Finite Group Databasebeta

← #13267623

Subgroups — #13267623

Order 1792.

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

normal subgroups only

This group has 350 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#11736881C211yesmaximal, characteristic
2C7×#1675C211yesmaximal, characteristic, complemented
3#11729142C211yesmaximal, characteristic, complemented
4#11729136C211yesmaximal, characteristic, complemented
5D4⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
6#11725280C211yesmaximal, characteristic, complemented
7#11725277C211yesmaximal, characteristic, complemented
8C2×#6467—21no—
9#90242—21no—
10C7×#500C2211yescharacteristic
11C7×#476—21no—
12C7×C8×D4C2211yescharacteristic
13#89888—21no—
14#89613—21no—
15#89505—22no—
16#89471C2211yescharacteristic
17C4×D56C2211yescharacteristic
18C7×#335C2211yescharacteristic
19C7×#333—21no—
20#89287—22no—
21#89286—21no—
22#89285—21no—
23C4⋊(C7⋊C16)C2211yescharacteristic
24C4×C7⋊C16C2211yescharacteristic
25#12375—71nomaximal, Sylow
26C2×#1157—22no—
27C22×C7×C8—21no—
28C7×C4⋊Q8—21no—
29C4×C7×D4D411yescharacteristic
30#6501—41no—
31#6487—22no—
32C2×Dic28—21no—
33C2×Dic28D412yes—
34#6467—42no—
35#6467—42no—
36C2×D56D411yescharacteristic
37C2×#1132—22no—
38D7×C4⋊C4—41no—
39#6446—42no—
40C4×D28—21no—
41#6432—21no—
42C7×#107—21no—
43C7×#107—21no—
44C7×#106—41no—
45C7×C4⋊C8—21no—
46C7×C4⋊C8D411yescharacteristic
47C7×#102—41no—
48C7×C22⋊C8—21no—
49C4×C7×C8C2311yescharacteristic
50#6395—21no—
51#6392D411yescharacteristic
52#6390—22no—
53C2×C7⋊C16—21no—
54C2×C7⋊C16—21no—
55#3283—71no—
56#2241—71no—
57#2226—71no—
58#1675D711yescharacteristic, complemented
59#1362—71no—
60#1331—71no—
61D4⋊C16—71no—
62C2×C7×Q8—41no—
63C2×C7×D4D811yescharacteristic
64C22×C4×C7—21no—
65C2×C22⋊D7—42no—
66#1157—44no—
67#1157—82no—
68C2×D28—21no—
69C2×C4×D7—42no—
70C2×C4×D7—41no—
71C2×Dic14—22no—
72C2×C7×C8—21no—
73C2×C7×C8—41no—
74C2×C7×C8—41no—
75C2×C7×C8—41no—
76C2×C7×C8C2×D411yescharacteristic
77C2×C7×C8C2×D411yescharacteristic
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4D811yescharacteristic
81C7×C22⋊C4—21no—
82C42×C7C2×D411yescharacteristic
83C22⋊Dic7—42no—
84(C2×D7)⋊C4—41no—
85C4⋊Dic7—21no—
86C4⋊Dic7—41no—
87C4⋊Dic7—41no—
88Dic7⋊C4—42no—
89Dic28—42no—
90Dic28—41no—
91Dic28—41no—
92D56—21no—
93D56SD1612yes—
94#1132—44no—
95C7⋊C16—41no—
96C7⋊C16—41no—
97C2×#135—141no—
98#547—141no—
99#500C2×D711yescharacteristic
100#494—71no—
101#476—21no—
102#467—141no—
103#461—141no—
104#452—142no—
105C4×D8—71no—
106C8×D4C2×D711yescharacteristic
107#367—141no—
108C4⋊C16—71no—
109#358—142no—
110#357—141no—
111C4×C16—71no—
112#335C2×D711yescharacteristic
113#333—21no—
114C23×C7—21no—
115C22×D7—41no—
116C7×Q8—161no—
117C7×D4—21no—
118C7×D4—21no—
119C2×C4×C7—21no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—81no—
123C2×C4×C7#12011yescharacteristic
124C2×C4×C7C2×D811yescharacteristic
125C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
126C22⋊D7—84no—
127C2×Dic7—41no—
128C2×Dic7—42no—
129D28—22no—
130D28—41no—
131C4×D7—84no—
132C4×D7—81no—
133C4×D7—81no—
134Dic14—42no—
135Dic14—42no—
136C7×C8—21no—
137C7×C8—41no—
138C7×C8—81no—
139C7×C8C2×SD1611yescharacteristic
140C7×C8AGL(1,ℤ/8)11yescharacteristic
141C2×#41—142no—
142#135—282no—
143#135—282no—
144C2×Q16—72no—
145C2×Q16—141no—
146C2×SD16—142no—
147C2×D8—71no—
148C22×C8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—141no—
151#123—282no—
152C22⋊Q8—142no—
153#121—281no—
154C4×D4—141no—
155C4×D4C22⋊D711yescharacteristic
156C2×C4⋊C4—281no—
157C2×C16—141no—
158C2×C16—141no—
159#107—21no—
160#107—21no—
161#107—71no—
162#106—41no—
163C4⋊C8—21no—
164C4⋊C8C22⋊D711yescharacteristic
165#103—142no—
166#102—41no—
167#102—141no—
168C22⋊C8—21no—
169C4×C8C22×D711yescharacteristic
170C22×C7—41no—
171C22×C7—41no—
172C22×C7—41no—
173C22×C7#44711yescharacteristic
174C2×D7—42no—
175C2×D7—81no—
176C2×D7—81no—
177C4×C7—21no—
178C4×C7—41no—
179C4×C7—161no—
180C4×C7#45011yescharacteristic
181C4×C7#45511yescharacteristic
182C4×C7#50811yescharacteristic
183C4×C7#51011yescharacteristic
184Dic7—82no—
185Dic7—81no—
186#41—284no—
187#41—562no—
188C2×Q8—41no—
189C2×Q8—142no—
190C2×D4—141no—
191C2×D4—282no—
192C2×D4#114111yescharacteristic
193C22×C4—21no—
194C22×C4—282no—
195C22×C4—281no—
196Q16—282no—
197Q16—281no—
198Q16—281no—
199SD16—284no—
200D8—72no—
201D8—141no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8D4×D711yescharacteristic
207C2×C8D4×D711yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—141no—
211C4⋊C4—282no—
212C4⋊C4—281no—
213C4⋊C4—281no—
214C4⋊C4#114111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—282no—
217C22⋊C4—281no—
218C42C2×C22⋊D711yescharacteristic
219C16—281no—
220C16—281no—
221C2×C7—81no—
222C2×C7#163911yescharacteristic
223C2×C7#218911yescharacteristic
224C2×C7#219311yescharacteristic
225D7—82no—
226C23—21no—
227C23—281no—
228Q8—161no—
229Q8—282no—
230Q8—282no—
231D4—21no—
232D4—21no—
233D4—142no—
234D4—281no—
235D4—564no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—81no—
240C2×C4—281no—
241C2×C4—282no—
242C2×C4—564no—
243C2×C4—561no—
244C2×C4—561no—
245C2×C4C2×#114111yescharacteristic
246C2×C4#650011yescharacteristic
247C2×C4#651211yescharacteristic
248C8—21no—
249C8—41no—
250C8—81no—
251C8D7×SD1611yescharacteristic
252C8#647711yescharacteristic
253C7#1237511yescharacteristic, complemented, Sylow
254C22—41no—
255C22—41no—
256C22—41no—
257C22—282no—
258C22—561no—
259C22—561no—
260C22#8980811yescharacteristic
261C4—21no—
262C4—41no—
263C4—161no—
264C4—562no—
265C4—561no—
266C4#8994111yescharacteristic
267C4#8995111yescharacteristic
268C4#8996611yescharacteristic
269C4#8997411yescharacteristic
270C2—81no—
271C2—562no—
272C2#1172879811yescharacteristic
273C2#1172884011yescharacteristic
274C2#1172884911yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1675C2
socleC22×C7#447
prime radical#11736881C2
hypercenter#1675D7
hypocenterC7#12375
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. C7×#1675C2semidirect
3. #11729142C2semidirect
4. #11729136C2semidirect
5. D4⋊(C7⋊C16)C2semidirect
6. #11725280C2semidirect
7. #11725277C2semidirect
58. #1675D7semidirect
253. C7#12375semidirect

Maximal subgroups

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

subgroupindex
#117368812
C7×#16752
#117291422
#117291362
D4⋊(C7⋊C16)2
#117252802
#117252772
#123757