Finite Group Databasebeta

← #13256460

Subgroups — #13256460

Order 1792.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#11736714C212yesmaximal
2#11730753C211yesmaximal, characteristic, complemented
3#11728534C212yesmaximal, complemented
4#11727149C211yesmaximal, characteristic
5C7×#1276C211yesmaximal, characteristic, complemented
6#90168C2211yescharacteristic
7C2×#6432C2212yes—
8C7×#431C2212yes—
9C43×C7C2211yescharacteristic
10#89708—22no—
11#89701—24no—
12C2×#6379—22no—
13#89696C2211yescharacteristic
14#89614—24no—
15#89507—24no—
16#89466—22no—
17#89452—22no—
18#89277—24no—
19#10285—71nomaximal, Sylow
20C2×#1157—21no—
21C22×Dic14—22no—
22C2×C42×C7—22no—
23C2×C42×C7D412yes—
24C2×C42×C7C2311yescharacteristic
25#6493—22no—
26C2×C4⋊Dic7—22no—
27C2×C4⋊Dic7D414yes—
28C2×#1136—21no—
29#6438—21no—
30#6437D411yescharacteristic
31C4×D28—22no—
32#6432—22no—
33#6432—22no—
34#6432—22no—
35#6432—22no—
36#6432D411yescharacteristic
37C7×C4⋊C8D414yes—
38C7×C22⋊C8—22no—
39C4×C7×C8—22no—
40C7⋊#95—28no—
41#6395—28no—
42#6391—28no—
43#6379—44no—
44C4⋊(C7⋊C8)—22no—
45#3227—72no—
46#2862—71no—
47#1962—71no—
48#1549—72no—
49#1276D711yescharacteristic, complemented
50C22×C4×C7—22no—
51C22×C4×C7C2×D412yes—
52C22×C4×C7C2×D411yescharacteristic
53C2×C22⋊D7—41no—
54C22×Dic7—24no—
55#1157—42no—
56#1157—42no—
57C2×D28—22no—
58C2×D28—21no—
59C2×C4×D7—41no—
60C2×Dic14—21no—
61C2×Dic14—48no—
62C2×Dic14—42no—
63C2×Dic14—42no—
64C2×C7×C8—24no—
65C42×C7—22no—
66C42×C7—44no—
67C42×C7SD1614yes—
68C42×C7C2×D412yes—
69C42×C7C2×D411yescharacteristic
70C42×C7#4111yescharacteristic
71(C2×D7)⋊C4—42no—
72C4⋊Dic7—24no—
73C4⋊Dic7—22no—
74C4⋊Dic7—24no—
75C4⋊Dic7—42no—
76C4⋊Dic7—42no—
77#1136—42no—
78C2×C7⋊C8—41no—
79#532—71no—
80C2×C4⋊Q8—72no—
81#498—142no—
82#486—142no—
83#481—144no—
84#476—144no—
85#431C2×D712yes—
86#423—71no—
87C2×#104—142no—
88#391—144no—
89#382—142no—
90C43C2×D711yescharacteristic
91#332—144no—
92C23×C7#12011yescharacteristic
93C22×D7—41no—
94C2×C4×C7—24no—
95C2×C4×C7—24no—
96C2×C4×C7—44no—
97C2×C4×C7#12012yes—
98C2×C4×C7#12112yes—
99C2×C4×C7#12111yescharacteristic
100C2×C4×C7#12711yescharacteristic
101C2×C4×C7C2×SD1612yes—
102C2×C4×C7#13712yes—
103C22⋊D7—82no—
104C2×Dic7—48no—
105C2×Dic7—44no—
106C2×Dic7—41no—
107D28—44no—
108D28—42no—
109C4×D7—81no—
110C4×D7—81no—
111Dic14—42no—
112Dic14—84no—
113Dic14—84no—
114C7×C8—44no—
115C7⋊C8—81no—
116C2×#41—141no—
117C22×Q8—142no—
118C2×M16—141no—
119C4⋊Q8—71no—
120C4⋊Q8—142no—
121C4⋊Q8—142no—
122C4⋊Q8—142no—
123C4⋊Q8—142no—
124#127—71no—
125#124—141no—
126#121—142no—
127C4×D4—142no—
128C2×C4⋊C4—74no—
129C2×C4⋊C4—142no—
130C2×C42—22no—
131C2×C42D2812yes—
132C2×C42C22×D711yescharacteristic
133#106—148no—
134C4⋊C8—142no—
135C4⋊C8D2814yes—
136#104—284no—
137#102—148no—
138C22⋊C8—22no—
139C4×C8—22no—
140#95—148no—
141C22×C7—22no—
142C22×C7#39211yescharacteristic
143C22×C7#44812yes—
144C2×D7—82no—
145C2×D7—81no—
146C4×C7—22no—
147C4×C7—44no—
148C4×C7#45414yes—
149C4×C7#46112yes—
150C4×C7#49211yescharacteristic
151C4×C7#49711yescharacteristic
152Dic7—84no—
153Dic7—81no—
154#41—282no—
155#41—282no—
156C2×Q8—141no—
157C2×Q8—288no—
158C2×Q8—282no—
159C2×Q8—282no—
160C2×D4—142no—
161C2×D4—141no—
162C2×D4—281no—
163C22×C4—22no—
164C22×C4—144no—
165C22×C4—281no—
166C22×C4C2×D2811yescharacteristic
167C22×C4D4×D712yes—
168M16—282no—
169C2×C8—24no—
170C2×C8—281no—
171C4⋊C4—144no—
172C4⋊C4—142no—
173C4⋊C4—144no—
174C4⋊C4—282no—
175C4⋊C4—282no—
176C22⋊C4—282no—
177C42—22no—
178C42—44no—
179C42#113214yes—
180C42C2×D2812yes—
181C42D4×D711yescharacteristic
182C42#116111yescharacteristic
183C2×C7—22no—
184C2×C7#200312yes—
185C2×C7#201311yescharacteristic
186D7—161no—
187C23—281no—
188C23#644511yescharacteristic
189Q8—282no—
190Q8—564no—
191Q8—564no—
192D4—282no—
193D4—284no—
194D4—562no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—288no—
199C2×C4—281no—
200C2×C4—284no—
201C2×C4—561no—
202C2×C4—561no—
203C2×C4#643711yescharacteristic
204C2×C4#644512yes—
205C2×C4#645812yes—
206C2×C4#645811yescharacteristic
207C2×C4C2×#113212yes—
208C2×C4#647212yes—
209C8—44no—
210C8—561no—
211C7#1028511yescharacteristic, complemented, Sylow
212C22—22no—
213C22—561no—
214C22—562no—
215C22#8944311yescharacteristic
216C22#8951012yes—
217C4—22no—
218C4—44no—
219C4—561no—
220C4—564no—
221C4#8946511yescharacteristic
222C4#8948711yescharacteristic
223C4#8959914yes—
224C4#8961312yes—
225C2—22no—
226C2—1121no—
227C2#1172644111yescharacteristic
228C2#1172652112yes—
subgroupG/H
centerC22#89443
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1276C2
socleC22×C7#392
prime radical#90168C22
hypercenter#1276D7
hypocenterC7#10285
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #11730753C2semidirect
3. #11728534C2semidirect
5. C7×#1276C2semidirect
49. #1276D7semidirect
211. C7#10285semidirect

Maximal subgroups

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

subgroupindex
#117367142
#117307532
#117285342
#117271492
C7×#12762
#102857