Finite Group Databasebeta

← #13247369

Subgroups — #13247369

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#8766—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—42no—
29Q8×Dic7—42no—
30C2×C4⋊Dic7—22no—
31C2×C4⋊Dic7—22no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4×Dic7—22no—
35C2×C4×Dic7—21no—
36C7×C4⋊C8—22no—
37C7×C4⋊C8—22no—
38C7×#101—42no—
39C7⋊#95—22no—
40C7⋊#95—21no—
41C7⋊#95C2×C411yescharacteristic
42C7⋊#95C2×C412yes—
43C7⋊#95C2×C411yescharacteristic
44C22⋊(C7⋊C8)—22no—
45C4⋊(C7⋊C8)—22no—
46C4×C7⋊C8—22no—
47#2351—71no—
48#1801D711yescharacteristic
49#1461—71no—
50#1303—72no—
51#1286—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—22no—
54C2×C7×Q8D812yes—
55C2×C7×Q8SD1612yes—
56C22×C4×C7C22⋊C412yes—
57C22×C4×C7C22⋊C412yes—
58C22×C4×C7C22×C411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D411yescharacteristic
61C22×Dic7—21no—
62C22×Dic7—21no—
63C22×Dic7—22no—
64C7×M16—42no—
65C7×M16—42no—
66C2×C7×C8—42no—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—22no—
69C42×C7—22no—
70C42×C7—22no—
71C42×C7#4112yes—
72C42×C7#4112yes—
73C4⋊Dic7—42no—
74C4⋊Dic7—44no—
75C4⋊Dic7—44no—
76C4⋊Dic7—41no—
77C4⋊Dic7—41no—
78C4×Dic7—44no—
79C4×Dic7—41no—
80C4×Dic7—41no—
81C2×C7⋊C8—42no—
82C2×C4×Q8—141no—
83C2×C4×Q8C2×D711yescharacteristic
84#431—72no—
85#423C2×D712yes—
86C2×#101—21no—
87#389—142no—
88#389—141no—
89#382—72no—
90#382—71no—
91C4×C4⋊C4—72no—
92C4×C4⋊C4—71no—
93#343—21no—
94#324—142no—
95C23×C7C2×C22⋊C411yescharacteristic
96C7×Q8—24no—
97C7×Q8—24no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—42no—
104C2×C4×C7#10214yes—
105C2×C4×C7#11711yescharacteristic
106C2×C4×C7#11711yescharacteristic
107C2×C4×C7#12312yes—
108C2×C4×C7#12312yes—
109C2×C4×C7C2×D811yescharacteristic
110C2×C4×C7C2×SD1611yescharacteristic
111C2×C4×C7#13712yes—
112C2×Dic7—22no—
113C2×Dic7—41no—
114C2×Dic7—41no—
115C2×Dic7—41no—
116C2×Dic7—42no—
117C2×Dic7—41no—
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—41no—
121C7×C8—82no—
122C7⋊C8—82no—
123C22×Q8C22⋊D711yescharacteristic
124C2×M16—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—282no—
128C4×Q8—282no—
129C2×C4⋊C4—142no—
130C2×C4⋊C4—142no—
131C2×C4⋊C4—141no—
132C2×C4⋊C4—141no—
133C2×C4⋊C4D2812yes—
134C2×C4⋊C4C22⋊D711yescharacteristic
135C2×C42—142no—
136C2×C42—141no—
137C2×C42C4×D712yes—
138C2×C42C22×D711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—142no—
142#101—42no—
143C22⋊C8—142no—
144C4×C8—142no—
145#95—71no—
146#95—72no—
147#95—71no—
148#95—142no—
149#95—141no—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7#38111yescharacteristic
153C22×C7C2×#10211yescharacteristic
154C22×C7#41911yescharacteristic
155C4×C7—24no—
156C4×C7—22no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—42no—
160C4×C7#48012yes—
161C4×C7#48312yes—
162Dic7—42no—
163Dic7—82no—
164Dic7—81no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8#114112yes—
168C2×Q8#114312yes—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4—142no—
172C22×C4(C2×D7)⋊C412yes—
173C22×C4(C2×D7)⋊C412yes—
174C22×C4C2×C4×D711yescharacteristic
175C22×C4C2×D2811yescharacteristic
176C22×C4C2×C22⋊D711yescharacteristic
177M16—42no—
178M16—42no—
179C2×C8—42no—
180C2×C8—282no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—282no—
184C4⋊C4—284no—
185C4⋊C4—281no—
186C4⋊C4—281no—
187C4⋊C4—284no—
188C42—22no—
189C42—22no—
190C42—281no—
191C42—281no—
192C42—284no—
193C42#115912yes—
194C42#115912yes—
195C2×C7—21no—
196C2×C7—21no—
197C2×C7#182311yescharacteristic
198C2×C7#182711yescharacteristic
199C2×C7#183011yescharacteristic
200C23C2×(C2×D7)⋊C411yescharacteristic
201Q8—24no—
202Q8—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—142no—
210C2×C4—281no—
211C2×C4—281no—
212C2×C4—281no—
213C2×C4—282no—
214C2×C4—281no—
215C2×C4—282no—
216C2×C4—281no—
217C2×C4—282no—
218C2×C4#638314yes—
219C2×C4#644011yescharacteristic
220C2×C4#644011yescharacteristic
221C2×C4#644912yes—
222C2×C4C2×#114111yescharacteristic
223C2×C4#649812yes—
224C2×C4C2×#114311yescharacteristic
225C2×C4#651412yes—
226C8—82no—
227C8—562no—
228C7#876611yescharacteristic, complemented, Sylow
229C22—21no—
230C22—21no—
231C22#8971511yescharacteristic
232C22C2×#638311yescharacteristic
233C22#8977611yescharacteristic
234C4—22no—
235C4—24no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—282no—
240C4—561no—
241C4—562no—
242C4#8980512yes—
243C4#8981512yes—
244C2—21no—
245C2—21no—
246C2#1172725511yescharacteristic
247C2#1172785511yescharacteristic
248C2#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#8766
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
228. C7#8766semidirect

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
#87667