Finite Group Databasebeta

← #13510240

Subgroups — #13510240

Order 1792.

488 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 127 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#11738187C212yesmaximal, complemented
2C7×#2219C211yesmaximal, characteristic, complemented
3#11729252C212yesmaximal, complemented
4#11729197C211yesmaximal, characteristic
5#11729175C211yesmaximal, characteristic, complemented
6#11728836C212yesmaximal, complemented
7#11727107C212yesmaximal, complemented
8#11726989C212yesmaximal, complemented
9#11726964C212yesmaximal, complemented
10#90334—22no—
11C2×C7×Q32C2211yescharacteristic
12C2×C7×M32C2211yescharacteristic
13C7×#467C2212yes—
14#89963C2212yes—
15Q16×Dic7C2212yes—
16C2×#6402C2212yes—
17#89887C2212yes—
18#89873C2211yescharacteristic
19C22×C7⋊C16C2211yescharacteristic
20#89821—22no—
21C2×D112C2211yescharacteristic
22#89646C2212yes—
23#89644C2212yes—
24#89589—22no—
25#89567—22no—
26C7×#367C2211yescharacteristic
27C7×#358C2212yes—
28#89359C2212yes—
29#89304C2212yes—
30#89300C2211yescharacteristic
31Dic7⋊C16C2212yes—
32Dic7⋊C16C2212yes—
33#89286C2212yes—
34#89285C2212yes—
35#33829—71nomaximal, Sylow
36C2×C7×Q16C2312yes—
37C22×C7×C8C2311yescharacteristic
38C7×C22⋊Q8—22no—
39#6510—22no—
40Q8×Dic7—22no—
41C2×#1143—22no—
42#6493—22no—
43#6489D411yescharacteristic
44C2×#1136D411yescharacteristic
45C2×D56C2312yes—
46#6461—42no—
47#6460—42no—
48#6456—22no—
49#6451—22no—
50#6448—42no—
51#6446—42no—
52C7×Q32—41no—
53C7×M32—41no—
54C2×C7×C16C2311yescharacteristic
55C2×C7×C16C2311yescharacteristic
56C7×#107C2312yes—
57C7×#103—22no—
58#6409—22no—
59#6402—42no—
60#6395—22no—
61#6392C2312yes—
62#6389D412yes—
63C8×Dic7C2312yes—
64C2×C7⋊C16—41no—
65C2×C7⋊C16C2312yes—
66#6383—22no—
67#6381—22no—
68D112—41no—
69#3175—72no—
70#2226—72no—
71#2225—72no—
72#2219D711yescharacteristic, complemented
73#2218—71no—
74#2217—72no—
75#2211—72no—
76#2210—72no—
77#2151—71no—
78C2×C7×Q8—22no—
79C22×C4×C7C2×D411yescharacteristic
80C2×C22⋊D7—42no—
81C2×D28—22no—
82C2×C4×D7—42no—
83C2×Dic14—42no—
84C7×Q16—22no—
85C7×Q16—22no—
86C2×C7×C8—21no—
87C2×C7×C8—21no—
88C2×C7×C8C2×D411yescharacteristic
89C2×C7×C8C2411yescharacteristic
90C7×C4⋊C4—22no—
91C7×C4⋊C4—42no—
92C7×C22⋊C4—42no—
93C22⋊Dic7D811yescharacteristic
94C22⋊Dic7D811yescharacteristic
95#1143—44no—
96(C2×D7)⋊C4—42no—
97(C2×D7)⋊C4—42no—
98(C2×D7)⋊C4—42no—
99(C2×D7)⋊C4—42no—
100C4⋊Dic7—22no—
101C4⋊Dic7—42no—
102C4⋊Dic7D812yes—
103Dic7⋊C4—42no—
104C4×Dic7—42no—
105C4×Dic7C2×D412yes—
106#1136—41no—
107C2×C7⋊C8C2×D412yes—
108D56—22no—
109D56—22no—
110C7×C16—21no—
111C7×C16—21no—
112C7⋊C16—22no—
113#542—142no—
114C2×Q32C2×D711yescharacteristic
115C2×SD32—72no—
116C2×D16—71no—
117C2×M32C2×D711yescharacteristic
118C22×C16—71no—
119#500—72no—
120#495—72no—
121#489—142no—
122#468—142no—
123#467C2×D712yes—
124#466—72no—
125#440—142no—
126C4×Q16—72no—
127C4×D8—72no—
128#405—71no—
129#367—71no—
130#367—72no—
131#367C2×D711yescharacteristic
132C4⋊C16—72no—
133C4⋊C16—72no—
134#358—72no—
135#358C2×D712yes—
136#357—72no—
137#357—72no—
138C23×C7C2×D811yescharacteristic
139C22×D7—42no—
140C7×Q8—42no—
141C7×Q8—42no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7C2×D811yescharacteristic
147C2×C4×C7C22×D411yescharacteristic
148C22⋊D7—162no—
149C2×Dic7—42no—
150C2×Dic7C2×D812yes—
151C2×Dic7C2×D812yes—
152D28—42no—
153D28—42no—
154C4×D7—162no—
155Dic14—162no—
156C7×C8—21no—
157C7×C8H(5,𝔽2)11yescharacteristic
158C7×C82−1+411yescharacteristic
159C7⋊C8—22no—
160C2×Q16C22×D712yes—
161C2×SD16—142no—
162C2×D8—72no—
163C2×M16—71no—
164C22×C8C22×D711yescharacteristic
165C4⋊Q8—142no—
166#126—282no—
167#124—142no—
168#124—282no—
169#123—282no—
170C22⋊Q8—22no—
171C22⋊Q8—282no—
172#121—142no—
173C4×Q8—142no—
174C4×D4—142no—
175#117—71no—
176Q32—41no—
177SD32—282no—
178D16—281no—
179M32—41no—
180C2×C16—72no—
181C2×C16—281no—
182C2×C16C22×D711yescharacteristic
183C2×C16C22×D711yescharacteristic
184#107—72no—
185#107—142no—
186#107C22×D712yes—
187#103—22no—
188#103—142no—
189#102—142no—
190#102—142no—
191#97—72no—
192C4×C8—72no—
193C22×C7—22no—
194C22×C7—21no—
195C22×C7C22×D811yescharacteristic
196C2×D7—82no—
197C2×D7—82no—
198C2×D7—82no—
199C4×C7—21no—
200C4×C7—82no—
201C4×C7—82no—
202C4×C7#53711yescharacteristic
203C4×C7#57511yescharacteristic
204Dic7—22no—
205Dic7—22no—
206Dic7—82no—
207C2×Q8—22no—
208C2×Q8—282no—
209C2×D4—142no—
210C2×D4—282no—
211C22×C4—282no—
212C22×C4D4×D711yescharacteristic
213Q16—22no—
214Q16—22no—
215SD16—284no—
216D8—142no—
217D8—142no—
218M16—281no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—72no—
222C2×C8D4×D711yescharacteristic
223C2×C8C23×D711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—72no—
227C4⋊C4—142no—
228C4⋊C4—282no—
229C4⋊C4—282no—
230C22⋊C4—42no—
231C22⋊C4—71no—
232C22⋊C4—71no—
233C22⋊C4—282no—
234C22⋊C4—282no—
235C22⋊C4—282no—
236C22⋊C4—282no—
237C42—72no—
238C42—282no—
239C16—21no—
240C16—21no—
241C16—142no—
242C2×C7—41no—
243C2×C7#324511yescharacteristic
244C2×C7#341612yes—
245D7—162no—
246C23—282no—
247C23D7×D811yescharacteristic
248Q8—42no—
249Q8—42no—
250Q8—1122no—
251D4—282no—
252D4—282no—
253D4—1122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—72no—
259C2×C4—72no—
260C2×C4—282no—
261C2×C4—1122no—
262C2×C4D7×D811yescharacteristic
263C2×C4C2×D4×D711yescharacteristic
264C8—21no—
265C8—142no—
266C8#654811yescharacteristic
267C8#655411yescharacteristic
268C7#3382911yescharacteristic, complemented, Sylow
269C22—21no—
270C22—22no—
271C22—562no—
272C22—562no—
273C22—562no—
274C22C2×D7×D811yescharacteristic
275C4—21no—
276C4—82no—
277C4—82no—
278C4—142no—
279C4—142no—
280C4—562no—
281C4#9028811yescharacteristic
282C4#9045911yescharacteristic
283C2—41no—
284C2—1122no—
285C2#1173814711yescharacteristic
286C2#1173948512yes—
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2219C2
socleC22×C7C22×D8
prime radical#11729197C2
hypercenter#2219D7
hypocenterC7#33829
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2219

Complements

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

NH ≅ G/Ntype
1. #11738187C2semidirect
2. C7×#2219C2semidirect
3. #11729252C2semidirect
5. #11729175C2semidirect
6. #11728836C2semidirect
7. #11727107C2semidirect
8. #11726989C2semidirect
9. #11726964C2semidirect
72. #2219D7semidirect
268. C7#33829semidirect

Maximal subgroups

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

subgroupindex
#117381872
C7×#22192
#117292522
#117291972
#117291752
#117288362
#117271072
#117269892
#117269642
#338297