Finite Group Databasebeta

← #13256870

Subgroups — #13256870

Order 1792.

400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 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#11727582C212yesmaximal
3#11727306C211yesmaximal, characteristic
4C7×#1295C211yesmaximal, characteristic
5#11725465C212yesmaximal
6C2×#6432C2211yescharacteristic
7C2×C7×#97C2212yes—
8C7×#382C2211yescharacteristic
9#89758—21no—
10#89751—21no—
11C2×#6384—21no—
12C2×#6384—21no—
13#89708C2212yes—
14C2×#6377C2211yescharacteristic
15#89432—22no—
16#89428—22no—
17#89346—22no—
18#89346—22no—
19#10359—71nomaximal, Sylow
20C22×Dic14—21no—
21C22×Dic14—21no—
22C22×C7×C8—22no—
23C2×C7×C4⋊C4D411yescharacteristic
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7—22no—
27C2×C4⋊Dic7—21no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×Dic7⋊C4—22no—
31C22×C7⋊C8—21no—
32#6432—22no—
33#6432—21no—
34#6432Q812yescomplemented
35C7×#97—22no—
36C7×#97—22no—
37C7×#95—22no—
38C7⋊#95—22no—
39C7⋊#95—21no—
40C7⋊#95—21no—
41C7⋊#95—22no—
42C7⋊#95—22no—
43#6384—42no—
44#6384—42no—
45#6377—22no—
46#2844—71no—
47#1993—72no—
48#1960—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—22no—
59C2×Dic14—41no—
60C2×Dic14—41no—
61C2×Dic14—41no—
62C2×Dic14—41no—
63C2×Dic14—41no—
64C2×Dic14—81no—
65C2×Dic14—82no—
66C2×C7×C8—24no—
67C2×C7×C8—42no—
68C2×C7×C8—42no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—41no—
71C42×C7C2×Q811yescharacteristic
72C42×C7#4112yes—
73C42×C7#4111yescharacteristic
74C4⋊Dic7—24no—
75C4⋊Dic7—42no—
76C4⋊Dic7—42no—
77C4⋊Dic7—42no—
78C4⋊Dic7—81no—
79Dic7⋊C4—82no—
80C2×C7⋊C8—22no—
81C2×C7⋊C8—41no—
82C2×C7⋊C8—41no—
83C2×C4⋊Q8—71no—
84C2×#103—141no—
85C2×#103—141no—
86C2×#97—71no—
87C2×#97C2×D712yes—
88#400—142no—
89#400—141no—
90#395—142no—
91#395—141no—
92#382—72no—
93#382C2×D711yescharacteristic
94#340—142no—
95#340—142no—
96C23×C7#12011yescharacteristic
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7C22⋊Q812yes—
106C2×C4×C7C22⋊Q811yescharacteristic
107C2×C4×C7#12411yescharacteristic
108C2×C4×C7#12412yes—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—42no—
112C2×Dic7—42no—
113C2×Dic7—42no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—81no—
117C2×Dic7—81no—
118Dic14—42no—
119Dic14—42no—
120Dic14—81no—
121Dic14—81no—
122Dic14—84no—
123C7×C8—44no—
124C7⋊C8—42no—
125C22×Q8—141no—
126C22×Q8—141no—
127C22×C8—22no—
128C22×C8—141no—
129C4⋊Q8—72no—
130C4⋊Q8—142no—
131C4⋊Q8—141no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4—142no—
135C2×C4⋊C4—142no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4D2811yescharacteristic
138C2×C4⋊C4D2811yescharacteristic
139C2×C42C22×D711yescharacteristic
140#103—282no—
141#103—282no—
142#97—22no—
143#97—22no—
144#97—142no—
145#95—22no—
146#95—142no—
147#95—141no—
148#95—141no—
149#95—142no—
150#95—142no—
151C22×C7#39311yescharacteristic
152C22×C7#45311yescharacteristic
153C22×C7#45411yescharacteristic
154C22×C7#45512yes—
155C22×C7#45612yes—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—22no—
159C4×C7—22no—
160C4×C7—81no—
161C4×C7—81no—
162Dic7—82no—
163Dic7—82no—
164Dic7—82no—
165C2×Q8—142no—
166C2×Q8—281no—
167C2×Q8—281no—
168C2×Q8—281no—
169C2×Q8—281no—
170C2×Q8—281no—
171C2×Q8—561no—
172C2×Q8—562no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—142no—
176C22×C4—142no—
177C22×C4—281no—
178C22×C4C2×D2811yescharacteristic
179C22×C4D4×D712yes—
180C2×C8—24no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—142no—
184C2×C8—281no—
185C2×C8—281no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—144no—
189C4⋊C4—282no—
190C4⋊C4—282no—
191C4⋊C4—282no—
192C4⋊C4—562no—
193C4⋊C4—561no—
194C42#115712yes—
195C42#115911yescharacteristic
196C42Q8×D711yescharacteristic
197C2×C7#165911yescharacteristic
198C2×C7#166011yescharacteristic
199C2×C7#166412yes—
200C2×C7#202711yescharacteristic
201C2×C7#202812yes—
202C23#644511yescharacteristic
203Q8—282no—
204Q8—282no—
205Q8—561no—
206Q8—561no—
207Q8—564no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—282no—
217C2×C4—282no—
218C2×C4—282no—
219C2×C4—282no—
220C2×C4—282no—
221C2×C4—282no—
222C2×C4—282no—
223C2×C4—561no—
224C2×C4—561no—
225C2×C4#643811yescharacteristic
226C2×C4#644812yes—
227C2×C4#645912yes—
228C2×C4#646011yescharacteristic
229C8—44no—
230C8—282no—
231C7#1035911yescharacteristic, complemented, Sylow
232C22#8944511yescharacteristic
233C22#8951911yescharacteristic
234C22#8952511yescharacteristic
235C22#8959712yes—
236C22#8959812yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—81no—
243C4—562no—
244C4—562no—
245C4—562no—
246C2#1172610711yescharacteristic
247C2#1172610911yescharacteristic
248C2#1172613312yes—
249C2#1172652312yes—
250C2#1172674911yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1295C2
socle, prime radicalC23×C7#120
hypercenter#1295D7
hypocenterC7#10359
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
34. #6432Q8semidirect
231. C7#10359semidirect

Maximal subgroups

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

subgroupindex
#117316632
#117275822
#117273062
C7×#12952
#117254652
#103597