Finite Group Databasebeta

← #13251280

Subgroups — #13251280

Order 1792.

380 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3185C211yesmaximal, characteristic
2#11740778C211yesmaximal, characteristic
3#11729591C211yesmaximal, characteristic
4#11729591C211yesmaximal, characteristic
5#11729577C211yesmaximal, characteristic
6#11725476C211yesmaximal, characteristic
7#11725472C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C7×#533C2211yescharacteristic
10C7×#498C411yescharacteristic, complemented
11C7×#494C411yescharacteristic, complemented
12C7×#487C411yescharacteristic, complemented
13C7×#487C411yescharacteristic, complemented
14C7×#470—22no—
15C7×#441—22no—
16C4×C7×M16C2211yescharacteristic
17#89796—22no—
18#89796—22no—
19#89772—22no—
20#89709C2211yescharacteristic
21#89698C2211yescharacteristic
22#89698C2211yescharacteristic
23#89698C2211yescharacteristic
24#89338—22no—
25#89333—22no—
26#89333—22no—
27#9416—71nomaximal, Sylow
28C2×C7×Q16—22no—
29C2×C7×M16—21no—
30C7×C4⋊Q8C2×C411yescharacteristic
31C7×C4⋊Q8C2×C411yescharacteristic
32C7×C4⋊Q8C2×C411yescharacteristic
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×#125—21no—
35C7×#125—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8D412yes—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8D412yes—
44C7×#117—21no—
45C7×#117—21no—
46C2×C42×C7C2311yescharacteristic
47C7×#106—22no—
48C7×#103—22no—
49C7×#103—22no—
50C7×#97—21no—
51C4×C7×C8C2×C411yescharacteristic
52C4×C7×C8C2×C411yescharacteristic
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C22⋊(C7⋊C8)—21no—
57C22⋊(C7⋊C8)—21no—
58C22⋊(C7⋊C8)D412yes—
59C4⋊(C7⋊C8)—24no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63#6377—22no—
64#6377—21no—
65#6377—21no—
66#6377D412yes—
67C4×C7⋊C8—22no—
68#3185D711yescharacteristic
69#2987—71no—
70#1537—71no—
71#1515—71no—
72#1515—71no—
73#1291—71no—
74#1288—71no—
75C2×C7×Q8—21no—
76C2×C7×Q8—21no—
77C2×C7×Q8—21no—
78C2×C7×Q8C22⋊C412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7#4111yescharacteristic
82C22×Dic7—41no—
83C7×Q16—44no—
84C7×M16—42no—
85C2×C7×C8—21no—
86C2×C7×C8—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—21no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4C22⋊C412yes—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—41no—
101C42×C7—22no—
102C42×C7—41no—
103C42×C7C22×C411yescharacteristic
104C42×C7C2×D411yescharacteristic
105C42×C7C2×D411yescharacteristic
106C42×C7#4111yescharacteristic
107C2×C7⋊C8—22no—
108C2×C7⋊C8—22no—
109C2×C7⋊C8—41no—
110#533C2×D711yescharacteristic
111#533C2×D711yescharacteristic
112#498Dic711yescharacteristic, complemented
113#494Dic711yescharacteristic, complemented
114#487Dic711yescharacteristic, complemented
115#487Dic711yescharacteristic, complemented
116#470—22no—
117#446—142no—
118#446—142no—
119#441—22no—
120#433—142no—
121#432—71no—
122#432—71no—
123#432—71no—
124C4×M16C2×D711yescharacteristic
125#383—71no—
126#330—142no—
127#328—142no—
128#328—142no—
129C23×C7#12111yescharacteristic
130C7×Q8—42no—
131C7×Q8—42no—
132C7×Q8—81no—
133C7×Q8—81no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7C2×C22⋊C411yescharacteristic
142C2×C4×C7C4×D411yescharacteristic
143C2×C4×C7C4×D411yescharacteristic
144C2×C4×C7#12011yescharacteristic
145C2×C4×C7#12111yescharacteristic
146C2×C4×C7#12311yescharacteristic
147C2×Dic7—81no—
148C2×Dic7—81no—
149C2×Dic7—81no—
150C7×C8—22no—
151C7×C8—41no—
152C7⋊C8—42no—
153C7⋊C8—42no—
154C7⋊C8—81no—
155C2×Q16—22no—
156C2×M16—21no—
157C4⋊Q8C2×Dic711yescharacteristic
158C4⋊Q8C2×Dic711yescharacteristic
159C4⋊Q8C2×Dic711yescharacteristic
160C4⋊Q8C2×Dic711yescharacteristic
161#125—21no—
162#125—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8C22⋊D712yes—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8C22⋊D712yes—
171#117—21no—
172#117—21no—
173C2×C42C22×D711yescharacteristic
174#106—22no—
175C4⋊C8—144no—
176C4⋊C8—141no—
177C4⋊C8—141no—
178C4⋊C8—141no—
179#103—22no—
180#103—22no—
181C22⋊C8—72no—
182C22⋊C8—141no—
183C22⋊C8—141no—
184#97—21no—
185#97—72no—
186#97—142no—
187#97—141no—
188#97—141no—
189C4×C8—142no—
190C4×C8C2×Dic711yescharacteristic
191C4×C8C2×Dic711yescharacteristic
192#95—141no—
193#95—141no—
194#95—141no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7#38611yescharacteristic
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—42no—
206C4×C7—42no—
207C4×C7—81no—
208C4×C7—81no—
209Dic7—161no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8C22⋊Dic712yes—
214C22×C4—281no—
215C22×C4D4×D711yescharacteristic
216C22×C4#115911yescharacteristic
217C22×C4C2×C22⋊D711yescharacteristic
218Q16—44no—
219M16—42no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—142no—
223C2×C8—142no—
224C2×C8—281no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4C22⋊Dic712yes—
237C22⋊C4—22no—
238C22⋊C4—41no—
239C42—22no—
240C42—41no—
241C42D4×D711yescharacteristic
242C42#115911yescharacteristic
243C42C22×Dic711yescharacteristic
244C42C2×C22⋊D711yescharacteristic
245C2×C7—41no—
246C2×C7#188411yescharacteristic
247C2×C7#190211yescharacteristic
248C2×C7#190511yescharacteristic
249C23#650211yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—81no—
253Q8—81no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—561no—
262C2×C4—561no—
263C2×C4—561no—
264C2×C4D4×Dic711yescharacteristic
265C2×C4D4×Dic711yescharacteristic
266C2×C4#649811yescharacteristic
267C2×C4#650011yescharacteristic
268C2×C4#650211yescharacteristic
269C2×C4C2×C22⋊Dic711yescharacteristic
270C8—22no—
271C8—41no—
272C8—282no—
273C8—282no—
274C8—561no—
275C7#941611yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—21no—
279C22#8999211yescharacteristic
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—1121no—
291C2—41no—
292C2#1172957111yescharacteristic
293C2#1172985511yescharacteristic
294C2#1172985611yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3185C2
socleC22×C7#386
prime radicalC23×C7#121
cosocleC2×C42×C7C23
hypercenter#3185D7
hypocenterC7#9416
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3185

Complements

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

NH ≅ G/Ntype
10. C7×#498C4semidirect
11. C7×#494C4semidirect
12. C7×#487C4semidirect
13. C7×#487C4semidirect
112. #498Dic7semidirect
113. #494Dic7semidirect
114. #487Dic7semidirect
115. #487Dic7semidirect
275. C7#9416semidirect

Maximal subgroups

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

subgroupindex
C7×#31852
#117407782
#117295912
#117295772
#117254762
#117254722
#94167