Finite Group Databasebeta

← #13249306

Subgroups — #13249306

Order 1792.

546 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C7×#2982C211yesmaximal, characteristic, complemented
2#11739900C211yesmaximal, characteristic
3#11729378C211yesmaximal, characteristic, complemented
4#11728511C212yesmaximal
5#11725543C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6510C411yescharacteristic
8C2×C4×D28C2211yescharacteristic
9C7×C8×Q8—24no—
10C7×#431—22no—
11C7×#431C2212yes—
12#90002C411yescharacteristic
13#89761—22no—
14#89760C411yescharacteristic
15#89748C411yescharacteristic
16#89712—22no—
17#89704C2211yescharacteristic
18#89697C2212yes—
19#89323—22no—
20#89254—24no—
21#9102—71nomaximal, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C22×D28C2×C411yescharacteristic
24C22×C4×D7—21no—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C2×C7×C4⋊C4C2×C411yescharacteristic
28C2×C7×C4⋊C4D412yes—
29C2×C42×C7D412yes—
30C2×C42×C7C2311yescharacteristic
31#6510—22no—
32#6510C814yescomplemented
33C2×(C2×D7)⋊C4—22no—
34C2×(C2×D7)⋊C4—21no—
35C2×(C2×D7)⋊C4—22no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—22no—
38C2×C4×Dic7C2×C411yescharacteristic
39C2×C4×Dic7C2×C411yescharacteristic
40C4×D28—22no—
41C4×D28—22no—
42C7×C4⋊C8—24no—
43C7×C4⋊C8—22no—
44C7×C22⋊C8—22no—
45C4×C7×C8—24no—
46C4×C7×C8D414yes—
47C7⋊#95—22no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C22⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52C4×C7⋊C8—22no—
53#2982D711yescharacteristic, complemented
54#2326—71no—
55#1478—72no—
56#1461—71no—
57#1274—72no—
58C23×D7—21no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8C2×C812yes—
62C2×C7×Q8M1612yes—
63C22×C4×C7C22⋊C412yes—
64C22×C4×C7C22⋊C412yes—
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71C2×D28—22no—
72C2×D28—22no—
73C2×D28—21no—
74C2×D28C2×C812yes—
75C2×C4×D7—42no—
76C2×C4×D7—82no—
77C2×C7×C8—24no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—24no—
80C42×C7—22no—
81C42×C7—22no—
82C42×C7C2×D412yes—
83C42×C7#4112yes—
84(C2×D7)⋊C4—28no—
85(C2×D7)⋊C4—41no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—82no—
88C4⋊Dic7—22no—
89Dic7⋊C4—82no—
90C4×Dic7—21no—
91C4×Dic7—22no—
92C4×Dic7C2×C812yes—
93C2×C7⋊C8—42no—
94C2×#124—71no—
95C2×C4×Q8C2×D711yescharacteristic
96C2×C4×D4—71no—
97C8×Q8—24no—
98#431—22no—
99#431—72no—
100#431C2×D712yes—
101#391—71no—
102#390—71no—
103#388—142no—
104#385—142no—
105#384—71no—
106#376—71no—
107D4⋊C8—144no—
108#324—142no—
109C23×C7C2×C22⋊C411yescharacteristic
110C22×D7—22no—
111C22×D7—42no—
112C22×D7—42no—
113C22×D7—41no—
114C22×D7—41no—
115C7×Q8—24no—
116C7×Q8—24no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7C22⋊C814yes—
124C2×C4×C7C4×D412yes—
125C2×C4×C7#12011yescharacteristic
126C2×C4×C7#12112yes—
127C2×C4×C7#12311yescharacteristic
128C2×C4×C7C22×C811yescharacteristic
129C2×C4×C7C2×M1611yescharacteristic
130C2×C4×C7#13112yes—
131C2×Dic7—22no—
132C2×Dic7—41no—
133C2×Dic7—41no—
134C2×Dic7—42no—
135C2×Dic7—41no—
136C2×Dic7—41no—
137C2×Dic7—41no—
138C2×Dic7—41no—
139D28—24no—
140D28—42no—
141C4×D7—84no—
142C7×C8—44no—
143C7⋊C8—82no—
144C22×Q8C4×D711yescharacteristic
145C22×D4—71no—
146C23×C4—141no—
147#124—74no—
148#124—142no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×D4—142no—
152C4×D4—142no—
153C2×C4⋊C4—71no—
154C2×C4⋊C4—142no—
155C2×C4⋊C4C4×D711yescharacteristic
156C2×C4⋊C4D2812yes—
157C2×C22⋊C4—142no—
158C2×C22⋊C4—141no—
159C2×C22⋊C4—142no—
160C2×C42—71no—
161C2×C42—71no—
162C2×C42C22⋊D712yes—
163C2×C42C22×D711yescharacteristic
164C4⋊C8—24no—
165C4⋊C8—22no—
166C4⋊C8—142no—
167C22⋊C8—22no—
168C22⋊C8—142no—
169C4×C8—24no—
170C4×C8—142no—
171C4×C8C22⋊D714yes—
172#95—142no—
173#95—141no—
174#95—141no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7#38611yescharacteristic
178C22×C7C2×C22⋊C811yescharacteristic
179C22×C7#40811yescharacteristic
180C2×D7—44no—
181C2×D7—42no—
182C2×D7—82no—
183C2×D7—81no—
184C2×D7—81no—
185C4×C7—24no—
186C4×C7—22no—
187C4×C7—21no—
188C4×C7—21no—
189C4×C7—42no—
190C4×C7C8×D412yes—
191C4×C7#43512yes—
192Dic7—42no—
193Dic7—81no—
194Dic7—81no—
195C24—141no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8C8×D712yes—
199C2×Q8#113112yes—
200C2×D4—72no—
201C2×D4—141no—
202C2×D4—142no—
203C2×D4—142no—
204C22×C4—141no—
205C22×C4—141no—
206C22×C4—141no—
207C22×C4—282no—
208C22×C4—562no—
209C22×C4(C2×D7)⋊C412yes—
210C22×C4(C2×D7)⋊C412yes—
211C22×C4C2×C4×D711yescharacteristic
212C22×C4C2×D2811yescharacteristic
213C22×C4C2×C22⋊D711yescharacteristic
214C2×C8—24no—
215C2×C8—282no—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—142no—
219C4⋊C4—562no—
220C22⋊C4—148no—
221C22⋊C4—281no—
222C22⋊C4—281no—
223C22⋊C4—562no—
224C42—22no—
225C42—22no—
226C42—72no—
227C42—141no—
228C42—142no—
229C42#115712yes—
230C42C2×C22⋊D712yes—
231C2×C7—21no—
232C2×C7—21no—
233C2×C7#186911yescharacteristic
234C2×C7#191011yescharacteristic
235C2×C7#191211yescharacteristic
236D7—82no—
237C23—142no—
238C23—282no—
239C23—282no—
240C23—281no—
241C23—281no—
242C23C2×(C2×D7)⋊C411yescharacteristic
243Q8—24no—
244Q8—24no—
245D4—144no—
246D4—282no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—142no—
254C2×C4—281no—
255C2×C4—281no—
256C2×C4—281no—
257C2×C4—282no—
258C2×C4—281no—
259C2×C4—281no—
260C2×C4—281no—
261C2×C4—564no—
262C2×C4(C2×D7)⋊C814yes—
263C2×C4C2×C8×D711yescharacteristic
264C2×C4C2×#113111yescharacteristic
265C2×C4#646412yes—
266C2×C4C4×C22⋊D712yes—
267C2×C4#649211yescharacteristic
268C2×C4#649312yes—
269C2×C4#651611yescharacteristic
270C8—44no—
271C8—562no—
272C7#910211yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22—284no—
276C22—282no—
277C22—562no—
278C22—561no—
279C22—561no—
280C22C2×(C2×D7)⋊C811yescharacteristic
281C22#8988211yescharacteristic
282C22#8998511yescharacteristic
283C4—21no—
284C4—21no—
285C4—24no—
286C4—22no—
287C4—42no—
288C4—282no—
289C4—561no—
290C4—561no—
291C4C8×C22⋊D712yes—
292C4#8988312yes—
293C2—21no—
294C2—21no—
295C2—562no—
296C2#1172941711yescharacteristic
297C2#1172972911yescharacteristic
298C2#1172977911yescharacteristic
subgroupG/H
centerC22#89985
commutatorC2×C4×C7C22×C8
FrattiniC2×C42C22×D7
FittingC7×#2982C2
socleC22×C7#386
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#2982D7
hypocenterC7#9102
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
3. #11729378C2semidirect
5. #11725543C2semidirect
32. #6510C8semidirect
53. #2982D7semidirect
272. C7#9102semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117399002
#117293782
#117285112
#117255432
#91027