Finite Group Databasebeta

← #13247374

Subgroups — #13247374

Order 1792.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#11739905C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3#11729378C211yesmaximal, characteristic
4#11725521C212yesmaximal
5#11725519C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×Q8×Dic7—21no—
8C7×#423C2212yes—
9C2×C7×#101—21no—
10#89757—21no—
11#89757—22no—
12#89708C412yes—
13#89708C2211yescharacteristic
14C4×C4⋊Dic7C412yes—
15C4×C4⋊Dic7C2211yescharacteristic
16#89697C2212yes—
17C7×#343—21no—
18#89323—22no—
19#8767—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C2×C7×M16—22no—
22C4×C7×Q8—22no—
23C4×C7×Q8—22no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C42×C7C2×C412yes—
27C2×C42×C7C2311yescharacteristic
28Q8×Dic7—44no—
29C2×C4⋊Dic7—22no—
30C2×C4⋊Dic7—22no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4×Dic7—22no—
34C2×C4×Dic7—21no—
35C7×C4⋊C8—22no—
36C7×C4⋊C8—22no—
37C7×#101—42no—
38C7⋊#95—22no—
39C7⋊#95—21no—
40C7⋊#95C2×C411yescharacteristic
41C7⋊#95C2×C412yes—
42C7⋊#95C2×C411yescharacteristic
43C22⋊(C7⋊C8)—22no—
44C4⋊(C7⋊C8)—22no—
45C4×C7⋊C8—22no—
46#2351—71no—
47#1801D711yescharacteristic
48#1461—71no—
49#1303—72no—
50#1286—72no—
51C2×C7×Q8—22no—
52C2×C7×Q8—22no—
53C2×C7×Q8SD1612yes—
54C2×C7×Q8Q1612yes—
55C22×C4×C7C22⋊C412yes—
56C22×C4×C7C22⋊C412yes—
57C22×C4×C7C22×C411yescharacteristic
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×Dic7—21no—
61C22×Dic7—21no—
62C22×Dic7—22no—
63C7×M16—42no—
64C7×M16—42no—
65C2×C7×C8—42no—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—22no—
68C42×C7—22no—
69C42×C7—22no—
70C42×C7#4112yes—
71C42×C7#4112yes—
72C4⋊Dic7—42no—
73C4⋊Dic7—44no—
74C4⋊Dic7—44no—
75C4⋊Dic7—42no—
76C4×Dic7—44no—
77C4×Dic7—42no—
78C2×C7⋊C8—42no—
79C2×C4×Q8—141no—
80C2×C4×Q8C2×D711yescharacteristic
81#431—72no—
82#423C2×D712yes—
83C2×#101—21no—
84#389—142no—
85#389—141no—
86#382—72no—
87#382—71no—
88C4×C4⋊C4—72no—
89C4×C4⋊C4—71no—
90#343—21no—
91#324—142no—
92C23×C7C2×C22⋊C411yescharacteristic
93C7×Q8—24no—
94C7×Q8—24no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—41no—
99C2×C4×C7—41no—
100C2×C4×C7—42no—
101C2×C4×C7#10314yes—
102C2×C4×C7#11711yescharacteristic
103C2×C4×C7#11711yescharacteristic
104C2×C4×C7#12312yes—
105C2×C4×C7#12312yes—
106C2×C4×C7C2×SD1611yescharacteristic
107C2×C4×C7C2×Q1611yescharacteristic
108C2×C4×C7AGL(1,ℤ/8)12yes—
109C2×Dic7—22no—
110C2×Dic7—41no—
111C2×Dic7—41no—
112C2×Dic7—41no—
113C2×Dic7—42no—
114C2×Dic7—41no—
115C2×Dic7—42no—
116C2×Dic7—42no—
117C2×Dic7—41no—
118C7×C8—82no—
119C7⋊C8—82no—
120C22×Q8C22⋊D711yescharacteristic
121C2×M16—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—284no—
125C2×C4⋊C4—142no—
126C2×C4⋊C4—142no—
127C2×C4⋊C4—141no—
128C2×C4⋊C4—141no—
129C2×C4⋊C4D2812yes—
130C2×C4⋊C4C22⋊D711yescharacteristic
131C2×C42—142no—
132C2×C42—141no—
133C2×C42C4×D712yes—
134C2×C42C22×D711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—142no—
138#101—42no—
139C22⋊C8—142no—
140C4×C8—142no—
141#95—71no—
142#95—72no—
143#95—71no—
144#95—142no—
145#95—141no—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#38111yescharacteristic
149C22×C7C2×#10311yescharacteristic
150C22×C7#41811yescharacteristic
151C4×C7—24no—
152C4×C7—22no—
153C4×C7—21no—
154C4×C7—21no—
155C4×C7—42no—
156C4×C7#48112yes—
157C4×C7#48412yes—
158Dic7—42no—
159Dic7—82no—
160Dic7—81no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8#114212yes—
164C2×Q8#114412yes—
165C22×C4—141no—
166C22×C4—141no—
167C22×C4—142no—
168C22×C4(C2×D7)⋊C412yes—
169C22×C4(C2×D7)⋊C412yes—
170C22×C4C2×C4×D711yescharacteristic
171C22×C4C2×D2811yescharacteristic
172C22×C4C2×C22⋊D711yescharacteristic
173M16—42no—
174M16—42no—
175C2×C8—42no—
176C2×C8—282no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—284no—
180C4⋊C4—282no—
181C4⋊C4—284no—
182C4⋊C4—282no—
183C42—22no—
184C42—22no—
185C42—284no—
186C42—282no—
187C42#115912yes—
188C42#115912yes—
189C2×C7—21no—
190C2×C7—21no—
191C2×C7#182411yescharacteristic
192C2×C7#182711yescharacteristic
193C2×C7#183011yescharacteristic
194C23C2×(C2×D7)⋊C411yescharacteristic
195Q8—24no—
196Q8—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—142no—
204C2×C4—281no—
205C2×C4—281no—
206C2×C4—281no—
207C2×C4—282no—
208C2×C4—281no—
209C2×C4—282no—
210C2×C4—282no—
211C2×C4—281no—
212C2×C4#638414yes—
213C2×C4#644011yescharacteristic
214C2×C4#644011yescharacteristic
215C2×C4#644912yes—
216C2×C4C2×#114211yescharacteristic
217C2×C4#649812yes—
218C2×C4C2×#114411yescharacteristic
219C2×C4#651212yes—
220C8—82no—
221C8—562no—
222C7#876711yescharacteristic, complemented, Sylow
223C22—21no—
224C22—21no—
225C22#8971511yescharacteristic
226C22C2×#638411yescharacteristic
227C22#8977311yescharacteristic
228C4—22no—
229C4—24no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—282no—
234C4—562no—
235C4—561no—
236C4#8980612yes—
237C4#8981612yes—
238C2—21no—
239C2—21no—
240C2#1172725611yescharacteristic
241C2#1172785511yescharacteristic
242C2#1172788511yescharacteristic
subgroupG/H
centerC22#89715
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1801C2
socleC22×C7#381
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1801D7
hypocenterC7#8767
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
222. C7#8767semidirect

Maximal subgroups

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

subgroupindex
#117399052
C7×#18012
#117293782
#117255212
#117255192
#87677