Finite Group Databasebeta

← #13267492

Subgroups — #13267492

Order 1792.

312 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 69 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#11737019C211yesmaximal, characteristic, complemented
2C7×#2206C211yesmaximal, characteristic
3#11729042C211yesmaximal, characteristic, complemented
4#11729040C211yesmaximal, characteristic, complemented
5#11725569C211yesmaximal, characteristic, complemented
6#11725371C211yesmaximal, characteristic, complemented
7#11725364C211yesmaximal, characteristic, complemented
8#90234—21no—
9C2×C7×D16—21no—
10C7×#493C2211yescharacteristic
11C4×C7×D8C2211yescharacteristic
12D8×Dic7—21no—
13#89847—21no—
14#89780—21no—
15#89650—21no—
16#89538—22no—
17#89454C2211yescharacteristic
18C8×Dic14C2211yescharacteristic
19C7×C4⋊C16C2211yescharacteristic
20C7×#357—21no—
21#89357—22no—
22#89339—21no—
23C4×C7⋊C16C2211yescharacteristic
24#89253C2211yescharacteristic
25#12355—71nomaximal, Sylow
26C2×C7×D8—21no—
27C2×C7×D8—21no—
28C2×C7×D8D411yescharacteristic
29C7×#127—21no—
30C4×C7×D4—21no—
31D4×Dic7—22no—
32C2×#1142—41no—
33C2×C4⋊Dic7—41no—
34#6487—41no—
35#6453—22no—
36#6441—42no—
37#6432—21no—
38C4×Dic14D411yescharacteristic
39C7×D16—42no—
40C2×C7×C16—21no—
41C7×#107D411yescharacteristic
42C7×#102—21no—
43C4×C7×C8C2311yescharacteristic
44#6406—22no—
45#6406—41no—
46#6392—21no—
47#6392D412yes—
48Dic7⋊C8—21no—
49C8×Dic7—21no—
50C2×C7⋊C16—21no—
51#6382—22no—
52C4⋊(C7⋊C8)—21no—
53C4⋊(C7⋊C8)D411yescharacteristic
54#3380—71no—
55#2240—71no—
56#2206D711yescharacteristic
57#1668—71no—
58#1365—71no—
59Q8⋊C16—71no—
60#1331—71no—
61C2×C7×D4—21no—
62C2×C7×D4—41no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C22×C4×C7—41no—
66C22×Dic7—42no—
67C2×Dic14—41no—
68C2×Dic14D811yescharacteristic
69C7×D8—21no—
70C7×D8—42no—
71C7×D8—41no—
72C7×D8—41no—
73C7×D8SD1612yescomplemented
74C2×C7×C8C2×D411yescharacteristic
75C2×C7×C8C2×D411yescharacteristic
76C7×C4⋊C4—21no—
77C7×C22⋊C4—41no—
78C42×C7C2×D411yescharacteristic
79C22⋊Dic7—42no—
80#1142—82no—
81C4⋊Dic7—22no—
82C4⋊Dic7—42no—
83C4⋊Dic7—41no—
84C4⋊Dic7D811yescharacteristic
85Dic7⋊C4—21no—
86Dic7⋊C4—42no—
87C4×Dic7—21no—
88C2×C7⋊C8—21no—
89C2×C7⋊C8—41no—
90C7×C16—41no—
91C7⋊C16—41no—
92#554—141no—
93C2×D16—21no—
94#500—71no—
95#499—141no—
96#493C2×D711yescharacteristic
97#478—142no—
98#476—141no—
99#460—141no—
100C8×Q8—71no—
101C4×D8—141no—
102C4×D8C2×D711yescharacteristic
103C4⋊C16C2×D711yescharacteristic
104#357—21no—
105#357—142no—
106C4×C16—71no—
107#335—71no—
108#331—141no—
109C23×C7—41no—
110C23×C7—81no—
111C7×D4—22no—
112C7×D4—41no—
113C7×D4—81no—
114C7×D4—81no—
115C7×D4—81no—
116C7×D4—81no—
117C7×D4—161no—
118C2×C4×C7—41no—
119C2×C4×C7—81no—
120C2×C4×C7—81no—
121C2×C4×C7#12011yescharacteristic
122C2×C4×C7C2×D811yescharacteristic
123C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
124C2×Dic7—21no—
125C2×Dic7—21no—
126C2×Dic7—42no—
127C2×Dic7—162no—
128Dic14—21no—
129Dic14—21no—
130Dic14—161no—
131C7×C8—21no—
132C7×C8C2×SD1611yescharacteristic
133C7×C8#13711yescharacteristic
134C7⋊C8—41no—
135C7⋊C8—81no—
136C2×SD16—281no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8C22⋊D711yescharacteristic
140C4⋊Q8—141no—
141#127—21no—
142#125—142no—
143C22⋊Q8—282no—
144C22⋊Q8—281no—
145C4×Q8—71no—
146C4×D4—21no—
147C4×D4—142no—
148C2×C4⋊C4—281no—
149D16—42no—
150C2×C16—21no—
151C2×C16—141no—
152#107—72no—
153#107—141no—
154#107C22⋊D711yescharacteristic
155#106—142no—
156C4⋊C8—71no—
157C4⋊C8—141no—
158C4⋊C8—141no—
159#102—21no—
160#102—142no—
161#102—281no—
162C4×C8—141no—
163C4×C8C22×D711yescharacteristic
164C22×C7—42no—
165C22×C7—81no—
166C22×C7—81no—
167C22×C7—161no—
168C22×C7—161no—
169C22×C7—161no—
170C22×C7#44711yescharacteristic
171C4×C7—21no—
172C4×C7—81no—
173C4×C7#44811yescharacteristic
174C4×C7#45411yescharacteristic
175C4×C7#50811yescharacteristic
176C4×C7#50911yescharacteristic
177Dic7—41no—
178Dic7—41no—
179Dic7—82no—
180C2×Q8—71no—
181C2×Q8—281no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C22×C4—41no—
187C22×C4—282no—
188SD16—562no—
189D8—21no—
190D8—42no—
191D8—41no—
192D8—41no—
193D8#114212yescomplemented
194C2×C8—141no—
195C2×C8—281no—
196C2×C8D4×D711yescharacteristic
197C2×C8C2×C22⋊D711yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—71no—
200C4⋊C4—141no—
201C4⋊C4—142no—
202C4⋊C4—282no—
203C4⋊C4—282no—
204C4⋊C4—281no—
205C22⋊C4—41no—
206C22⋊C4—282no—
207C42—141no—
208C42D4×D711yescharacteristic
209C16—41no—
210C16—281no—
211C2×C7—82no—
212C2×C7—321no—
213C2×C7#162711yescharacteristic
214C2×C7#218611yescharacteristic
215C2×C7#219211yescharacteristic
216C23—41no—
217C23—81no—
218Q8—141no—
219Q8—141no—
220Q8—1121no—
221D4—22no—
222D4—41no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—81no—
227D4—161no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—141no—
232C2×C4—141no—
233C2×C4—282no—
234C2×C4—1122no—
235C2×C4D7×D811yescharacteristic
236C2×C4#647411yescharacteristic
237C2×C4#650011yescharacteristic
238C8—21no—
239C8—281no—
240C8—561no—
241C8C2×#114211yescharacteristic
242C8#651411yescharacteristic
243C7#1235511yescharacteristic, complemented, Sylow
244C22—42no—
245C22—81no—
246C22—81no—
247C22—161no—
248C22—161no—
249C22—161no—
250C22#8992811yescharacteristic
251C4—21no—
252C4—81no—
253C4—281no—
254C4—281no—
255C4—562no—
256C4#8968311yescharacteristic
257C4#8968411yescharacteristic
258C4#8981211yescharacteristic
259C4#8997311yescharacteristic
260C2—82no—
261C2—321no—
262C2#1172876811yescharacteristic
263C2#1172922211yescharacteristic
264C2#1172927111yescharacteristic
subgroupG/H
centerC22#89928
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
Fitting, prime radicalC7×#2206C2
socleC22×C7#447
hypercenter#2206D7
hypocenterC7#12355
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2206

Complements

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

NH ≅ G/Ntype
1. #11737019C2semidirect
3. #11729042C2semidirect
4. #11729040C2semidirect
5. #11725569C2semidirect
6. #11725371C2semidirect
7. #11725364C2semidirect
73. C7×D8SD16semidirect
193. D8#1142semidirect
243. C7#12355semidirect

Maximal subgroups

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

subgroupindex
#117370192
C7×#22062
#117290422
#117290402
#117255692
#117253712
#117253642
#123557