Finite Group Databasebeta

← #13266231

Subgroups — #13266231

Order 1792.

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

normal subgroups only

This group has 372 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#11731662C211yesmaximal, characteristic, complemented
2C7×#2040C211yesmaximal, characteristic
3#11729628C212yesmaximal
4#11725438C211yesmaximal, characteristic, complemented
5#11725301C212yesmaximal, complemented
6C7×#537C2211yescharacteristic
7C2×#6433—21no—
8C2×C7×#104—21no—
9C7×#417—21no—
10C7×#382C2211yescharacteristic
11#89805—22no—
12#89755—21no—
13#89755C2212yes—
14#89433—22no—
15#89431—22no—
16C7×#343C2211yescharacteristic
17#89332—22no—
18#89323C2212yes—
19#12157—71nomaximal, Sylow
20C2×C7×#41—21no—
21C2×C7×M16—21no—
22C7×C22⋊Q8—41no—
23C7×#121—22no—
24C7×#121—41no—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7D411yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7D412yes—
32C2×Dic7⋊C4—22no—
33C2×Dic7⋊C4—22no—
34#6433—42no—
35#6433—42no—
36C7×#104—42no—
37C7×#103—41no—
38C7×#102—41no—
39C7×#95—22no—
40#6406—42no—
41C7⋊#95—21no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95D412yes—
45C22⋊(C7⋊C8)—22no—
46#6381—42no—
47#2822—71no—
48#2040D711yescharacteristic
49#1610—72no—
50#1346—72no—
51#1324—71no—
52C7×#41—41no—
53C7×#41—41no—
54C7×#41—41no—
55C7×#41—41no—
56C2×C7×Q8—21no—
57C2×C7×D4—21no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—41no—
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D412yes—
65C22×Dic7—22no—
66C22×Dic7—22no—
67C7×M16—41no—
68C7×M16—41no—
69C2×C7×C8—41no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C22⋊C4—42no—
74C42×C7—21no—
75C42×C7D812yes—
76C4⋊Dic7—42no—
77C4⋊Dic7—42no—
78C4⋊Dic7—42no—
79C4⋊Dic7—81no—
80Dic7⋊C4—44no—
81Dic7⋊C4—42no—
82Dic7⋊C4—42no—
83C2×C7⋊C8—42no—
84#537C2×D711yescharacteristic
85C2×#125—141no—
86#480—142no—
87C2×#104—21no—
88#417—21no—
89#400—142no—
90#398—72no—
91#398—142no—
92#398—141no—
93#382C2×D711yescharacteristic
94#343C2×D711yescharacteristic
95#327—142no—
96#324—72no—
97C23×C7—41no—
98C23×C7#12011yescharacteristic
99C7×Q8—41no—
100C7×Q8—41no—
101C7×D4—41no—
102C7×D4—41no—
103C7×D4—82no—
104C7×D4—81no—
105C7×D4—81no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—81no—
117C2×C4×C7—81no—
118C2×C4×C7C2×D811yescharacteristic
119C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
120C2×Dic7—42no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124C2×Dic7—42no—
125C2×Dic7—42no—
126C7×C8—81no—
127C7⋊C8—82no—
128C2×#41—21no—
129C2×M16—21no—
130#125—282no—
131#125—282no—
132C22⋊Q8—41no—
133#121—22no—
134#121—41no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—142no—
137C2×C4⋊C4—141no—
138C2×C4⋊C4—141no—
139C2×C4⋊C4—142no—
140C2×C4⋊C4—141no—
141C2×C4⋊C4C22⋊D711yescharacteristic
142C2×C4⋊C4C22×D711yescharacteristic
143C2×C42C22⋊D711yescharacteristic
144#107—282no—
145#104—42no—
146#103—41no—
147#102—41no—
148#102—282no—
149C22⋊C8—142no—
150#95—22no—
151#95—72no—
152#95—142no—
153#95—141no—
154#95—142no—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7—81no—
158C22×C7—81no—
159C22×C7—81no—
160C22×C7#44711yescharacteristic
161C22×C7#45411yescharacteristic
162C22×C7B(4,𝔽2)11yescharacteristic
163C4×C7—21no—
164C4×C7—21no—
165C4×C7—42no—
166C4×C7—42no—
167C4×C7—81no—
168C4×C7—81no—
169Dic7—82no—
170Dic7—82no—
171#41—41no—
172#41—41no—
173#41—41no—
174#41—41no—
175C2×Q8—21no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—41no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—142no—
183C22×C4—142no—
184C22×C4D4×D712yes—
185C22×C4C2×C22⋊D711yescharacteristic
186M16—41no—
187M16—41no—
188C2×C8—41no—
189C2×C8—282no—
190C4⋊C4—22no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—282no—
194C4⋊C4—284no—
195C4⋊C4—282no—
196C4⋊C4—282no—
197C4⋊C4—282no—
198C4⋊C4—282no—
199C4⋊C4—561no—
200C22⋊C4—42no—
201C42—21no—
202C42#114112yes—
203C2×C7—21no—
204C2×C7—21no—
205C2×C7—161no—
206C2×C7#159811yescharacteristic
207C2×C7#220111yescharacteristic
208C2×C7#220311yescharacteristic
209C23—41no—
210C23#650011yescharacteristic
211Q8—41no—
212Q8—41no—
213D4—41no—
214D4—41no—
215D4—82no—
216D4—81no—
217D4—81no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—282no—
231C2×C4—282no—
232C2×C4—282no—
233C2×C4—282no—
234C2×C4—282no—
235C2×C4—282no—
236C2×C4C2×#114111yescharacteristic
237C2×C4#651211yescharacteristic
238C8—81no—
239C8—562no—
240C7#1215711yescharacteristic, complemented, Sylow
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#8980411yescharacteristic
247C22#8980811yescharacteristic
248C22#8983411yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—562no—
256C4—562no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2#1172745711yescharacteristic
261C2#1172874311yescharacteristic
262C2#1172874611yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2040C2
socleC22×C7B(4,𝔽2)
prime radicalC7×#537C22
hypercenter#2040D7
hypocenterC7#12157
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #11731662C2semidirect
4. #11725438C2semidirect
5. #11725301C2semidirect
240. C7#12157semidirect

Maximal subgroups

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

subgroupindex
#117316622
C7×#20402
#117296282
#117254382
#117253012
#121577