Finite Group Databasebeta

← #13263285

Subgroups — #13263285

Order 1792.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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#11733929C211yesmaximal, characteristic
2#11725899C212yesmaximal
3C7×#1319C211yesmaximal, characteristic, complemented
4#11725449C211yesmaximal, characteristic, complemented
5#11725259C212yesmaximal, complemented
6C2×#6510C411yescharacteristic
7C2×Q8×Dic7C411yescharacteristic
8C2×#6456C2211yescharacteristic
9C7×#391C2211yescharacteristic
10#89748C2211yescharacteristic
11#89711C411yescharacteristic
12#89704C411yescharacteristic
13#89498—22no—
14#89441—22no—
15#89440—22no—
16C7×#324C2212yes—
17#89323C2212yes—
18#89263—22no—
19#11535—71nomaximal, Sylow
20C22×C7×Q8C2×C411yescharacteristic
21C22×D28C2×C411yescharacteristic
22C22×C4×D7D412yes—
23C2×C7×C4⋊C4C2311yescharacteristic
24C2×C42×C7C2×C411yescharacteristic
25#6510—22no—
26#6510—22no—
27Q8×Dic7—42no—
28C2×(C2×D7)⋊C4—22no—
29C2×(C2×D7)⋊C4—22no—
30C2×(C2×D7)⋊C4D412yes—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7C2×C411yescharacteristic
33C2×Dic7⋊C4—22no—
34C2×C4×Dic7—21no—
35C2×C4×Dic7C2×C411yescharacteristic
36C2×C4×Dic7C2×C411yescharacteristic
37C2×#1131—42no—
38#6456—42no—
39C7×C22⋊C8—22no—
40C7×#95—22no—
41C7⋊#95—21no—
42C7⋊#95—21no—
43C22⋊(C7⋊C8)—22no—
44(C2×D7)⋊C8—42no—
45#2326—71no—
46#1465—72no—
47#1325—72no—
48#1319—71no—
49#1319D711yescharacteristic, complemented
50C23×D7C22⋊C412yes—
51C2×C7×Q8—21no—
52C2×C7×Q8—41no—
53C2×C7×Q8—41no—
54C2×C7×Q8M1612yescomplemented
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7C22×C411yescharacteristic
58C22×C4×C7C2×D412yes—
59C22×Dic7—21no—
60C22×Dic7—21no—
61C22×Dic7C22⋊C411yescharacteristic
62C22×Dic7C22⋊C411yescharacteristic
63C2×D28—21no—
64C2×D28—21no—
65C2×D28—41no—
66C2×D28—41no—
67C2×C4×D7—22no—
68C2×C4×D7—22no—
69C2×C4×D7—82no—
70C2×C7×C8—42no—
71C7×C4⋊C4—41no—
72C42×C7—41no—
73(C2×D7)⋊C4—28no—
74(C2×D7)⋊C4—42no—
75(C2×D7)⋊C4—82no—
76C4⋊Dic7—42no—
77C4⋊Dic7—41no—
78Dic7⋊C4—82no—
79C4×Dic7—21no—
80C4×Dic7—22no—
81C4×Dic7—42no—
82C4×Dic7M1612yescomplemented
83C2×C7⋊C8—42no—
84#1131—84no—
85C2×#124—71no—
86C2×C4×Q8—71no—
87C2×C4×D4—71no—
88#407—142no—
89#391C2×D711yescharacteristic
90#390—71no—
91#388—142no—
92#385—142no—
93#384—71no—
94#376—71no—
95#324—72no—
96#324C2×D712yes—
97#323—142no—
98C23×C7C2×C22⋊C411yescharacteristic
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—42no—
103C22×D7—42no—
104C7×Q8—42no—
105C7×Q8—42no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7C2×M1611yescharacteristic
117C2×C4×C7C2×M1611yescharacteristic
118C2×Dic7—22no—
119C2×Dic7—21no—
120C2×Dic7—41no—
121C2×Dic7—41no—
122C2×Dic7—41no—
123C2×Dic7—41no—
124C2×Dic7—42no—
125C2×Dic7—41no—
126C2×Dic7#9912yes—
127D28—42no—
128D28—42no—
129C4×D7—44no—
130C4×D7—82no—
131C7×C8—82no—
132C7⋊C8—82no—
133C22×Q8C4×D711yescharacteristic
134C22×D4—71no—
135C23×C4—72no—
136C2×M16—282no—
137#124—142no—
138#124—142no—
139C4×Q8—282no—
140C4×D4—282no—
141C2×C4⋊C4—71no—
142C2×C4⋊C4—141no—
143C2×C4⋊C4—142no—
144C2×C4⋊C4C22×D711yescharacteristic
145C2×C22⋊C4—72no—
146C2×C22⋊C4—142no—
147C2×C22⋊C4—142no—
148C2×C42—71no—
149C2×C42—71no—
150C2×C42—141no—
151C2×C42C4×D711yescharacteristic
152C22⋊C8—22no—
153C22⋊C8—142no—
154C22⋊C8—282no—
155#95—22no—
156#95—141no—
157#95—141no—
158C22×C7—21no—
159C22×C7—21no—
160C22×C7#40711yescharacteristic
161C22×C7C2×#9911yescharacteristic
162C22×C7#41311yescharacteristic
163C2×D7—44no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—42no—
167C2×D7—82no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—42no—
171C4×C7—42no—
172C4×C7—81no—
173Dic7—22no—
174Dic7—42no—
175Dic7—41no—
176Dic7—81no—
177C24—72no—
178C2×Q8—21no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8#113112yescomplemented
182C2×D4—141no—
183C2×D4—141no—
184C2×D4—281no—
185C2×D4—281no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—142no—
191C22×C4—142no—
192C22×C4—141no—
193C22×C4—141no—
194C22×C4—562no—
195C22×C4C2×C4×D711yescharacteristic
196C22×C4D4×D712yes—
197M16—564no—
198C2×C8—42no—
199C2×C8—282no—
200C4⋊C4—41no—
201C4⋊C4—281no—
202C4⋊C4—282no—
203C4⋊C4—562no—
204C22⋊C4—148no—
205C22⋊C4—282no—
206C22⋊C4—562no—
207C42—41no—
208C42—72no—
209C42—142no—
210C42—141no—
211C42—282no—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#146511yescharacteristic
215C2×C7#212711yescharacteristic
216C2×C7#213511yescharacteristic
217D7—82no—
218C23—142no—
219C23—142no—
220C23—142no—
221C23—282no—
222C23—282no—
223C23D7×C22⋊C411yescharacteristic
224Q8—42no—
225Q8—42no—
226D4—282no—
227D4—282no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—72no—
239C2×C4—142no—
240C2×C4—141no—
241C2×C4—284no—
242C2×C4—281no—
243C2×C4—281no—
244C2×C4—281no—
245C2×C4—281no—
246C2×C4—282no—
247C2×C4—281no—
248C2×C4—562no—
249C2×C4C2×#113111yescharacteristic
250C2×C4D7×M1611yescharacteristic
251C8—82no—
252C8—562no—
253C7#1153511yescharacteristic, complemented, Sylow
254C22—21no—
255C22—21no—
256C22—284no—
257C22—282no—
258C22—282no—
259C22—282no—
260C22—562no—
261C22#8949811yescharacteristic
262C22D7×#9911yescharacteristic
263C22#8951711yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—142no—
270C4—282no—
271C4—281no—
272C4—561no—
273C2—21no—
274C2—21no—
275C2—562no—
276C2#1172587511yescharacteristic
277C2#1172691111yescharacteristic
278C2#1172692711yescharacteristic
subgroupG/H
centerC22D7×#99
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1319C2
socleC22×C7C2×#99
prime radicalC22×D28C2×C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1319D7
hypocenterC7#11535
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. C7×#1319C2semidirect
4. #11725449C2semidirect
5. #11725259C2semidirect
49. #1319D7semidirect
54. C2×C7×Q8M16semidirect
82. C4×Dic7M16semidirect
181. C2×Q8#1131semidirect
253. C7#11535semidirect

Maximal subgroups

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

subgroupindex
#117339292
#117258992
C7×#13192
#117254492
#117252592
#115357