Finite Group Databasebeta

← #13242586

Subgroups — #13242586

Order 1792.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#11733843C211yesmaximal, characteristic
2C7×#2013C211yesmaximal, characteristic
3#11729536C212yesmaximal, complemented
4#11725445C211yesmaximal, characteristic
5#11725284C212yesmaximal, complemented
6C22×C7×SD16—21no—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×#6498—21no—
9C2×C7×#103—21no—
10C7×#390C2211yescharacteristic
11#89755—21no—
12C2×#6381—22no—
13#89725—22no—
14#89722—22no—
15#89700C2212yes—
16C7×#336C2211yescharacteristic
17#89324—22no—
18#89319C2212yes—
19#7751—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4D411yescharacteristic
22C23×Dic7—21no—
23C2×C7×SD16—42no—
24C2×C7×SD16—42no—
25C2×C7×SD16—42no—
26C22×C7×C8—21no—
27C7×C4⋊Q8—42no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C22⋊C4—22no—
31C2×C42×C7C2311yescharacteristic
32C2×C22⋊Dic7—22no—
33C2×C22⋊Dic7—21no—
34#6498—42no—
35#6498—42no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7D412yes—
38C2×C4⋊Dic7D412yes—
39C2×Dic7⋊C4—22no—
40C22×C7⋊C8—22no—
41C7×#103—42no—
42C7⋊#95—22no—
43C7⋊#95—21no—
44C7⋊#95—22no—
45#6381—44no—
46#2724—71no—
47#2057—72no—
48#2013D711yescharacteristic
49#1271—72no—
50#1270—71no—
51C24×C7—21no—
52C2×C7×Q8—22no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C2×C7×D4—21no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4D812yes—
62C22×C4×C7—21no—
63C22×C4×C7—41no—
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D412yes—
66C22×Dic7—22no—
67C22×Dic7—22no—
68C22×Dic7—42no—
69C22×Dic7—81no—
70C22×Dic7—81no—
71C7×SD16—48no—
72C2×C7×C8—22no—
73C2×C7×C8—41no—
74C2×C7×C8—41no—
75C7×C4⋊C4—24no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C22⋊C4—42no—
79C7×C22⋊C4—42no—
80C42×C7—41no—
81C22⋊Dic7—42no—
82C22⋊Dic7—44no—
83C4⋊Dic7—22no—
84C4⋊Dic7—22no—
85C4⋊Dic7—42no—
86C4⋊Dic7—81no—
87Dic7⋊C4—42no—
88Dic7⋊C4—42no—
89C2×C7⋊C8—24no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—42no—
92C22×SD16—21no—
93C2×C4⋊Q8C2×D711yescharacteristic
94C2×#123—141no—
95C2×#107—142no—
96C2×#103—21no—
97#398—141no—
98#397—142no—
99#390C2×D711yescharacteristic
100#389—72no—
101#385—142no—
102#340—142no—
103#336—72no—
104#336C2×D711yescharacteristic
105C23×C7—22no—
106C23×C7—42no—
107C23×C7—41no—
108C23×C7—42no—
109C23×C7—41no—
110C23×C7#12011yescharacteristic
111C7×Q8—42no—
112C7×Q8—42no—
113C7×Q8—81no—
114C7×Q8—81no—
115C7×D4—42no—
116C7×D4—81no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—21no—
121C2×C4×C7—42no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—81no—
125C2×C4×C7—81no—
126C2×C4×C7C2×D811yescharacteristic
127C2×C4×C7#13512yes—
128C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C7#13712yes—
130C2×Dic7—42no—
131C2×Dic7—42no—
132C2×Dic7—42no—
133C2×Dic7—42no—
134C2×Dic7—42no—
135C2×Dic7—42no—
136C2×Dic7—84no—
137C7×C8—42no—
138C7⋊C8—44no—
139C22×Q8—21no—
140C22×D4C22⋊D711yescharacteristic
141C23×C4—141no—
142C2×SD16—42no—
143C2×SD16—42no—
144C2×SD16—42no—
145C22×C8—21no—
146C22×C8—142no—
147C4⋊Q8—42no—
148#123—282no—
149#123—282no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—72no—
153C2×C4⋊C4—142no—
154C2×C4⋊C4—141no—
155C2×C4⋊C4C22⋊D711yescharacteristic
156C2×C22⋊C4—22no—
157C2×C22⋊C4—142no—
158C2×C22⋊C4—141no—
159C2×C42C22×D711yescharacteristic
160#107—284no—
161#103—42no—
162#95—142no—
163#95—141no—
164#95—142no—
165C22×C7—42no—
166C22×C7—44no—
167C22×C7—82no—
168C22×C7—81no—
169C22×C7—81no—
170C22×C7#44711yescharacteristic
171C22×C7#45212yes—
172C22×C7#45311yescharacteristic
173C22×C7#45612yes—
174C22×C7B(4,𝔽2)11yescharacteristic
175C4×C7—21no—
176C4×C7—21no—
177C4×C7—22no—
178C4×C7—22no—
179C4×C7—82no—
180C4×C7—81no—
181Dic7—82no—
182Dic7—82no—
183C24—21no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4#114112yes—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—142no—
197C22×C4—142no—
198C22×C4—282no—
199C22×C4—561no—
200C22×C4—561no—
201C22×C4D4×D712yes—
202C22×C4C2×C22⋊D711yescharacteristic
203SD16—48no—
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—144no—
208C2×C8—282no—
209C2×C8—282no—
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—142no—
214C4⋊C4—142no—
215C4⋊C4—282no—
216C4⋊C4—282no—
217C4⋊C4—282no—
218C4⋊C4—561no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—282no—
222C22⋊C4—284no—
223C42—41no—
224C2×C7—82no—
225C2×C7#159511yescharacteristic
226C2×C7#160312yes—
227C2×C7#163011yescharacteristic
228C2×C7#164712yes—
229C2×C7#166011yescharacteristic
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#650011yescharacteristic
236Q8—42no—
237Q8—42no—
238Q8—81no—
239Q8—81no—
240D4—42no—
241D4—81no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—282no—
252C2×C4—282no—
253C2×C4—282no—
254C2×C4—282no—
255C2×C4—282no—
256C2×C4—282no—
257C2×C4—564no—
258C2×C4#647812yes—
259C2×C4#647912yes—
260C2×C4C2×#114111yescharacteristic
261C2×C4#651211yescharacteristic
262C8—42no—
263C8—284no—
264C7#775111yescharacteristic, complemented, Sylow
265C22—42no—
266C22—44no—
267C22—82no—
268C22—81no—
269C22—81no—
270C22#8980411yescharacteristic
271C22#8980811yescharacteristic
272C22#8984911yescharacteristic
273C22#8994512yes—
274C22#8997712yes—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—82no—
280C4—81no—
281C4—562no—
282C4—562no—
283C2—82no—
284C2#1172745011yescharacteristic
285C2#1172872812yes—
286C2#1172880212yes—
287C2#1172902411yescharacteristic
288C2#1172908411yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2013C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenter#2013D7
hypocenterC7#7751
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
3. #11729536C2semidirect
5. #11725284C2semidirect
264. C7#7751semidirect

Maximal subgroups

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

subgroupindex
#117338432
C7×#20132
#117295362
#117254452
#117252842
#77517