Finite Group Databasebeta

← #13268572

Subgroups — #13268572

Order 1792.

318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 79 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×#2240C211yesmaximal, characteristic, complemented
2#11729135C211yesmaximal, characteristic, complemented
3#11729114C211yesmaximal, characteristic, complemented
4#11729045C211yesmaximal, characteristic, complemented
5#11729030C211yesmaximal, characteristic
6#11726815C211yesmaximal, characteristic, complemented
7#11725305C211yesmaximal, characteristic, complemented
8C7×#500C2211yescharacteristic
9C7×#493C2211yescharacteristic
10C2×#6401—22no—
11#89853—21no—
12#89851—21no—
13#89845—21no—
14#89845—21no—
15C2×#6374—22no—
16#89471C2211yescharacteristic
17#89467C2211yescharacteristic
18C7×#357—22no—
19C4×C7×C16C2211yescharacteristic
20#89286—22no—
21C4×C7⋊C16C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12529—71nomaximal, Sylow
24C2×C7×D8—21no—
25C2×C7×D8—21no—
26C7×C4⋊Q8—21no—
27C7×#127—21no—
28C2×#1141—42no—
29C2×#1141—42no—
30C2×Dic28—21no—
31C2×Dic28—21no—
32C2×D56—21no—
33C2×D56—21no—
34#6437—21no—
35#6432—21no—
36C2×C7×C16D412yes—
37C7×#107—21no—
38C7×#107—21no—
39C4×C7×C8C2311yescharacteristic
40#6409—42no—
41#6401—44no—
42C2×C7⋊C16D412yes—
43#6382—42no—
44C4×C7⋊C8D411yescharacteristic
45C4×C7⋊C8D411yescharacteristic
46#6374—44no—
47#2248—71no—
48#2248—71no—
49#2240—71no—
50#2240D711yescharacteristic, complemented
51#1712—71no—
52#1707—71no—
53#1312—71no—
54C2×C7×Q8—41no—
55C2×C7×D4—41no—
56C2×C7×D4—41no—
57C2×C7×D4—41no—
58C2×D28—41no—
59C2×D28—41no—
60C2×D28—41no—
61C2×Dic14—41no—
62C7×D8—42no—
63C7×D8—42no—
64C2×C7×C8C2×D411yescharacteristic
65C2×C7×C8C2×D411yescharacteristic
66C7×C4⋊C4—41no—
67C7×C4⋊C4—41no—
68C42×C7C2×D411yescharacteristic
69#1141—84no—
70#1141—84no—
71C4⋊Dic7—41no—
72C4⋊Dic7—41no—
73C2×C7⋊C8D812yes—
74C2×C7⋊C8D812yes—
75Dic28—42no—
76Dic28—42no—
77D56—42no—
78D56—42no—
79C7×C16—22no—
80C7⋊C16—22no—
81C2×SD32—142no—
82C2×SD32—142no—
83#500C2×D711yescharacteristic
84#498—141no—
85#494—71no—
86#493—71no—
87#493—141no—
88#493—141no—
89#493C2×D711yescharacteristic
90#487—141no—
91#357—22no—
92#357—142no—
93C4×C16—71no—
94C4×C16C2×D711yescharacteristic
95C82—71no—
96C23×C7—81no—
97C22×D7—81no—
98C7×Q8—82no—
99C7×D4—81no—
100C7×D4—81no—
101C7×D4—82no—
102C7×D4—82no—
103C2×C4×C7—81no—
104C2×C4×C7#12711yescharacteristic
105C2×C4×C7C2×D811yescharacteristic
106C2×C4×C7C2×D811yescharacteristic
107C2×Dic7—81no—
108D28—81no—
109D28—81no—
110D28—82no—
111D28—82no—
112Dic14—81no—
113Dic14—81no—
114C7×C8AGL(1,ℤ/8)12yes—
115C7×C8AGL(1,ℤ/8)12yes—
116C7⋊C8—22no—
117C7⋊C8SD3214yescomplemented
118C2×Q16—141no—
119C2×Q16—141no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—141no—
123C2×D8—141no—
124C2×D8—282no—
125C2×D8—282no—
126C4⋊Q8—21no—
127C4⋊Q8—141no—
128#127—21no—
129#127—141no—
130SD32—284no—
131SD32—284no—
132C2×C16—72no—
133C2×C16C22⋊D712yes—
134#107—21no—
135#107—21no—
136#106—282no—
137#103—282no—
138C4×C8—71no—
139C4×C8—71no—
140C4×C8C22×D711yescharacteristic
141C22×C7—161no—
142C22×C7—161no—
143C22×C7—161no—
144C22×C7#49311yescharacteristic
145C2×D7—161no—
146C2×D7—161no—
147C2×D7—161no—
148C4×C7—161no—
149C4×C7#49611yescharacteristic
150C4×C7#49611yescharacteristic
151C4×C7C2×SD3212yes—
152C4×C7#50912yes—
153Dic7—161no—
154C2×Q8—41no—
155C2×Q8—281no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—281no—
160C2×D4—281no—
161C2×D4—281no—
162Q16—282no—
163Q16—282no—
164D8—42no—
165D8—42no—
166D8—282no—
167D8—282no—
168D8—564no—
169D8—564no—
170C2×C8—72no—
171C2×C8—72no—
172C2×C8D4×D711yescharacteristic
173C2×C8C2×C22⋊D711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—281no—
177C4⋊C4—281no—
178C42D4×D711yescharacteristic
179C16—22no—
180C16—142no—
181C2×C7—321no—
182C2×C7#172111yescharacteristic
183C2×C7#224811yescharacteristic
184C2×C7#225011yescharacteristic
185D7—321no—
186C23—81no—
187C23—561no—
188Q8—82no—
189Q8—561no—
190Q8—561no—
191D4—82no—
192D4—82no—
193D4—81no—
194D4—81no—
195D4—562no—
196D4—562no—
197D4—561no—
198D4—561no—
199C2×C4—81no—
200C2×C4—561no—
201C2×C4D7×D811yescharacteristic
202C2×C4D7×D811yescharacteristic
203C2×C4#650311yescharacteristic
204C8—74no—
205C8—142no—
206C8#647712yes—
207C8#651212yes—
208C7#1252911yescharacteristic, complemented, Sylow
209C22—161no—
210C22—161no—
211C22—161no—
212C22—1121no—
213C22—1121no—
214C22—1121no—
215C22#8992311yescharacteristic
216C4—161no—
217C4—1121no—
218C4#8968312yes—
219C4D7×SD3212yes—
220C4#8983011yescharacteristic
221C4#8994811yescharacteristic
222C2—321no—
223C2—2241no—
224C2#1172894811yescharacteristic
225C2#1172922011yescharacteristic
226C2#1172923311yescharacteristic
subgroupG/H
centerC22#89923
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#2240C2
socleC22×C7#493
prime radical#11729030C2
hypercenter#2240D7
hypocenterC7#12529
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C7×#2240C2semidirect
2. #11729135C2semidirect
3. #11729114C2semidirect
4. #11729045C2semidirect
6. #11726815C2semidirect
7. #11725305C2semidirect
50. #2240D7semidirect
117. C7⋊C8SD32semidirect
208. C7#12529semidirect

Maximal subgroups

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

subgroupindex
C7×#22402
#117291352
#117291142
#117290452
#117290302
#117268152
#117253052
#125297