Finite Group Databasebeta

← #13266869

Subgroups — #13266869

Order 1792.

332 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2843C211yesmaximal, characteristic
2#11727473C211yesmaximal, characteristic
3#11727461C211yesmaximal, characteristic
4#11727327C211yesmaximal, characteristic
5#11725458C211yesmaximal, characteristic
6#11725454C211yesmaximal, characteristic
7#11725441C211yesmaximal, characteristic
8#90191C2211yescharacteristic
9C7×#400—21no—
10C7×#400C2211yescharacteristic
11C7×#398—21no—
12C7×#398—21no—
13C7×#398—21no—
14C7×#395—21no—
15C7×#395C2211yescharacteristic
16C7×#384C2211yescharacteristic
17C7×#383—21no—
18#90000—21no—
19#89742—21no—
20#89736—21no—
21#89325C2211yescharacteristic
22#89323C2211yescharacteristic
23#89323C2211yescharacteristic
24#89318—21no—
25#89247—21no—
26#89247—21no—
27#12259—71nomaximal, Sylow
28C22×C7×Q8D411yescharacteristic
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4C2311yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×#1144—41no—
40#6489—21no—
41#6487—21no—
42#6487—21no—
43C2×#1136—21no—
44#6453—41no—
45#6451—41no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95D411yescharacteristic
55C7×#95D411yescharacteristic
56#6409—41no—
57C22⋊(C7⋊C8)—21no—
58C22⋊(C7⋊C8)—21no—
59#6384—41no—
60#6382—41no—
61#6381—41no—
62#6381—41no—
63#2843D711yescharacteristic
64#2060—71no—
65#1617—71no—
66#1605—71no—
67#1354—71no—
68#1353—71no—
69#1325—71no—
70C2×C7×Q8—41no—
71C2×C7×Q8—81no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C2×Dic14—21no—
82C2×Dic14—21no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—81no—
87C7×C4⋊C4—81no—
88C7×C4⋊C4—81no—
89C7×C4⋊C4—81no—
90C7×C4⋊C4—81no—
91C7×C4⋊C4—81no—
92C42×C7—21no—
93C42×C7Q1612yescomplemented
94C22⋊Dic7—41no—
95#1144—82no—
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100Dic7⋊C4—41no—
101Dic7⋊C4—41no—
102C4×Dic7—41no—
103#1136—22no—
104#1136—41no—
105C2×C7⋊C8—41no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108#544—71no—
109#484—141no—
110#451—141no—
111#428—141no—
112#400—21no—
113#400C2×D711yescharacteristic
114#398—21no—
115#398—21no—
116#398—21no—
117#395—21no—
118#395C2×D711yescharacteristic
119#384C2×D711yescharacteristic
120#383—21no—
121#343—71no—
122#339—141no—
123#328—141no—
124#328—141no—
125#324—71no—
126#324—71no—
127C23×C7#12011yescharacteristic
128C7×Q8—42no—
129C7×Q8—81no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7C2×Q1611yescharacteristic
154C2×C4×C7#13711yescharacteristic
155C2×Dic7—41no—
156C2×Dic7—41no—
157Dic14—81no—
158Dic14—81no—
159C7⋊C8—42no—
160C7⋊C8—81no—
161C7⋊C8—81no—
162C22×Q8C22⋊D711yescharacteristic
163C2×Q16—281no—
164C2×M16—141no—
165C4⋊Q8—281no—
166#125—281no—
167C22⋊Q8—141no—
168C22⋊Q8—141no—
169#117—141no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C22⋊D711yescharacteristic
177C2×C4⋊C4C22⋊D711yescharacteristic
178C2×C4⋊C4C22×D711yescharacteristic
179C2×C42C22⋊D711yescharacteristic
180#107—281no—
181#107—281no—
182#106—281no—
183#103—281no—
184#103—281no—
185C22⋊C8—141no—
186C22⋊C8—141no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C22⋊D711yescharacteristic
196#95C22⋊D711yescharacteristic
197C22×C7—21no—
198C22×C7—21no—
199C22×C7#45111yescharacteristic
200C22×C7#45611yescharacteristic
201C22×C7B(4,𝔽2)11yescharacteristic
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—42no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—81no—
208C4×C7—81no—
209C4×C7—81no—
210C4×C7—81no—
211C4×C7—81no—
212Dic7—81no—
213Dic7—81no—
214C2×Q8—41no—
215C2×Q8—81no—
216C2×Q8—141no—
217C2×Q8—141no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4C2×C22⋊D711yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C22×C4C2×C22⋊D711yescharacteristic
227Q16—562no—
228M16—142no—
229M16—281no—
230C2×C8—281no—
231C2×C8—281no—
232C2×C8—281no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—281no—
247C4⋊C4—281no—
248C22⋊C4—281no—
249C42—21no—
250C42—281no—
251C42#114412yescomplemented
252C2×C7—21no—
253C2×C7—21no—
254C2×C7#160511yescharacteristic
255C2×C7#220011yescharacteristic
256C2×C7#220311yescharacteristic
257C23#651611yescharacteristic
258Q8—42no—
259Q8—81no—
260Q8—561no—
261Q8—561no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4#650511yescharacteristic
288C2×C4C2×#114411yescharacteristic
289C8—282no—
290C8—561no—
291C8—561no—
292C7#1225911yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22#9000011yescharacteristic
296C22#9001611yescharacteristic
297C22#9001811yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—561no—
309C4—561no—
310C2—21no—
311C2—21no—
312C2#1172965811yescharacteristic
313C2#1172970111yescharacteristic
314C2#1172971311yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2843C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2843D7
hypocenterC7#12259
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2843

Complements

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

NH ≅ G/Ntype
93. C42×C7Q16semidirect
251. C42#1144semidirect
292. C7#12259semidirect

Maximal subgroups

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

subgroupindex
C7×#28432
#117274732
#117274612
#117273272
#117254582
#117254542
#117254412
#122597