic
80(C2×D7)⋊C4—42no—
81(C2×D7)⋊C4—41no—
82C4⋊Dic7—22no—
83C4⋊Dic7—21no—
84C4⋊Dic7—41no—
85Dic7⋊C4—42no—
86Dic7⋊C4—42no—
87D56—21no—
88D56SD1612yes—
89C7⋊C16—41no—
90C7⋊C16—41no—
91#554—141no—
92C2×SD32—141no—
93#500—71no—
94#499—141no—
95#494C2×D711yescharacteristic
96#478—142no—
97#476—141no—
98#475—21no—
99C8×Q8C2×D711yescharacteristic
100C4×D8—71no—
101C4×D8—141no—
102C4⋊C16—71no—
103#358—141no—
104#358—142no—
105C4×C16—71no—
106#335C2×D711yescharacteristic
107#332—21no—
108C22×D7—41no—
109C7×Q8—21no—
110C7×Q8—21no—
111C7×Q8—81no—
112C7×Q8—81no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—81no—
116C2×C4×C7#12011yescharacteristic
117C2×C4×C7C2×D811yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×Dic7—41no—
120C2×Dic7—42no—
121D28—22no—
122D28—41no—
123C4×D7—81no—
124C4×D7—81no—
125C4×D7—162no—
126Dic14—161no—
127C7×C8—21no—
128C7×C8—41no—
129C7×C8—81no—
130C7×C8C2×SD1611yescharacteristic
131C7×C8#13711yescharacteristic
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—71no—
135C4⋊Q8—21no—
136C4⋊Q8—141no—
137#125—142no—
138C22⋊Q8—282no—
139C22⋊Q8—281no—
140C4×Q8C22⋊D711yescharacteristic
141C4×D4—142no—
142C4×D4—141no—
143C2×C4⋊C4—281no—
144SD32—282no—
145C2×C16—141no—
146C2×C16—141no—
147#107—71no—
148#107—72no—
149#107—141no—
150#106—41no—
151#106—142no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8C22⋊D711yescharacteristic
155#103—41no—
156#102—142no—
157#102—141no—
158C4×C8—21no—
159C4×C8C22×D711yescharacteristic
160C22×C7#44711yescharacteristic
161C2×D7—42no—
162C2×D7—81no—
163C2×D7—81no—
164C4×C7—21no—
165C4×C7—41no—
166C4×C7—41no—
167C4×C7—161no—
168C4×C7#44811yescharacteristic
169C4×C7#45411yescharacteristic
170C4×C7#50811yescharacteristic
171C4×C7#51011yescharacteristic
172Dic7—81no—
173Dic7—82no—
174C2×Q8—41no—
175C2×Q8—281no—
176C2×Q8#114111yescharacteristic
177C2×D4—141no—
178C22×C4—282no—
179C22×C4—281no—
180Q16—42no—
181Q16—41no—
182Q16—41no—
183D8—72no—
184D8—141no—
185C2×C8—21no—
186C2×C8—41no—
187C2×C8D4×D711yescharacteristic
188C2×C8D4×D711yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—141no—
193C4⋊C4—142no—
194C4⋊C4—282no—
195C4⋊C4—282no—
196C4⋊C4—281no—
197C4⋊C4#114111yescharacteristic
198C22⋊C4—282no—
199C22⋊C4—281no—
200C42—21no—
201C42C2×C22⋊D711yescharacteristic
202C16—281no—
203C16—281no—
204C2×C7#162711yescharacteristic
205C2×C7#218911yescharacteristic
206C2×C7#219411yescharacteristic
207D7—82no—
208C23—281no—
209Q8—21no—
210Q8—21no—
211Q8—81no—
212Q8—81no—
213Q8—1121no—
214D4—142no—
215D4—281no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—81no—
219C2×C4—281no—
220C2×C4—282no—
221C2×C4—561no—
222C2×C4—561no—
223C2×C4—1122no—
224C2×C4C2×#114111yescharacteristic
225C2×C4#650011yescharacteristic
226C2×C4#651211yescharacteristic
227C8—21no—
228C8—41no—
229C8—81no—
230C8D7×SD1611yescharacteristic
231C8#647811yescharacteristic
232C7#1235911yescharacteristic, complemented, Sylow
233C22—282no—
234C22—561no—
235C22—561no—
236C22#8980811yescharacteristic
237C4—21no—
238C4—41no—
239C4—41no—
240C4—161no—
241C4—561no—
242C4—562no—
243C4#8994211yescharacteristic
244C4#8995111yescharacteristic
245C4#8996611yescharacteristic
246C4#8997311yescharacteristic
247C2—562no—
248C2#1172878911yescharacteristic
249C2#1172884011yescharacteristic
250C2#1172885211yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1670C2
socleC22×C7#447
prime radicalC2×D56D4
hypercenter#1670D7
hypocenterC7#12359
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. C7×#1670C2semidirect
4. #11729064C2semidirect
5. Q8⋊(C7⋊C16)C2semidirect
6. #11725282C2semidirect
57. #1670D7semidirect
232. C7#12359semidirect

Maximal subgroups

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

subgroupindex
#117373752
C7×#16702
#117290692
#117290642
Q8⋊(C7⋊C16)2
#117252822
#117252772
#123597