Finite Group Databasebeta

← #13266057

Subgroups — #13266057

Order 1792.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#11734047C211yesmaximal, characteristic, complemented
2C7×#2007C211yesmaximal, characteristic
3#11729629C212yesmaximal
4#11725449C211yesmaximal, characteristic, complemented
5#11725302C212yesmaximal, complemented
6C2×C7×#137—21no—
7C7×#537C2211yescharacteristic
8C2×C7×#104—21no—
9C7×#391C2211yescharacteristic
10#90002—21no—
11#89806—22no—
12#89758C2212yes—
13#89700—21no—
14#89433—22no—
15#89427—22no—
16C7×#343C2211yescharacteristic
17#89332—22no—
18#89323C2212yes—
19#12131—71nomaximal, Sylow
20C2×C7×#41—21no—
21C22×C7×Q8D411yescharacteristic
22C7×#137—42no—
23C7×#137—42no—
24C2×C7×Q16—41no—
25C2×C7×SD16—41no—
26C2×C7×M16—21no—
27C7×C22⋊Q8—41no—
28C7×#121—22no—
29C7×#121—41no—
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7D412yes—
36C2×Dic7⋊C4—22no—
37C2×C4×Dic7—21no—
38C7×#104—42no—
39C7×#95—22no—
40#6406—42no—
41C7⋊#95—22no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95D412yes—
45C22⋊(C7⋊C8)—22no—
46#6382—42no—
47#2809—71no—
48#2007D711yescharacteristic
49#1616—72no—
50#1353—72no—
51#1319—71no—
52C7×#41—41no—
53C7×#41—41no—
54C7×#41—41no—
55C7×#41—41no—
56C2×C7×Q8—21no—
57C2×C7×Q8—21no—
58C2×C7×Q8—41no—
59C2×C7×Q8—41no—
60C2×C7×Q8SD1612yes—
61C2×C7×D4—21no—
62C2×C7×D4—22no—
63C2×C7×D4—41no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—41no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D412yes—
69C22×Dic7—22no—
70C22×Dic7—22no—
71C7×Q16—44no—
72C7×SD16—44no—
73C7×M16—22no—
74C7×M16—41no—
75C2×C7×C8—41no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—41no—
78C7×C22⋊C4—42no—
79C42×C7—41no—
80C4⋊Dic7—42no—
81C4⋊Dic7—42no—
82C4⋊Dic7—81no—
83C4⋊Dic7—81no—
84Dic7⋊C4—44no—
85C4×Dic7—81no—
86C2×C7⋊C8—42no—
87C2×#137—21no—
88#537C2×D711yescharacteristic
89#481—142no—
90C2×#104—21no—
91#400—72no—
92#398—142no—
93#391—141no—
94#391C2×D711yescharacteristic
95#389—141no—
96#387—142no—
97#343C2×D711yescharacteristic
98#327—142no—
99#324—72no—
100C23×C7—41no—
101C23×C7#12011yescharacteristic
102C7×Q8—42no—
103C7×Q8—41no—
104C7×Q8—41no—
105C7×Q8—81no—
106C7×D4—41no—
107C7×D4—41no—
108C7×D4—82no—
109C7×D4—81no—
110C7×D4—81no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—41no—
122C2×C4×C7—81no—
123C2×C4×C7—81no—
124C2×C4×C7C2×SD1611yescharacteristic
125C2×C4×C7#13711yescharacteristic
126C2×Dic7—42no—
127C2×Dic7—42no—
128C2×Dic7—42no—
129C2×Dic7—42no—
130C2×Dic7—42no—
131C2×Dic7—42no—
132C7×C8—42no—
133C7⋊C8—82no—
134C2×#41—21no—
135C22×Q8C22⋊D711yescharacteristic
136#137—42no—
137#137—42no—
138C2×Q16—41no—
139C2×SD16—41no—
140C2×M16—21no—
141C22⋊Q8—41no—
142#121—22no—
143#121—41no—
144C2×C4⋊C4—72no—
145C2×C4⋊C4—142no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4—141no—
148C2×C4⋊C4—141no—
149C2×C4⋊C4C22×D711yescharacteristic
150C2×C42—141no—
151C2×C42C22⋊D711yescharacteristic
152#106—282no—
153#104—42no—
154#102—282no—
155C22⋊C8—142no—
156#95—22no—
157#95—72no—
158#95—142no—
159#95—142no—
160#95—142no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7—81no—
164C22×C7—81no—
165C22×C7—81no—
166C22×C7#44811yescharacteristic
167C22×C7#45311yescharacteristic
168C22×C7B(4,𝔽2)11yescharacteristic
169C4×C7—21no—
170C4×C7—21no—
171C4×C7—42no—
172C4×C7—42no—
173C4×C7—42no—
174C4×C7—81no—
175Dic7—82no—
176Dic7—82no—
177#41—41no—
178#41—41no—
179#41—41no—
180#41—41no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8#114212yes—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—41no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—142no—
193C22×C4—142no—
194C22×C4D4×D712yes—
195C22×C4C2×C22⋊D711yescharacteristic
196Q16—44no—
197SD16—44no—
198M16—22no—
199M16—41no—
200C2×C8—41no—
201C2×C8—282no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—282no—
205C4⋊C4—284no—
206C4⋊C4—282no—
207C4⋊C4—561no—
208C4⋊C4—561no—
209C22⋊C4—42no—
210C42—41no—
211C42—561no—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7—161no—
215C2×C7#159611yescharacteristic
216C2×C7#220111yescharacteristic
217C2×C7#220411yescharacteristic
218C23—41no—
219C23#650011yescharacteristic
220Q8—42no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224D4—41no—
225D4—41no—
226D4—82no—
227D4—81no—
228D4—81no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—282no—
243C2×C4—282no—
244C2×C4—282no—
245C2×C4—282no—
246C2×C4—282no—
247C2×C4—282no—
248C2×C4C2×#114211yescharacteristic
249C2×C4#651411yescharacteristic
250C8—42no—
251C8—562no—
252C7#1213111yescharacteristic, complemented, Sylow
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#8980411yescharacteristic
259C22#8981211yescharacteristic
260C22#8984911yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—562no—
268C4—562no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2#1172745211yescharacteristic
273C2#1172874611yescharacteristic
274C2#1172874811yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2007C2
socleC22×C7B(4,𝔽2)
prime radicalC7×#537C22
hypercenter#2007D7
hypocenterC7#12131
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #11734047C2semidirect
4. #11725449C2semidirect
5. #11725302C2semidirect
252. C7#12131semidirect

Maximal subgroups

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

subgroupindex
#117340472
C7×#20072
#117296292
#117254492
#117253022
#121317