Finite Group Databasebeta

← #13263227

Subgroups — #13263227

Order 1792.

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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#11736483C211yesmaximal, characteristic
2#11726868C212yesmaximal, complemented
3C7×#1327C211yesmaximal, characteristic, complemented
4#11725523C211yesmaximal, characteristic, complemented
5#11725336C212yesmaximal
6C2×C42×D7C411yescharacteristic
7C7×#423C2211yescharacteristic
8#89898C411yescharacteristic
9#89898C2211yescharacteristic
10M16×Dic7C411yescharacteristic
11M16×Dic7C2211yescharacteristic
12#89696C411yescharacteristic
13#89607—22no—
14#89605—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11521—71nomaximal, Sylow
20C22×C4×D7—21no—
21C22×C4×D7C2×C411yescharacteristic
22C2×C7×M16C2×C411yescharacteristic
23C2×C7×M16C2311yescharacteristic
24C2×C42×C7C2×C411yescharacteristic
25C2×C4×Dic7—21no—
26C2×C4×Dic7C2×C411yescharacteristic
27C2×#1136C2×C411yescharacteristic
28C2×#1136C2×C411yescharacteristic
29C42×D7—21no—
30C42×D7—21no—
31C42×D7—21no—
32C42×D7—21no—
33C7×M32—22no—
34C7×C4⋊C8—22no—
35(C2×D7)⋊C8—22no—
36(C2×D7)⋊C8—22no—
37#6389—21no—
38#6389D412yes—
39C8×Dic7—21no—
40C8×Dic7D412yes—
41#6386—22no—
42C4⋊(C7⋊C8)—22no—
43#2920—71no—
44#2165—72no—
45#1327—71no—
46#1327D711yescharacteristic, complemented
47#1320—72no—
48C23×D7M1611yescharacteristic
49C22×C4×C7—21no—
50C22×C4×C7C22×C411yescharacteristic
51C22×Dic7—21no—
52C22×Dic7M1611yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7—21no—
55C2×C4×D7—21no—
56C2×C4×D7—21no—
57C2×C4×D7—21no—
58C2×C4×D7—21no—
59C2×C4×D7—44no—
60C2×C4×D7—41no—
61C2×C4×D7—41no—
62C7×M16—41no—
63C7×M16—41no—
64C2×C7×C8—21no—
65C2×C7×C8C2×D412yes—
66C42×C7—21no—
67C42×C7M1611yescharacteristic
68C42×C7M1611yescharacteristic
69C4×Dic7—21no—
70C4×Dic7—21no—
71C4×Dic7—21no—
72C4×Dic7—21no—
73C4×Dic7—21no—
74C4×Dic7C22⋊C411yescharacteristic
75C4×Dic7C22⋊C411yescharacteristic
76#1136—41no—
77#1136—41no—
78C2×C7⋊C8—21no—
79C2×C7⋊C8C22⋊C412yes—
80C7×C16—42no—
81C7⋊C16—42no—
82C22×C42—71no—
83#432—142no—
84#431—142no—
85#423—71no—
86#423C2×D711yescharacteristic
87#407—71no—
88#407—71no—
89C4×M16—71no—
90C4×M16—71no—
91#364—142no—
92#349—72no—
93#349C2×D712yes—
94#347—142no—
95C23×C7C2×M1611yescharacteristic
96C22×D7—21no—
97C22×D7—42no—
98C22×D7—41no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—42no—
104C2×C4×C7C2×C22⋊C411yescharacteristic
105C2×C4×C7C2×M1611yescharacteristic
106C2×Dic7—21no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7—42no—
110C2×Dic7—41no—
111C4×D7—24no—
112C4×D7—42no—
113C4×D7—42no—
114C4×D7—42no—
115C4×D7—42no—
116C4×D7—41no—
117C4×D7—41no—
118C7×C8—22no—
119C7×C8—41no—
120C7⋊C8—22no—
121C7⋊C8—41no—
122C23×C4—71no—
123C23×C4—141no—
124C2×M16—71no—
125C2×M16—71no—
126C2×M16C4×D711yescharacteristic
127C2×M16C22×D711yescharacteristic
128C2×C42—71no—
129C2×C42—141no—
130C2×C42—141no—
131C2×C42—141no—
132C2×C42—141no—
133C2×C42—141no—
134C2×C42C4×D711yescharacteristic
135M32—22no—
136M32—142no—
137C4⋊C8—22no—
138C4⋊C8—142no—
139C22⋊C8—142no—
140C22⋊C8—142no—
141#97—72no—
142#97—141no—
143C4×C8—72no—
144C4×C8—141no—
145C22×C7—21no—
146C22×C7—41no—
147C22×C7#40711yescharacteristic
148C2×D7—22no—
149C2×D7—41no—
150C2×D7—42no—
151C2×D7—41no—
152C2×D7—42no—
153C4×C7—22no—
154C4×C7—21no—
155C4×C7—41no—
156C4×C7#41011yescharacteristic
157C4×C7#41311yescharacteristic
158Dic7—22no—
159Dic7—41no—
160Dic7—41no—
161C24—71no—
162C22×C4—21no—
163C22×C4—71no—
164C22×C4—141no—
165C22×C4—141no—
166C22×C4—141no—
167C22×C4—141no—
168C22×C4—141no—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4—281no—
172C22×C4—281no—
173C22×C4—284no—
174C22×C4C2×C4×D711yescharacteristic
175M16—41no—
176M16—41no—
177M16—281no—
178M16—281no—
179C2×C8—21no—
180C2×C8—72no—
181C2×C8—141no—
182C2×C8D4×D712yes—
183C42—21no—
184C42—71no—
185C42—71no—
186C42—141no—
187C42—141no—
188C42—141no—
189C42—141no—
190C42—141no—
191C42#113111yescharacteristic
192C42#113111yescharacteristic
193C16—42no—
194C16—282no—
195C2×C7—21no—
196C2×C7—41no—
197C2×C7#146411yescharacteristic
198D7—42no—
199C23—141no—
200C23—281no—
201C23—282no—
202C23D7×M1611yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—144no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—141no—
212C2×C4—282no—
213C2×C4—282no—
214C2×C4—282no—
215C2×C4—282no—
216C2×C4—282no—
217C2×C4—281no—
218C2×C4—281no—
219C2×C4—281no—
220C2×C4D7×C22⋊C411yescharacteristic
221C2×C4C2×#113111yescharacteristic
222C8—22no—
223C8—41no—
224C8—142no—
225C8—281no—
226C7#1152111yescharacteristic, complemented, Sylow
227C22—21no—
228C22—41no—
229C22—142no—
230C22—281no—
231C22—282no—
232C22—282no—
233C22—281no—
234C22#8949811yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—142no—
239C4—281no—
240C4—281no—
241C4#8951211yescharacteristic
242C4#8952311yescharacteristic
243C2—21no—
244C2—41no—
245C2—282no—
246C2#1172588011yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1327C2
socleC2×C7#1464
prime radicalC23×D7M16
cosocleC2×C7×M16C23
hypercenter#1327D7
hypocenterC7#11521
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #11726868C2semidirect
3. C7×#1327C2semidirect
4. #11725523C2semidirect
46. #1327D7semidirect
226. C7#11521semidirect

Maximal subgroups

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

subgroupindex
#117364832
#117268682
C7×#13272
#117255232
#117253362
#115217