Finite Group Databasebeta

← #13243101

Subgroups — #13243101

Order 1792.

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

normal subgroups only

This group has 494 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×#2733C211yesmaximal, characteristic
2#11727438C211yesmaximal, characteristic, complemented
3#11727410C212yesmaximal, complemented
4#11725448C211yesmaximal, characteristic, complemented
5#11725445C212yesmaximal, complemented
6C2×C7×#124—21no—
7C7×#394—21no—
8C7×#392—22no—
9C7×#391C2211yescharacteristic
10C7×#390C2212yes—
11C7×#388—22no—
12C2×#6409—21no—
13C2×#6406—22no—
14C2×#6382—22no—
15C2×#6381—21no—
16#89707C2211yescharacteristic
17#89319C2212yes—
18#89319C2211yescharacteristic
19#7830—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×C7×D4—21no—
22C22×C7×D4—21no—
23C22×C7×D4D412yes—
24C7×#124—42no—
25C7×#124—42no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4D412yes—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C42×C7—21no—
33C2×C42×C7C2311yescharacteristic
34C2×C4⋊Dic7—22no—
35C2×C4⋊Dic7—21no—
36C22×C7⋊C8—22no—
37C22×C7⋊C8—21no—
38C7×#95—21no—
39C7×#95—22no—
40#6409—42no—
41#6406—44no—
42#6382—44no—
43#6381—42no—
44#2733D711yescharacteristic
45#2095—71no—
46#2094—72no—
47#1272—71no—
48#1270—72no—
49C24×C7—22no—
50C2×C7×Q8—21no—
51C2×C7×Q8—41no—
52C2×C7×Q8—41no—
53C2×C7×Q8SD1612yes—
54C2×C7×D4—22no—
55C2×C7×D4—22no—
56C2×C7×D4—42no—
57C2×C7×D4—42no—
58C2×C7×D4—42no—
59C2×C7×D4—81no—
60C2×C7×D4—81no—
61C2×C7×D4—82no—
62C2×C7×D4—81no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D412yes—
68C22×Dic7—41no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—42no—
71C7×C22⋊C4—44no—
72C7×C22⋊C4—42no—
73C7×C22⋊C4—42no—
74C7×C22⋊C4—42no—
75C7×C22⋊C4—42no—
76C7×C22⋊C4—82no—
77C42×C7—41no—
78C42×C7—42no—
79C4⋊Dic7—42no—
80C4⋊Dic7—41no—
81C4⋊Dic7—41no—
82C4⋊Dic7—42no—
83C2×C7⋊C8—24no—
84C2×C7⋊C8—22no—
85C2×C7⋊C8—42no—
86C2×C7⋊C8—41no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—41no—
89C2×#124—21no—
90C2×#107—141no—
91C2×#106—142no—
92C2×#103—141no—
93C2×#102—142no—
94#394—21no—
95#392—22no—
96#391C2×D711yescharacteristic
97#390C2×D712yes—
98#388—22no—
99#384—71no—
100#336—72no—
101#336—71no—
102C23×C7—42no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7—42no—
109C23×C7#12011yescharacteristic
110C7×Q8—42no—
111C7×Q8—81no—
112C7×D4—44no—
113C7×D4—82no—
114C7×D4—84no—
115C7×D4—84no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7C2×SD1611yescharacteristic
129C2×C4×C7#13512yes—
130C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C7#13712yes—
132C2×Dic7—81no—
133C2×Dic7—82no—
134C7⋊C8—42no—
135C7⋊C8—44no—
136C22×Q8C22⋊D711yescharacteristic
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22⋊D712yes—
140C22×C8—142no—
141C22×C8—141no—
142#124—42no—
143#124—42no—
144C2×C4⋊C4—142no—
145C2×C4⋊C4—141no—
146C2×C4⋊C4C22⋊D712yes—
147C2×C4⋊C4C22⋊D711yescharacteristic
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C42—21no—
153C2×C42C22×D711yescharacteristic
154#107—282no—
155#106—284no—
156#103—282no—
157#102—284no—
158#95—21no—
159#95—22no—
160C22×C7—82no—
161C22×C7—82no—
162C22×C7—82no—
163C22×C7—82no—
164C22×C7—82no—
165C22×C7—82no—
166C22×C7—82no—
167C22×C7#45011yescharacteristic
168C22×C7#45212yes—
169C22×C7#45311yescharacteristic
170C22×C7#45312yes—
171C22×C7#45811yescharacteristic
172C4×C7—21no—
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—42no—
177C4×C7—82no—
178C4×C7—81no—
179Dic7—161no—
180C24—22no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8#114212yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—82no—
193C2×D4—81no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—281no—
198C22×C4C2×C22⋊D711yescharacteristic
199C22×C4C2×C22⋊D712yes—
200C2×C8—144no—
201C2×C8—142no—
202C2×C8—282no—
203C2×C8—281no—
204C2×C8—281no—
205C2×C8—282no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—282no—
209C4⋊C4—281no—
210C4⋊C4—282no—
211C4⋊C4—281no—
212C22⋊C4—44no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—82no—
218C42—41no—
219C42—42no—
220C2×C7—162no—
221C2×C7#161311yescharacteristic
222C2×C7#161812yes—
223C2×C7#163112yes—
224C2×C7#163411yescharacteristic
225C2×C7#165511yescharacteristic
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23#651611yescharacteristic
234Q8—42no—
235Q8—81no—
236D4—44no—
237D4—82no—
238D4—84no—
239D4—84no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—562no—
253C2×C4—561no—
254C2×C4#649511yescharacteristic
255C2×C4C2×#114211yescharacteristic
256C2×C4#651312yes—
257C2×C4#651412yes—
258C8—282no—
259C8—284no—
260C7#783011yescharacteristic, complemented, Sylow
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#8999011yescharacteristic
269C22#9001112yes—
270C22#9001312yes—
271C22#9001311yescharacteristic
272C22#9001511yescharacteristic
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—82no—
279C4—81no—
280C4—1121no—
281C2—162no—
282C2#1172959811yescharacteristic
283C2#1172963712yes—
284C2#1172967312yes—
285C2#1172968211yescharacteristic
286C2#1172970211yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#2733C2
socleC23×C7#120
hypercenter#2733D7
hypocenterC7#7830
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2733

Complements

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

NH ≅ G/Ntype
2. #11727438C2semidirect
3. #11727410C2semidirect
4. #11725448C2semidirect
5. #11725445C2semidirect
260. C7#7830semidirect

Maximal subgroups

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

subgroupindex
C7×#27332
#117274382
#117274102
#117254482
#117254452
#78307