Finite Group Databasebeta

← #13267058

Subgroups — #13267058

Order 1792.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#11736471C211yesmaximal, characteristic
2C7×#2086C211yesmaximal, characteristic, complemented
3#11728353C212yesmaximal, complemented
4#11725515C211yesmaximal, characteristic, complemented
5#11725291C212yesmaximal, complemented
6#90168—21no—
7C2×C7×#107C2211yescharacteristic
8C2×C7×C4⋊C8C2211yescharacteristic
9C7×#389—21no—
10#89887—21no—
11#89887C2212yes—
12#89618—22no—
13#89615—22no—
14C7×#340—21no—
15C7×#340C2211yescharacteristic
16C22⋊(C7⋊C16)C2212yes—
17#89286—22no—
18#89284—22no—
19#12288—71nomaximal, Sylow
20C2×#1157—41no—
21C22×C7×C8D411yescharacteristic
22C22×C7×C8C2311yescharacteristic
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—41no—
26C2×C42×C7D411yescharacteristic
27#6493—22no—
28#6493—41no—
29C2×D56—21no—
30C2×D56D412yes—
31#6438—41no—
32#6437—21no—
33C4×D28—22no—
34C4×D28—41no—
35#6432—21no—
36C7×#107—22no—
37C7×C4⋊C8—22no—
38C7×#95—41no—
39#6395—22no—
40#6395—22no—
41#6392—21no—
42#6392D412yes—
43C2×C7⋊C16—22no—
44#3245—71no—
45#2232—72no—
46#2086D711yescharacteristic, complemented
47#1347—72no—
48#1322—71no—
49C22×C4×C7—41no—
50C22×C4×C7—41no—
51C22×C4×C7D811yescharacteristic
52C22×C4×C7D811yescharacteristic
53C22×C4×C7C2×D411yescharacteristic
54C2×C22⋊D7—42no—
55#1157—82no—
56#1157—82no—
57C2×D28—22no—
58C2×D28—41no—
59C2×D28—41no—
60C2×C4×D7—42no—
61C2×Dic14—41no—
62C2×C7×C8—21no—
63C2×C7×C8—41no—
64C2×C7×C8—41no—
65C2×C7×C8C2×D412yes—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—82no—
71C42×C7—21no—
72C42×C7—21no—
73(C2×D7)⋊C4—42no—
74(C2×D7)⋊C4—42no—
75C4⋊Dic7—22no—
76C4⋊Dic7—41no—
77D56—42no—
78D56—81no—
79C7⋊C16—42no—
80#532—141no—
81#474—142no—
82#466—72no—
83#466—141no—
84#459—142no—
85C2×#107C2×D711yescharacteristic
86C2×C4⋊C8C2×D711yescharacteristic
87#389—21no—
88#366—142no—
89#357—142no—
90C22⋊C16—72no—
91#340—21no—
92#340C2×D711yescharacteristic
93C23×C7C2×D811yescharacteristic
94C22×D7—42no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—41no—
101C2×C4×C7—42no—
102C2×C4×C7—81no—
103C2×C4×C7—81no—
104C2×C4×C7—81no—
105C2×C4×C7—81no—
106C2×C4×C7—81no—
107C2×C4×C7D1612yes—
108C2×C4×C7#12011yescharacteristic
109C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
110C22⋊D7—82no—
111C22⋊D7—82no—
112C2×Dic7—42no—
113D28—42no—
114D28—42no—
115D28—82no—
116D28—82no—
117C4×D7—84no—
118Dic14—82no—
119C7×C8—22no—
120C7×C8—41no—
121C2×#41—281no—
122C2×D8—72no—
123C2×D8—141no—
124C22×C8C22⋊D711yescharacteristic
125C22×C8C22×D711yescharacteristic
126C4⋊Q8—141no—
127#127—141no—
128#124—281no—
129#121—142no—
130#121—281no—
131C4×D4—142no—
132C4×D4—281no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C42C22⋊D711yescharacteristic
137C2×C16—142no—
138#107—22no—
139#107—72no—
140#107—141no—
141C4⋊C8—22no—
142#102—142no—
143#102—142no—
144#95—41no—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7#44711yescharacteristic
148C22×C7C2×D1611yescharacteristic
149C22×C7#50911yescharacteristic
150C2×D7—82no—
151C2×D7—82no—
152C2×D7—82no—
153C4×C7—21no—
154C4×C7—22no—
155C4×C7—41no—
156C4×C7—82no—
157C4×C7—161no—
158C4×C7#45611yescharacteristic
159C4×C7#45811yescharacteristic
160Dic7—82no—
161#41—562no—
162#41—562no—
163C2×Q8—281no—
164C2×D4—142no—
165C2×D4—281no—
166C2×D4—281no—
167C2×D4—282no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—282no—
171C22×C4#114111yescharacteristic
172C22×C4#114111yescharacteristic
173C22×C4C2×C22⋊D711yescharacteristic
174D8—282no—
175D8—561no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8D4×D712yes—
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—82no—
185C4⋊C4—142no—
186C4⋊C4—281no—
187C22⋊C4—282no—
188C22⋊C4—282no—
189C42—21no—
190C42—21no—
191C16—282no—
192C2×C7—21no—
193C2×C7—21no—
194C2×C7#161011yescharacteristic
195C2×C7#218411yescharacteristic
196C2×C7#219111yescharacteristic
197D7—162no—
198C23—282no—
199C23C2×#114111yescharacteristic
200Q8—562no—
201D4—282no—
202D4—282no—
203D4—562no—
204D4—562no—
205D4—562no—
206D4—562no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—282no—
220C2×C4—564no—
221C2×C4#640012yes—
222C2×C4#650011yescharacteristic
223C2×C4#651211yescharacteristic
224C8—22no—
225C8—41no—
226C7#1228811yescharacteristic, complemented, Sylow
227C22—21no—
228C22—21no—
229C22—562no—
230C22—562no—
231C22—562no—
232C22#8980811yescharacteristic
233C22C2×#640011yescharacteristic
234C22#8996511yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—562no—
241C4#8982511yescharacteristic
242C4#8986711yescharacteristic
243C2—21no—
244C2—21no—
245C2—1122no—
246C2#1172746611yescharacteristic
247C2#1172882911yescharacteristic
248C2#1172884511yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#2086C2
socleC22×C7#447
prime radical#11736471C2
hypercenter#2086D7
hypocenterC7#12288
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
2. C7×#2086C2semidirect
3. #11728353C2semidirect
4. #11725515C2semidirect
5. #11725291C2semidirect
46. #2086D7semidirect
226. C7#12288semidirect

Maximal subgroups

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

subgroupindex
#117364712
C7×#20862
#117283532
#117255152
#117252912
#122887