Finite Group Databasebeta

← #13256088

Subgroups — #13256088

Order 1792.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic
2#11740768C212yesmaximal
3#11729592C212yesmaximal
4#11727576C211yesmaximal, characteristic
5#11725467C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×#6432C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#427—22no—
11C2×C7×#97C2211yescharacteristic
12C2×#6390—22no—
13#89853—22no—
14#89851—22no—
15#89815—24no—
16#89792—24no—
17#89708C2211yescharacteristic
18#89697C2212yes—
19#89333—24no—
20#10227—71nomaximal, Sylow
21C22×Dic14—21no—
22C22×C7×C8D412yes—
23C7×C4⋊Q8D412yes—
24C7×C4⋊Q8D412yes—
25C7×#125—22no—
26C7×C22⋊Q8—22no—
27C7×C22⋊Q8—22no—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C7×#117—22no—
31C2×C42×C7C2311yescharacteristic
32C2×C4⋊Dic7—22no—
33C2×C4⋊Dic7—21no—
34#6432—22no—
35#6432—21no—
36#6432D412yes—
37C7×#107—24no—
38C7×#106—22no—
39C7×#106—22no—
40C7×#97—22no—
41#6409—28no—
42C7⋊#95—22no—
43C22⋊(C7⋊C8)D414yes—
44#6390—44no—
45#6382—28no—
46C4⋊(C7⋊C8)—22no—
47C4×C7⋊C8—22no—
48#3229—72no—
49#3162D711yescharacteristic
50#1960—71no—
51#1518—72no—
52#1281—71no—
53C2×C7×Q8—22no—
54C2×C7×Q8—22no—
55C22×C4×C7C2×D412yes—
56C22×C4×C7C2×D411yescharacteristic
57C22×Dic7—41no—
58C22×Dic7—41no—
59C2×Dic14—22no—
60C2×Dic14—42no—
61C2×Dic14—42no—
62C2×Dic14—41no—
63C2×C7×C8—22no—
64C2×C7×C8—22no—
65C2×C7×C8SD1614yes—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C22⋊C4—42no—
73C42×C7—42no—
74C42×C7C2×D411yescharacteristic
75C42×C7C2×D412yes—
76C42×C7#4111yescharacteristic
77C4⋊Dic7—24no—
78C4⋊Dic7—42no—
79C4⋊Dic7—42no—
80C2×C7⋊C8—24no—
81#533C2×D712yes—
82C2×C4⋊Q8—71no—
83#501—22no—
84#501—22no—
85#498—142no—
86#487—142no—
87#483—144no—
88#475—144no—
89#431—72no—
90#427—22no—
91C2×#103—142no—
92C2×#97C2×D711yescharacteristic
93#382—71no—
94#328—144no—
95C23×C7#12711yescharacteristic
96C7×Q8—44no—
97C7×Q8—82no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7#12012yes—
102C2×C4×C7#12011yescharacteristic
103C2×C4×C7#12111yescharacteristic
104C2×C4×C7#12112yes—
105C2×C4×C7C2×SD1612yes—
106C2×C4×C7#13712yes—
107C2×Dic7—42no—
108C2×Dic7—81no—
109C2×Dic7—81no—
110C2×Dic7—82no—
111C2×Dic7—81no—
112Dic14—44no—
113Dic14—82no—
114C7×C8—24no—
115C7⋊C8—44no—
116C22×Q8—141no—
117C22×C8C22⋊D712yes—
118C4⋊Q8—72no—
119C4⋊Q8—141no—
120C4⋊Q8—142no—
121C4⋊Q8C22⋊D712yes—
122C4⋊Q8C22⋊D712yes—
123#125—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—22no—
129C2×C4⋊C4—142no—
130C2×C4⋊C4—141no—
131C2×C42C22×D711yescharacteristic
132#107—24no—
133#106—22no—
134#106—22no—
135#106—148no—
136C4⋊C8—142no—
137#103—148no—
138#103—284no—
139C22⋊C8—74no—
140#97—22no—
141C4×C8—142no—
142#95—142no—
143C22×C7—22no—
144C22×C7#39211yescharacteristic
145C22×C7#49211yescharacteristic
146C22×C7#49711yescharacteristic
147C4×C7—22no—
148C4×C7—22no—
149C4×C7—82no—
150C4×C7—82no—
151C4×C7#44812yes—
152C4×C7#46112yes—
153Dic7—82no—
154Dic7—161no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—142no—
158C2×Q8—281no—
159C2×Q8—282no—
160C2×Q8—282no—
161C22×C4—281no—
162C22×C4—281no—
163C22×C4D4×D712yes—
164C22×C4C2×C22⋊D711yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—144no—
168C2×C8#114314yes—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—144no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C22⋊C4—42no—
179C42—42no—
180C42D4×D711yescharacteristic
181C42#115911yescharacteristic
182C42C2×C22⋊D712yes—
183C2×C7—21no—
184C2×C7—21no—
185C2×C7#200912yes—
186C2×C7#201311yescharacteristic
187C23#650311yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—284no—
191Q8—562no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—282no—
196C2×C4—561no—
197C2×C4—561no—
198C2×C4—562no—
199C2×C4—561no—
200C2×C4#650012yes—
201C2×C4#650111yescharacteristic
202C2×C4#650212yes—
203C2×C4C2×#114312yes—
204C2×C4#650512yes—
205C2×C4#651611yescharacteristic
206C8—24no—
207C8—284no—
208C7#1022711yescharacteristic, complemented, Sylow
209C22—22no—
210C22#8985711yescharacteristic
211C22#8985811yescharacteristic
212C22#8999511yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—562no—
218C4—1121no—
219C4#8985512yes—
220C4#8999912yes—
221C2—21no—
222C2—21no—
223C2#1172954511yescharacteristic
224C2#1172987412yes—
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3162C2
socleC22×C7#392
prime radicalC23×C7#127
hypercenter#3162D7
hypocenterC7#10227
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
208. C7#10227semidirect

Maximal subgroups

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

subgroupindex
C7×#31622
#117407682
#117295922
#117275762
#117254672
#102277