Finite Group Databasebeta

← #13510236

Subgroups — #13510236

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×#2216C211yesmaximal, characteristic, complemented
3#11729252C212yesmaximal, complemented
4#11729167C211yesmaximal, characteristic
5#11729155C211yesmaximal, characteristic, complemented
6#11728838C212yesmaximal, complemented
7#11727087C212yesmaximal, complemented
8#11726989C212yesmaximal, complemented
9#11726965C212yesmaximal, complemented
10#90334—22no—
11C2×C7×Q32C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13C7×#467C2212yes—
14#89963C2212yes—
15Q16×Dic7C2212yes—
16C2×#6402C2212yes—
17#89887C2212yes—
18#89873C2211yescharacteristic
19C2×#6386C2211yescharacteristic
20#89821—22no—
21C2×D112C2211yescharacteristic
22#89646C2212yes—
23#89644C2212yes—
24#89589—22no—
25#89567—22no—
26C7×#366C2211yescharacteristic
27C7×#358C2212yes—
28#89359C2212yes—
29#89304C2212yes—
30#89299C2211yescharacteristic
31Dic7⋊C16C2212yes—
32Dic7⋊C16C2212yes—
33#89286C2212yes—
34#89284C2212yes—
35#33828—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—
53C2×C7×C16—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—
64#6386—41no—
65C2×C7⋊C16C2312yes—
66#6383—22no—
67#6381—22no—
68D112—41no—
69#3175—72no—
70#2227—72no—
71#2224—72no—
72#2220—72no—
73#2216D711yescharacteristic, complemented
74#2215—71no—
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⋊Dic7D812yes—
94#1143—44no—
95(C2×D7)⋊C4—42no—
96(C2×D7)⋊C4—42no—
97(C2×D7)⋊C4—42no—
98(C2×D7)⋊C4—42no—
99C4⋊Dic7—22no—
100C4⋊Dic7—42no—
101C4⋊Dic7D811yescharacteristic
102C4⋊Dic7D811yescharacteristic
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×M32—71no—
118C22×C16C2×D711yescharacteristic
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#366—71no—
130#366—72no—
131#366C2×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—281no—
180C2×C16—41no—
181C2×C16—72no—
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—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7C22×D811yescharacteristic
197C2×D7—82no—
198C2×D7—82no—
199C2×D7—82no—
200C4×C7—21no—
201C4×C7—82no—
202C4×C7—82no—
203C4×C7#53711yescharacteristic
204C4×C7#57511yescharacteristic
205Dic7—22no—
206Dic7—22no—
207Dic7—82no—
208C2×Q8—22no—
209C2×Q8—282no—
210C2×D4—142no—
211C2×D4—282no—
212C22×C4—282no—
213C22×C4D4×D711yescharacteristic
214Q16—22no—
215Q16—22no—
216SD16—284no—
217D8—142no—
218D8—142no—
219M16—281no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—72no—
223C2×C8D4×D711yescharacteristic
224C2×C8C23×D711yescharacteristic
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—71no—
228C4⋊C4—71no—
229C4⋊C4—142no—
230C4⋊C4—282no—
231C4⋊C4—282no—
232C22⋊C4—42no—
233C22⋊C4—72no—
234C22⋊C4—282no—
235C22⋊C4—282no—
236C22⋊C4—282no—
237C22⋊C4—282no—
238C42—72no—
239C42—282no—
240C16—21no—
241C16—21no—
242C16—142no—
243C2×C7—41no—
244C2×C7#324511yescharacteristic
245C2×C7#341511yescharacteristic
246C2×C7#341711yescharacteristic
247D7—162no—
248C23—282no—
249C23D7×D811yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—1122no—
253D4—282no—
254D4—282no—
255D4—1122no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—72no—
261C2×C4—72no—
262C2×C4—282no—
263C2×C4—1122no—
264C2×C4D7×D811yescharacteristic
265C2×C4C2×D4×D711yescharacteristic
266C8—21no—
267C8—142no—
268C8#654811yescharacteristic
269C8#655411yescharacteristic
270C7#3382811yescharacteristic, complemented, Sylow
271C22—21no—
272C22—21no—
273C22—21no—
274C22—562no—
275C22—562no—
276C22—562no—
277C22C2×D7×D811yescharacteristic
278C4—21no—
279C4—82no—
280C4—82no—
281C4—142no—
282C4—142no—
283C4—562no—
284C4#9028811yescharacteristic
285C4#9045911yescharacteristic
286C2—41no—
287C2—1122no—
288C2#1173814711yescharacteristic
289C2#1173947311yescharacteristic
290C2#1173947611yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2216C2
socleC22×C7C22×D8
prime radical#11729167C2
hypercenter#2216D7
hypocenterC7#33828
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
1. #11738187C2semidirect
2. C7×#2216C2semidirect
3. #11729252C2semidirect
5. #11729155C2semidirect
6. #11728838C2semidirect
7. #11727087C2semidirect
8. #11726989C2semidirect
9. #11726965C2semidirect
73. #2216D7semidirect
270. C7#33828semidirect

Maximal subgroups

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

subgroupindex
#117381872
C7×#22162
#117292522
#117291672
#117291552
#117288382
#117270872
#117269892
#117269652
#338287