Finite Group Databasebeta

← #13256458

Subgroups — #13256458

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
1C7×#2862C211yesmaximal, characteristic, complemented
2#11740661C212yesmaximal
3#11729529C212yesmaximal, complemented
4#11727149C211yesmaximal, characteristic
5#11725417C211yesmaximal, characteristic, complemented
6C2×C7×C4⋊Q8C2212yes—
7#90168C2211yescharacteristic
8C7×#391—24no—
9C7×#382—22no—
10C43×C7C2211yescharacteristic
11#89860—22no—
12#89856—24no—
13#89851—22no—
14#89740—24no—
15C2×#6379—22no—
16#89697C2212yes—
17#89696C2211yescharacteristic
18#89342—24no—
19#10285—71nomaximal, Sylow
20C22×C7×Q8—22no—
21C2×#1157—21no—
22C7×C4⋊Q8—22no—
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—22no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4D414yes—
28C2×C42×C7—22no—
29C2×C42×C7D412yes—
30C2×C42×C7C2311yescharacteristic
31#6493—22no—
32C2×#1136—21no—
33#6438—21no—
34#6437D411yescharacteristic
35C4×D28—22no—
36#6432D411yescharacteristic
37C7×#95—28no—
38C22⋊(C7⋊C8)—22no—
39#6383—28no—
40#6382—28no—
41#6379—44no—
42C4⋊(C7⋊C8)—22no—
43C4⋊(C7⋊C8)D414yes—
44C4×C7⋊C8—22no—
45#3227—72no—
46#2862D711yescharacteristic, complemented
47#1962—71no—
48#1549—72no—
49#1276—71no—
50C2×C7×Q8—48no—
51C2×C7×Q8—42no—
52C2×C7×Q8—42no—
53C22×C4×C7—22no—
54C22×C4×C7—24no—
55C22×C4×C7C2×D412yes—
56C22×C4×C7C2×D411yescharacteristic
57C2×C22⋊D7—41no—
58#1157—42no—
59#1157—42no—
60C2×D28—22no—
61C2×D28—21no—
62C2×C4×D7—41no—
63C2×Dic14—21no—
64C7×C4⋊C4—24no—
65C7×C4⋊C4—24no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C42×C7—22no—
69C42×C7—44no—
70C42×C7SD1614yes—
71C42×C7C2×D411yescharacteristic
72C42×C7C2×D412yes—
73C42×C7#4111yescharacteristic
74(C2×D7)⋊C4—42no—
75C4⋊Dic7—22no—
76#1136—42no—
77C2×C7⋊C8—24no—
78C2×C7⋊C8—41no—
79#532—71no—
80C2×C4⋊Q8C2×D712yes—
81#498—142no—
82#486—142no—
83#481—144no—
84#476—144no—
85#431—72no—
86#423—71no—
87C2×#104—142no—
88#391—24no—
89#382—22no—
90C43C2×D711yescharacteristic
91#332—144no—
92C23×C7#12011yescharacteristic
93C22×D7—41no—
94C7×Q8—84no—
95C7×Q8—84no—
96C2×C4×C7—24no—
97C2×C4×C7—24no—
98C2×C4×C7—44no—
99C2×C4×C7—48no—
100C2×C4×C7—44no—
101C2×C4×C7#12012yes—
102C2×C4×C7#12111yescharacteristic
103C2×C4×C7#12112yes—
104C2×C4×C7#12711yescharacteristic
105C2×C4×C7C2×SD1612yes—
106C2×C4×C7#13712yes—
107C22⋊D7—82no—
108C2×Dic7—41no—
109D28—44no—
110D28—42no—
111C4×D7—81no—
112C4×D7—81no—
113Dic14—42no—
114C7⋊C8—44no—
115C7⋊C8—81no—
116C2×#41—141no—
117C22×Q8—22no—
118C2×M16—141no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—71no—
124#127—71no—
125#124—141no—
126#121—142no—
127C4×D4—142no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4C22⋊D714yes—
130C2×C42—22no—
131C2×C42C22⋊D712yes—
132C2×C42C22×D711yescharacteristic
133#106—148no—
134C4⋊C8—74no—
135C4⋊C8—142no—
136#104—284no—
137#102—148no—
138C22⋊C8—142no—
139C4×C8—142no—
140#95—28no—
141C22×C7—22no—
142C22×C7#39211yescharacteristic
143C22×C7#44812yes—
144C2×D7—82no—
145C2×D7—81no—
146C4×C7—22no—
147C4×C7—44no—
148C4×C7—84no—
149C4×C7#45414yes—
150C4×C7#46112yes—
151C4×C7#49211yescharacteristic
152C4×C7#49711yescharacteristic
153Dic7—81no—
154#41—282no—
155#41—282no—
156C2×Q8—48no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—141no—
160C2×D4—142no—
161C2×D4—141no—
162C2×D4—281no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—281no—
166C22×C4C2×C22⋊D712yes—
167C22×C4C2×C22⋊D711yescharacteristic
168M16—282no—
169C2×C8—144no—
170C2×C8—281no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—142no—
176C22⋊C4—282no—
177C42—22no—
178C42—44no—
179C42#114314yes—
180C42D4×D711yescharacteristic
181C42D4×D712yes—
182C42#115911yescharacteristic
183C2×C7—22no—
184C2×C7#200312yes—
185C2×C7#201311yescharacteristic
186D7—161no—
187C23—281no—
188C23#651611yescharacteristic
189Q8—84no—
190Q8—84no—
191Q8—282no—
192D4—284no—
193D4—282no—
194D4—562no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—48no—
199C2×C4—44no—
200C2×C4—281no—
201C2×C4—561no—
202C2×C4—561no—
203C2×C4#650012yes—
204C2×C4#650111yescharacteristic
205C2×C4#650212yes—
206C2×C4#650311yescharacteristic
207C2×C4C2×#114312yes—
208C2×C4#650512yes—
209C8—284no—
210C8—561no—
211C7#1028511yescharacteristic, complemented, Sylow
212C22—22no—
213C22—562no—
214C22—561no—
215C22#8999511yescharacteristic
216C22#8999912yes—
217C4—22no—
218C4—44no—
219C4—84no—
220C4—561no—
221C4#8983414yes—
222C4#8985512yes—
223C4#8985711yescharacteristic
224C4#8985811yescharacteristic
225C2—22no—
226C2—1121no—
227C2#1172948412yes—
228C2#1172954511yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2862C2
socleC22×C7#392
prime radical#90168C22
hypercenter#2862D7
hypocenterC7#10285
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C7×#2862C2semidirect
3. #11729529C2semidirect
5. #11725417C2semidirect
46. #2862D7semidirect
211. C7#10285semidirect

Maximal subgroups

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

subgroupindex
C7×#28622
#117406612
#117295292
#117271492
#117254172
#102857