Finite Group Databasebeta

← #13256798

Subgroups — #13256798

Order 1792.

500 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 109 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×#2646C211yesmaximal, characteristic
2#11727256C212yesmaximal
3#11727177C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725419C211yesmaximal, characteristic
6C22×C4⋊Dic7C2211yescharacteristic
7C7×#400—22no—
8C7×#398—22no—
9C7×#385—22no—
10C7×#380C2212yes—
11C2×C7×#95C2211yescharacteristic
12C2×C7⋊#95—22no—
13C2×C7⋊#95C2211yescharacteristic
14C2×C22⋊(C7⋊C8)C2212yes—
15#89324—22no—
16#89324—22no—
17#10347—71nomaximal, Sylow
18C23×C4×C7D412yes—
19C23×C4×C7C2311yescharacteristic
20C23×Dic7—21no—
21C23×Dic7—21no—
22C2×C7×C4⋊C4—22no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C4⋊C4D412yes—
26C2×C7×C22⋊C4—22no—
27C2×C4⋊Dic7—22no—
28C2×C4⋊Dic7—22no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7Q812yes—
31C22×C7⋊C8—22no—
32C7×#95—22no—
33C7×#95—22no—
34C7×#95—22no—
35C7×#95—22no—
36C7⋊#95—22no—
37C7⋊#95—44no—
38C22⋊(C7⋊C8)—22no—
39C22⋊(C7⋊C8)—22no—
40#2646D711yescharacteristic
41#1824—72no—
42#1445—71no—
43#1294—72no—
44#1280—71no—
45C24×C7C2×D411yescharacteristic
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7—42no—
52C22×C4×C7—42no—
53C22×C4×C7—42no—
54C22×C4×C7C2×D412yes—
55C22×C4×C7C2×Q811yescharacteristic
56C22×C4×C7#4112yes—
57C22×C4×C7#4111yescharacteristic
58C22×Dic7—22no—
59C22×Dic7—22no—
60C22×Dic7—42no—
61C22×Dic7—41no—
62C22×Dic7—42no—
63C22×Dic7—42no—
64C22×Dic7—41no—
65C22×Dic7—41no—
66C22×Dic7—41no—
67C22×Dic7—42no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—82no—
71C7×C4⋊C4—82no—
72C7×C22⋊C4—42no—
73C7×C22⋊C4—42no—
74C4⋊Dic7—24no—
75C4⋊Dic7—42no—
76C2×C7⋊C8—42no—
77C2×C7⋊C8—42no—
78C2×C7⋊C8—42no—
79C22×C4⋊C4—71no—
80C2×C22⋊C8—72no—
81#400—22no—
82#398—22no—
83#385—22no—
84#380C2×D712yes—
85C2×#95—71no—
86C2×#95—142no—
87C2×#95C2×D711yescharacteristic
88#340—142no—
89#340—142no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—22no—
93C23×C7#12011yescharacteristic
94C23×C7C22⋊Q811yescharacteristic
95C23×C7#12411yescharacteristic
96C23×C7#13512yes—
97C23×C7#13712yes—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—42no—
103C2×C4×C7—44no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—82no—
114C2×C4×C7C22⋊Q812yes—
115C2×C4×C7#12412yes—
116C2×Dic7—44no—
117C2×Dic7—42no—
118C2×Dic7—44no—
119C2×Dic7—42no—
120C2×Dic7—82no—
121C2×Dic7—81no—
122C2×Dic7—82no—
123C2×Dic7—81no—
124C2×Dic7—81no—
125C2×Dic7—81no—
126C7⋊C8—82no—
127C23×C4—141no—
128C23×C4—141no—
129C23×C4C22⋊D712yes—
130C23×C4C22×D711yescharacteristic
131C22×C8—142no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—72no—
135C2×C4⋊C4—142no—
136C2×C4⋊C4—142no—
137C2×C4⋊C4—141no—
138C2×C4⋊C4C22⋊D712yes—
139C2×C4⋊C4C22⋊D712yes—
140C2×C22⋊C4—22no—
141C22⋊C8—142no—
142C22⋊C8—142no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—142no—
148#95—284no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7—22no—
152C22×C7—22no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7#39311yescharacteristic
158C22×C7#45212yes—
159C22×C7#45311yescharacteristic
160C22×C7#45411yescharacteristic
161C22×C7#45812yes—
162C22×C7#47912yes—
163C22×C7#48811yescharacteristic
164C22×C7#49111yescharacteristic
165C4×C7—22no—
166C4×C7—22no—
167C4×C7—82no—
168C4×C7—82no—
169C4×C7—82no—
170Dic7—82no—
171Dic7—82no—
172C24C2×C22⋊D711yescharacteristic
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—142no—
182C22×C4—142no—
183C22×C4—282no—
184C22×C4—282no—
185C22×C4—281no—
186C22×C4—282no—
187C22×C4—281no—
188C22×C4—281no—
189C22×C4—282no—
190C22×C4—281no—
191C22×C4Q8×D711yescharacteristic
192C22×C4#116112yes—
193C22×C4#116111yescharacteristic
194C22×C4C2×C22⋊D712yes—
195C2×C8—282no—
196C2×C8—282no—
197C2×C8—282no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—144no—
203C4⋊C4—282no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C2×C7—21no—
207C2×C7—22no—
208C2×C7—21no—
209C2×C7#161512yes—
210C2×C7#161812yes—
211C2×C7#202611yescharacteristic
212C2×C7#202711yescharacteristic
213C2×C7#203011yescharacteristic
214C23—21no—
215C23—22no—
216C23—21no—
217C23#650911yescharacteristic
218C23#651011yescharacteristic
219C23#651312yes—
220C23#651412yes—
221C23#651611yescharacteristic
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—284no—
239C2×C4—282no—
240C2×C4—284no—
241C2×C4—282no—
242C2×C4—562no—
243C2×C4—561no—
244C2×C4—562no—
245C2×C4—561no—
246C2×C4—561no—
247C2×C4—561no—
248C2×C4#650912yes—
249C2×C4#651012yes—
250C8—562no—
251C7#1034711yescharacteristic, complemented, Sylow
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22#8983512yes—
261C22#8984011yescharacteristic
262C22#8984111yescharacteristic
263C22#9000311yescharacteristic
264C22#9001112yes—
265C22#9001311yescharacteristic
266C22#9001512yes—
267C22#9001711yescharacteristic
268C4—22no—
269C4—22no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—562no—
274C4—562no—
275C2—21no—
276C2—22no—
277C2—21no—
278C2#1172949811yescharacteristic
279C2#1172963712yes—
280C2#1172964112yes—
281C2#1172964811yescharacteristic
282C2#1172965711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2646C2
socleC23×C7#120
prime radicalC24×C7C2×D4
hypercenter#2646D7
hypocenterC7#10347
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
251. C7#10347semidirect

Maximal subgroups

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

subgroupindex
C7×#26462
#117272562
#117271772
#117254302
#117254192
#103477