Finite Group Databasebeta

← #13267588

Subgroups — #13267588

Order 1792.

350 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 73 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
1D4×Dic28C211yesmaximal, characteristic
2C7×#1673C211yesmaximal, characteristic
3#11729143C211yesmaximal, characteristic, complemented
4#11729136C211yesmaximal, characteristic, complemented
5D4⋊(C7⋊C16)C211yesmaximal, characteristic
6#11725280C211yesmaximal, characteristic, complemented
7#11725278C211yesmaximal, characteristic, complemented
8C22×Dic28—21no—
9D4×Dic14—21no—
10C7×#500C2211yescharacteristic
11C7×#461—21no—
12C7×C8×D4C2211yescharacteristic
13#89888—21no—
14#89621—21no—
15#89511—22no—
16#89471C2211yescharacteristic
17C4×Dic28C2211yescharacteristic
18C7×#335C2211yescharacteristic
19C7×#332—21no—
20#89287—22no—
21#89287—21no—
22#89285—21no—
23C4⋊(C7⋊C16)C2211yescharacteristic
24C4×C7⋊C16C2211yescharacteristic
25#12370—71nomaximal, Sylow
26C22×Dic14—22no—
27C2×C7×SD16—41no—
28C22×C7×C8—21no—
29C7×C4⋊Q8—21no—
30C4×C7×D4D411yescharacteristic
31D4×Dic7—41no—
32#6487—22no—
33C2×Dic28—22no—
34C2×Dic28—21no—
35C2×Dic28—42no—
36C2×Dic28—42no—
37C2×Dic28D411yescharacteristic
38C2×Dic28D412yes—
39#6451—41no—
40#6441—42no—
41#6432—21no—
42C4×Dic14—21no—
43C7×#107—21no—
44C7×#107—21no—
45C7×C4⋊C8—21no—
46C7×C4⋊C8D411yescharacteristic
47C7×#103—41no—
48C7×C22⋊C8—21no—
49C4×C7×C8C2311yescharacteristic
50#6392D411yescharacteristic
51#6390—22no—
52#6390—21no—
53C2×C7⋊C16—21no—
54C2×C7⋊C16—21no—
55D4×Q16—71no—
56#2241—71no—
57#2225—71no—
58#1673D711yescharacteristic
59#1362—71no—
60#1332—71no—
61D4⋊C16—71no—
62C2×C7×Q8—41no—
63C2×C7×D4D811yescharacteristic
64C22×C4×C7—21no—
65C22×Dic7—42no—
66C2×Dic14—21no—
67C2×Dic14—22no—
68C2×Dic14—44no—
69C2×Dic14—42no—
70C2×Dic14—82no—
71C7×SD16—82no—
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—
84C4⋊Dic7—21no—
85C4⋊Dic7—41no—
86C4⋊Dic7—41no—
87Dic7⋊C4—42no—
88Dic7⋊C4—41no—
89C4×Dic7—41no—
90Dic28—21no—
91Dic28—42no—
92Dic28—44no—
93Dic28—41no—
94Dic28—41no—
95Dic28SD1612yescomplemented
96C7⋊C16—41no—
97C7⋊C16—41no—
98C22×Q16—141no—
99D4×Q8—141no—
100#500C2×D711yescharacteristic
101#494—71no—
102#467—141no—
103#462—141no—
104#461—21no—
105#451—142no—
106C4×Q16—71no—
107C8×D4C2×D711yescharacteristic
108#367—141no—
109C4⋊C16—71no—
110#358—141no—
111#358—142no—
112C4×C16—71no—
113#335C2×D711yescharacteristic
114#332—21no—
115C23×C7—21no—
116C7×Q8—81no—
117C7×Q8—81no—
118C7×D4—21no—
119C7×D4SD3212yes—
120C2×C4×C7—21no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—81no—
124C2×C4×C7#12011yescharacteristic
125C2×C4×C7C2×D811yescharacteristic
126C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
127C2×Dic7—41no—
128C2×Dic7—41no—
129C2×Dic7—42no—
130C2×Dic7—84no—
131Dic14—22no—
132Dic14—41no—
133Dic14—42no—
134Dic14—42no—
135Dic14—84no—
136C7×C8—21no—
137C7×C8—41no—
138C7×C8—81no—
139C7×C8C2×SD1611yescharacteristic
140C7×C8AGL(1,ℤ/8)11yescharacteristic
141C22×Q8—142no—
142C2×Q16—71no—
143C2×Q16—72no—
144C2×Q16—142no—
145C2×Q16—141no—
146C2×Q16—282no—
147C2×Q16—282no—
148C2×SD16—41no—
149C22×C8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—141no—
152C4⋊Q8—281no—
153C22⋊Q8—142no—
154C22⋊Q8—282no—
155C4×Q8—141no—
156C4×D4—281no—
157C4×D4C22⋊D711yescharacteristic
158C2×C16—141no—
159C2×C16—141no—
160#107—21no—
161#107—21no—
162#107—71no—
163C4⋊C8—21no—
164C4⋊C8C22⋊D711yescharacteristic
165#103—41no—
166#103—142no—
167#103—141no—
168C22⋊C8—21no—
169C4×C8C22×D711yescharacteristic
170C22×C7—22no—
171C22×C7—41no—
172C22×C7—41no—
173C22×C7#44711yescharacteristic
174C4×C7—21no—
175C4×C7—41no—
176C4×C7—161no—
177C4×C7#45011yescharacteristic
178C4×C7#45511yescharacteristic
179C4×C7C2×SD3211yescharacteristic
180C4×C7#51011yescharacteristic
181Dic7—42no—
182Dic7—82no—
183Dic7—81no—
184C2×Q8—41no—
185C2×Q8—141no—
186C2×Q8—142no—
187C2×Q8—282no—
188C2×Q8—284no—
189C2×Q8—562no—
190C2×D4#114111yescharacteristic
191C22×C4—21no—
192C22×C4—282no—
193Q16—72no—
194Q16—141no—
195Q16—282no—
196Q16—284no—
197Q16—281no—
198Q16—281no—
199SD16—82no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8D4×D711yescharacteristic
205C2×C8D4×D711yescharacteristic
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—141no—
209C4⋊C4—282no—
210C4⋊C4—281no—
211C4⋊C4—281no—
212C4⋊C4—281no—
213C4⋊C4#114111yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—282no—
216C42—281no—
217C42C2×C22⋊D711yescharacteristic
218C16—281no—
219C16—281no—
220C2×C7—42no—
221C2×C7#163911yescharacteristic
222C2×C7#218511yescharacteristic
223C2×C7#219511yescharacteristic
224C23—21no—
225Q8—81no—
226Q8—81no—
227Q8—142no—
228Q8—281no—
229Q8—282no—
230Q8—282no—
231Q8—564no—
232D4—21no—
233D4#640112yes—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—281no—
239C2×C4—281no—
240C2×C4—282no—
241C2×C4—564no—
242C2×C4C2×#114111yescharacteristic
243C2×C4#650011yescharacteristic
244C2×C4#651211yescharacteristic
245C8—21no—
246C8—41no—
247C8—81no—
248C8D7×SD1611yescharacteristic
249C8#647711yescharacteristic
250C7#1237011yescharacteristic, complemented, Sylow
251C22—22no—
252C22—41no—
253C22—41no—
254C22#8980811yescharacteristic
255C4—21no—
256C4—41no—
257C4—161no—
258C4—282no—
259C4—562no—
260C4—561no—
261C4C2×#640111yescharacteristic
262C4#8994111yescharacteristic
263C4#8996711yescharacteristic
264C4#8997411yescharacteristic
265C2—42no—
266C2#1172879811yescharacteristic
267C2#1172883111yescharacteristic
268C2#1172885311yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1673C2
socleC22×C7#447
prime radicalC2×C7×D4D8
hypercenter#1673D7
hypocenterC7#12370
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1673

Complements

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

NH ≅ G/Ntype
3. #11729143C2semidirect
4. #11729136C2semidirect
6. #11725280C2semidirect
7. #11725278C2semidirect
95. Dic28SD16semidirect
250. C7#12370semidirect

Maximal subgroups

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

subgroupindex
D4×Dic282
C7×#16732
#117291432
#117291362
D4⋊(C7⋊C16)2
#117252802
#117252782
#123707