Finite Group Databasebeta

← #13263194

Subgroups — #13263194

Order 1792.

578 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 81 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
1D7×#384C211yesmaximal, characteristic
2#11725898C212yesmaximal
3C7×#1325C211yesmaximal, characteristic, complemented
4#11725441C211yesmaximal, characteristic, complemented
5#11725259C212yesmaximal, complemented
6C2×D7×C4⋊C4—22no—
7C2×D7×C4⋊C4C411yescharacteristic
8C2×D7×C4⋊C4C2211yescharacteristic
9C2×C42×D7C411yescharacteristic
10C7×#384C2211yescharacteristic
11#89748—22no—
12#89748C411yescharacteristic
13#89748C2211yescharacteristic
14#89707C411yescharacteristic
15#89498—22no—
16C7×#324C2212yes—
17#89323C2212yes—
18#89264—22no—
19#11512—71nomaximal, Sylow
20C22×C4×D7—21no—
21C22×C4×D7—21no—
22C22×C4×D7C2×C411yescharacteristic
23C22×C4×D7D412yes—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4C2×C411yescharacteristic
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C2×C4⋊Dic7—22no—
29C2×C4⋊Dic7C2×C411yescharacteristic
30C2×C4⋊Dic7C2×C411yescharacteristic
31C2×Dic7⋊C4—22no—
32C2×Dic7⋊C4—22no—
33C2×Dic7⋊C4—21no—
34C2×Dic7⋊C4D412yes—
35C2×C4×Dic7—21no—
36C2×C4×Dic7C2×C411yescharacteristic
37C2×#1131—42no—
38D7×C4⋊C4—41no—
39D7×C4⋊C4—41no—
40D7×C4⋊C4—41no—
41D7×C4⋊C4—41no—
42D7×C4⋊C4—48no—
43C42×D7—22no—
44C42×D7—22no—
45C7×C22⋊C8—22no—
46C22⋊(C7⋊C8)—22no—
47(C2×D7)⋊C8—42no—
48C2×#384—71no—
49#1463—72no—
50#1325—71no—
51#1325—72no—
52#1325D711yescharacteristic, complemented
53C23×D7C22⋊C411yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7C22×C411yescharacteristic
57C22×C4×C7C2×D412yes—
58C22×Dic7—21no—
59C22×Dic7—21no—
60C22×Dic7C22⋊C411yescharacteristic
61C22×Dic7C22⋊C412yes—
62C2×C4×D7—24no—
63C2×C4×D7—24no—
64C2×C4×D7—24no—
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—22no—
70C2×C4×D7—22no—
71C2×C4×D7—22no—
72C2×C4×D7—44no—
73C2×C4×D7—44no—
74C2×C4×D7—41no—
75C2×C4×D7—41no—
76C2×C7×C8—42no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—44no—
80C42×C7—21no—
81C42×C7M1612yescomplemented
82C4⋊Dic7—41no—
83C4⋊Dic7—44no—
84C4⋊Dic7—41no—
85Dic7⋊C4—42no—
86Dic7⋊C4—44no—
87Dic7⋊C4—44no—
88Dic7⋊C4—42no—
89C4×Dic7—24no—
90C4×Dic7—21no—
91C4×Dic7M1612yescomplemented
92C2×C7⋊C8—42no—
93#1131—84no—
94C22×C4⋊C4—71no—
95C22×C4⋊C4—71no—
96C22×C4⋊C4—142no—
97C22×C42—71no—
98#407—142no—
99#384—71no—
100#384—71no—
101#384—71no—
102#384—142no—
103#384C2×D711yescharacteristic
104#324—72no—
105#324—142no—
106#324C2×D712yes—
107C23×C7C2×C22⋊C411yescharacteristic
108C22×D7—21no—
109C22×D7—21no—
110C22×D7—22no—
111C22×D7—22no—
112C22×D7#9911yescharacteristic
113C22×D7#9911yescharacteristic
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—22no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—41no—
123C2×C4×C7C2×M1611yescharacteristic
124C2×C4×C7C2×M1611yescharacteristic
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—22no—
131C2×Dic7—42no—
132C2×Dic7—42no—
133C2×Dic7—41no—
134C2×Dic7#10112yes—
135C4×D7—216no—
136C4×D7—22no—
137C4×D7—22no—
138C4×D7—22no—
139C4×D7—22no—
140C4×D7—44no—
141C4×D7—82no—
142C4×D7—82no—
143C4×D7—82no—
144C7×C8—82no—
145C7⋊C8—82no—
146C23×C4—71no—
147C23×C4—72no—
148C23×C4—141no—
149C23×C4—141no—
150C2×M16—282no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—71no—
153C2×C4⋊C4—72no—
154C2×C4⋊C4—71no—
155C2×C4⋊C4—141no—
156C2×C4⋊C4—142no—
157C2×C4⋊C4—142no—
158C2×C4⋊C4—142no—
159C2×C4⋊C4—281no—
160C2×C4⋊C4—281no—
161C2×C4⋊C4—281no—
162C2×C4⋊C4—281no—
163C2×C4⋊C4—288no—
164C2×C4⋊C4C4×D711yescharacteristic
165C2×C4⋊C4C22×D711yescharacteristic
166C2×C42—71no—
167C2×C42—142no—
168C2×C42—141no—
169C2×C42—142no—
170C2×C42C4×D711yescharacteristic
171C22⋊C8—22no—
172C22⋊C8—142no—
173C22⋊C8—282no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7#40711yescharacteristic
177C22×C7C2×#9911yescharacteristic
178C22×C7C2×#10111yescharacteristic
179C2×D7—24no—
180C2×D7—24no—
181C2×D7—21no—
182C2×D7—21no—
183C2×D7—21no—
184C2×D7—21no—
185C2×D7#35312yes—
186C2×D7#35612yes—
187C4×C7—24no—
188C4×C7—21no—
189C4×C7—21no—
190C4×C7—42no—
191C4×C7—81no—
192Dic7—24no—
193Dic7—22no—
194Dic7—42no—
195Dic7—81no—
196C24—71no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—71no—
200C22×C4—72no—
201C22×C4—142no—
202C22×C4—141no—
203C22×C4—141no—
204C22×C4—144no—
205C22×C4—144no—
206C22×C4—144no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—141no—
210C22×C4—141no—
211C22×C4—141no—
212C22×C4—141no—
213C22×C4—284no—
214C22×C4—281no—
215C22×C4—281no—
216C22×C4—284no—
217C22×C4C2×C4×D711yescharacteristic
218C22×C4D4×D712yes—
219M16—564no—
220C2×C8—42no—
221C2×C8—282no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—44no—
225C4⋊C4—282no—
226C4⋊C4—281no—
227C4⋊C4—281no—
228C4⋊C4—284no—
229C4⋊C4—284no—
230C4⋊C4—284no—
231C4⋊C4—282no—
232C42—21no—
233C42—72no—
234C42—144no—
235C42—141no—
236C42#113112yescomplemented
237C2×C7—21no—
238C2×C7—21no—
239C2×C7#146311yescharacteristic
240C2×C7C2×#35311yescharacteristic
241C2×C7C2×#35611yescharacteristic
242D7—22no—
243D7—22no—
244C23—71no—
245C23—71no—
246C23—142no—
247C23—142no—
248C23—141no—
249C23—141no—
250C23D7×C22⋊C411yescharacteristic
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—72no—
261C2×C4—1416no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—141no—
270C2×C4—141no—
271C2×C4—142no—
272C2×C4—284no—
273C2×C4—282no—
274C2×C4—282no—
275C2×C4—281no—
276C2×C4—562no—
277C2×C4—562no—
278C2×C4—562no—
279C2×C4C2×#113111yescharacteristic
280C2×C4D7×M1611yescharacteristic
281C8—82no—
282C8—562no—
283C7#1151211yescharacteristic, complemented, Sylow
284C22—21no—
285C22—21no—
286C22—72no—
287C22—72no—
288C22—144no—
289C22—144no—
290C22—141no—
291C22—141no—
292C22—141no—
293C22—141no—
294C22#8949811yescharacteristic
295C22D7×#9911yescharacteristic
296C22D7×#10111yescharacteristic
297C4—24no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—81no—
302C4—144no—
303C4—142no—
304C4—282no—
305C4—561no—
306C2—21no—
307C2—21no—
308C2—142no—
309C2—142no—
310C2#1172589811yescharacteristic
311C2D7×#35311yescharacteristic
312C2D7×#35611yescharacteristic
subgroupG/H
centerC22D7×#99
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1325C2
socleC22×C7C2×#99
prime radicalC23×D7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1325D7
hypocenterC7#11512
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. C7×#1325C2semidirect
4. #11725441C2semidirect
5. #11725259C2semidirect
52. #1325D7semidirect
81. C42×C7M16semidirect
91. C4×Dic7M16semidirect
236. C42#1131semidirect
283. C7#11512semidirect

Maximal subgroups

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

subgroupindex
D7×#3842
#117258982
C7×#13252
#117254412
#117252592
#115127