Finite Group Databasebeta

← #13258294

Subgroups — #13258294

Order 1792.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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#11731663C211yesmaximal, characteristic
2#11727570C212yesmaximal
3#11727304C211yesmaximal, characteristic
4C7×#1295C211yesmaximal, characteristic
5#11725465C212yesmaximal
6C2×C7×#97C2212yes—
7C7×#382C2211yescharacteristic
8C2×#6392—22no—
9C2×#6391—22no—
10#89758—21no—
11#89751—21no—
12C2×#6382—21no—
13C2×#6381—21no—
14#89708C2212yes—
15#89707C2211yescharacteristic
16C2×#6377C2211yescharacteristic
17#89432—22no—
18#89428—22no—
19#10583—71nomaximal, Sylow
20C22×Dic14—21no—
21C22×C7×C8—22no—
22C2×C7×C4⋊C4D411yescharacteristic
23C2×C7×C4⋊C4D411yescharacteristic
24C2×C42×C7C2311yescharacteristic
25C2×C4⋊Dic7—22no—
26C2×C4⋊Dic7—21no—
27C2×C4⋊Dic7—21no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×Dic7⋊C4—22no—
31C22×C7⋊C8—21no—
32C7×#97—22no—
33C7×#97Q814yescomplemented
34C7×#95—22no—
35C7⋊#95—22no—
36C7⋊#95—21no—
37C7⋊#95—21no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40#6392—44no—
41#6391—44no—
42#6382—42no—
43#6381—42no—
44#6377—21no—
45#6377Q812yescomplemented
46#2844—71no—
47#1847—72no—
48#1847—71no—
49#1295—72no—
50#1295D711yescharacteristic
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D412yes—
55C22×Dic7—22no—
56C22×Dic7—22no—
57C22×Dic7—41no—
58C2×Dic14—82no—
59C2×Dic14—81no—
60C2×C7×C8—24no—
61C2×C7×C8—24no—
62C2×C7×C8—24no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—41no—
65C42×C7C2×Q812yes—
66C42×C7C2×Q811yescharacteristic
67C42×C7#4111yescharacteristic
68C4⋊Dic7—42no—
69C4⋊Dic7—42no—
70C4⋊Dic7—41no—
71C4⋊Dic7—41no—
72C4⋊Dic7—42no—
73C4⋊Dic7—42no—
74C4⋊Dic7—81no—
75Dic7⋊C4—82no—
76C2×C7⋊C8—22no—
77C2×C7⋊C8—22no—
78C2×C7⋊C8—22no—
79C2×#107—142no—
80C2×#107—141no—
81C2×#106—142no—
82C2×#106—141no—
83C2×#97—71no—
84C2×#97C2×D712yes—
85#400—142no—
86#400—141no—
87#395—142no—
88#395—141no—
89#384—71no—
90#382—72no—
91#382C2×D711yescharacteristic
92C23×C7#12011yescharacteristic
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—42no—
98C2×C4×C7—41no—
99C2×C4×C7—41no—
100C2×C4×C7—42no—
101C2×C4×C7C22⋊Q812yes—
102C2×C4×C7C22⋊Q811yescharacteristic
103C2×C4×C7C4⋊Q811yescharacteristic
104C2×C4×C7C4⋊Q812yes—
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—81no—
112C2×Dic7—82no—
113Dic14—84no—
114C7×C8—28no—
115C7⋊C8—24no—
116C22×Q8—141no—
117C22×C8—22no—
118C22×C8—141no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—141no—
122C2×C4⋊C4—142no—
123C2×C4⋊C4—142no—
124C2×C4⋊C4—141no—
125C2×C4⋊C4D2811yescharacteristic
126C2×C4⋊C4D2811yescharacteristic
127C2×C42C22×D711yescharacteristic
128#107—282no—
129#107—284no—
130#106—282no—
131#106—284no—
132#97—22no—
133#97—72no—
134#97—141no—
135#97Dic1414yescomplemented
136#95—22no—
137#95—142no—
138#95—141no—
139#95—141no—
140#95—142no—
141#95—142no—
142C22×C7#39511yescharacteristic
143C22×C7#45311yescharacteristic
144C22×C7#45312yes—
145C22×C7#45611yescharacteristic
146C22×C7#45612yes—
147C4×C7—21no—
148C4×C7—22no—
149C4×C7—21no—
150C4×C7—22no—
151C4×C7—81no—
152C4×C7—81no—
153Dic7—82no—
154Dic7—82no—
155Dic7—161no—
156C2×Q8—561no—
157C2×Q8—562no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—142no—
161C22×C4—142no—
162C22×C4—281no—
163C22×C4C2×D2811yescharacteristic
164C22×C4D4×D712yes—
165C2×C8—24no—
166C2×C8—24no—
167C2×C8—24no—
168C2×C8—142no—
169C2×C8—142no—
170C2×C8—142no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—282no—
174C4⋊C4—282no—
175C4⋊C4—281no—
176C4⋊C4—282no—
177C4⋊C4—281no—
178C4⋊C4—282no—
179C4⋊C4—562no—
180C4⋊C4—561no—
181C42C2×Dic1412yes—
182C42#115911yescharacteristic
183C42Q8×D711yescharacteristic
184C2×C7#165511yescharacteristic
185C2×C7#166011yescharacteristic
186C2×C7#166012yes—
187C2×C7#206011yescharacteristic
188C2×C7#206012yes—
189C23#644511yescharacteristic
190Q8—564no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—282no—
200C2×C4—282no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4—282no—
205C2×C4—561no—
206C2×C4—562no—
207C2×C4#643211yescharacteristic
208C2×C4#644112yes—
209C2×C4#645112yes—
210C2×C4#646011yescharacteristic
211C8—28no—
212C8—144no—
213C7#1058311yescharacteristic, complemented, Sylow
214C22#8942811yescharacteristic
215C22#8951911yescharacteristic
216C22#8952411yescharacteristic
217C22#8959412yes—
218C22#8959812yes—
219C4—21no—
220C4—21no—
221C4—22no—
222C4—22no—
223C4—81no—
224C4—81no—
225C4—562no—
226C4—562no—
227C4—1121no—
228C2#1172610411yescharacteristic
229C2#1172610911yescharacteristic
230C2#1172612612yes—
231C2#1172648912yes—
232C2#1172673211yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1295C2
socle, prime radicalC23×C7#120
hypercenter#1295D7
hypocenterC7#10583
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
33. C7×#97Q8semidirect
45. #6377Q8semidirect
135. #97Dic14semidirect
213. C7#10583semidirect

Maximal subgroups

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

subgroupindex
#117316632
#117275702
#117273042
C7×#12952
#117254652
#105837