Finite Group Databasebeta

← #13243114

Subgroups — #13243114

Order 1792.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2804C211yesmaximal, characteristic
2#11727438C212yesmaximal, complemented
3#11727410C211yesmaximal, characteristic
4#11725448C212yesmaximal, complemented
5#11725445C211yesmaximal, characteristic
6C7×#400—21no—
7C7×#393—22no—
8C7×#391C2212yes—
9C7×#390C2211yescharacteristic
10C7×#386—21no—
11C7×#385—22no—
12C2×#6409—22no—
13C2×#6406—21no—
14C2×#6382—21no—
15C2×#6381—22no—
16#89707C2211yescharacteristic
17#89319C2211yescharacteristic
18#89319C2212yes—
19#7833—71nomaximal, Sylow
20C22×C7×Q8D412yes—
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C22⋊C4—22no—
27C2×C7×C22⋊C4—22no—
28C2×C7×C22⋊C4—21no—
29C2×C42×C7C2311yescharacteristic
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—22no—
32C22×C7⋊C8—21no—
33C22×C7⋊C8—22no—
34C7×#95—22no—
35C7×#95—21no—
36C7×#95—22no—
37C7×#95—22no—
38#6409—44no—
39#6406—42no—
40#6382—42no—
41#6381—44no—
42#2804D711yescharacteristic
43#2095—72no—
44#2094—71no—
45#1272—72no—
46#1270—71no—
47C24×C7—21no—
48C2×C7×Q8—22no—
49C2×C7×Q8—22no—
50C2×C7×Q8—42no—
51C2×C7×Q8—42no—
52C2×C7×D4—21no—
53C2×C7×D4—41no—
54C2×C7×D4—41no—
55C2×C7×D4Q1612yes—
56C22×C4×C7—22no—
57C22×C4×C7—21no—
58C22×C4×C7—22no—
59C22×C4×C7—82no—
60C22×C4×C7—81no—
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D412yes—
63C22×Dic7—41no—
64C7×C4⋊C4—42no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—81no—
67C7×C22⋊C4—42no—
68C7×C22⋊C4—42no—
69C7×C22⋊C4—82no—
70C7×C22⋊C4—81no—
71C42×C7—41no—
72C4⋊Dic7—41no—
73C4⋊Dic7—42no—
74C4⋊Dic7—42no—
75C4⋊Dic7—41no—
76C2×C7⋊C8—22no—
77C2×C7⋊C8—24no—
78C2×C7⋊C8—41no—
79C2×C7⋊C8—42no—
80C2×C7⋊C8—42no—
81C2×C7⋊C8—41no—
82C2×#107—142no—
83C2×#106—141no—
84C2×#103—142no—
85C2×#102—141no—
86#400—21no—
87#393—22no—
88#391C2×D712yes—
89#390C2×D711yescharacteristic
90#386—21no—
91#385—22no—
92#384—71no—
93#336—71no—
94#336—72no—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—42no—
98C23×C7—41no—
99C23×C7—41no—
100C23×C7#12011yescharacteristic
101C7×Q8—44no—
102C7×Q8—82no—
103C7×D4—42no—
104C7×D4—81no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—84no—
118C2×C4×C7C2×Q1611yescharacteristic
119C2×C4×C7#13512yes—
120C2×C4×C7#13711yescharacteristic
121C2×C4×C7#13712yes—
122C2×Dic7—81no—
123C2×Dic7—82no—
124C7⋊C8—42no—
125C7⋊C8—44no—
126C22×Q8C22⋊D712yes—
127C22×D4C22⋊D711yescharacteristic
128C23×C4—21no—
129C22×C8—141no—
130C22×C8—142no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—141no—
133C2×C4⋊C4—142no—
134C2×C4⋊C4C22⋊D712yes—
135C2×C4⋊C4C22⋊D711yescharacteristic
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C42C22×D711yescharacteristic
140#107—284no—
141#106—282no—
142#103—284no—
143#102—282no—
144#95—22no—
145#95—21no—
146#95—22no—
147#95—22no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7—82no—
151C22×C7—81no—
152C22×C7—81no—
153C22×C7#45111yescharacteristic
154C22×C7#45212yes—
155C22×C7#45311yescharacteristic
156C22×C7#45312yes—
157C22×C7#45811yescharacteristic
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—82no—
163C4×C7—81no—
164C4×C7—82no—
165Dic7—161no—
166C24—21no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4#114412yes—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—82no—
179C22×C4—81no—
180C22×C4—281no—
181C22×C4C2×C22⋊D711yescharacteristic
182C22×C4C2×C22⋊D712yes—
183C2×C8—142no—
184C2×C8—144no—
185C2×C8—281no—
186C2×C8—282no—
187C2×C8—281no—
188C2×C8—282no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—281no—
193C4⋊C4—282no—
194C4⋊C4—281no—
195C4⋊C4—282no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—82no—
199C22⋊C4—81no—
200C42—41no—
201C2×C7—82no—
202C2×C7#161411yescharacteristic
203C2×C7#161812yes—
204C2×C7#163011yescharacteristic
205C2×C7#163212yes—
206C2×C7#165511yescharacteristic
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#651611yescharacteristic
213Q8—44no—
214Q8—82no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4—562no—
231C2×C4—561no—
232C2×C4#650511yescharacteristic
233C2×C4C2×#114411yescharacteristic
234C2×C4#651312yes—
235C2×C4#651412yes—
236C8—284no—
237C8—282no—
238C7#783311yescharacteristic, complemented, Sylow
239C22—44no—
240C22—42no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#9000011yescharacteristic
245C22#9001112yes—
246C22#9001312yes—
247C22#9001311yescharacteristic
248C22#9001511yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—82no—
255C4—82no—
256C4—1121no—
257C2—82no—
258C2#1172963712yes—
259C2#1172964511yescharacteristic
260C2#1172967812yes—
261C2#1172969811yescharacteristic
262C2#1172970211yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2804C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenter#2804D7
hypocenterC7#7833
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #11727438C2semidirect
4. #11725448C2semidirect
238. C7#7833semidirect

Maximal subgroups

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

subgroupindex
C7×#28042
#117274382
#117274102
#117254482
#117254452
#78337