Finite Group Databasebeta

← #13255132

Subgroups — #13255132

Order 1792.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2822C211yesmaximal, characteristic
2C2×#89837C212yesmaximal
3C2×#89338C212yesmaximal
4#11727306C211yesmaximal, characteristic
5#11725436C211yesmaximal, characteristic
6C2×C7×#125C2212yes—
7C2×#6432C2211yescharacteristic
8C7×#400—22no—
9C7×#398—22no—
10C7×#398—22no—
11C7×#382C2211yescharacteristic
12#89837—24no—
13#89837—24no—
14C2×#6384—22no—
15C2×#6382—22no—
16C2×#6381—22no—
17C2×#6377C2211yescharacteristic
18C2×#6377C2212yes—
19#89338—24no—
20#10087—71nomaximal, Sylow
21C22×Dic14—21no—
22C7×#125—22no—
23C7×#125—22no—
24C7×#125—22no—
25C7×#125—22no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4D412yes—
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7—22no—
34C22×C7⋊C8—22no—
35C22×C7⋊C8—21no—
36#6432—22no—
37#6432—21no—
38#6432D412yes—
39C7×#95—24no—
40C7×#95—22no—
41C7×#95—22no—
42#6384—44no—
43#6382—28no—
44#6381—28no—
45#6377—22no—
46#6377—22no—
47#6377D414yes—
48C2×#501—72no—
49#2822D711yescharacteristic
50#1960—71no—
51C2×#330—72no—
52#1295—71no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7C2×D412yes—
57C22×C4×C7C2×D411yescharacteristic
58C22×Dic7—41no—
59C2×Dic14—22no—
60C2×Dic14—42no—
61C2×Dic14—42no—
62C2×Dic14—41no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—82no—
73C42×C7C2×D411yescharacteristic
74C42×C7C2×D412yes—
75C42×C7#4111yescharacteristic
76C4⋊Dic7—24no—
77C4⋊Dic7—42no—
78C2×C7⋊C8—28no—
79C2×C7⋊C8—24no—
80C2×C7⋊C8—22no—
81C2×C7⋊C8—41no—
82C2×C7⋊C8—41no—
83C2×C4⋊Q8—71no—
84C2×#125C2×D712yes—
85#501—144no—
86#501—144no—
87C2×#107—142no—
88C2×#106—142no—
89C2×#103—142no—
90C2×#97—72no—
91C2×#97—71no—
92#400—22no—
93#398—22no—
94#398—22no—
95#382C2×D711yescharacteristic
96#330—144no—
97C23×C7#12011yescharacteristic
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—24no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—44no—
107C2×C4×C7—44no—
108C2×C4×C7#12012yes—
109C2×C4×C7#12111yescharacteristic
110C2×C4×C7#12112yes—
111C2×C4×C7#12711yescharacteristic
112C2×Dic7—42no—
113C2×Dic7—81no—
114C2×Dic7—81no—
115Dic14—44no—
116Dic14—82no—
117C7⋊C8—28no—
118C7⋊C8—42no—
119C22×Q8—141no—
120C22×C8—142no—
121C22×C8—141no—
122C4⋊Q8—72no—
123C4⋊Q8—142no—
124C4⋊Q8—141no—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—142no—
133C2×C4⋊C4C22⋊D712yes—
134C2×C4⋊C4C22⋊D712yes—
135C2×C4⋊C4C22⋊D712yes—
136C2×C42C22×D711yescharacteristic
137#107—148no—
138#106—148no—
139#103—284no—
140#97—74no—
141#97—142no—
142#97—142no—
143#95—24no—
144#95—22no—
145#95—22no—
146C22×C7#39211yescharacteristic
147C22×C7#45311yescharacteristic
148C22×C7#45411yescharacteristic
149C22×C7#45614yes—
150C4×C7—22no—
151C4×C7—22no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—82no—
155C4×C7—82no—
156C4×C7—82no—
157Dic7—82no—
158C2×Q8—142no—
159C2×Q8—282no—
160C2×Q8—282no—
161C2×Q8—281no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—281no—
166C22×C4C2×C22⋊D712yes—
167C22×C4C2×C22⋊D711yescharacteristic
168C2×C8—148no—
169C2×C8—144no—
170C2×C8—142no—
171C2×C8—281no—
172C2×C8—281no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—144no—
184C4⋊C4—282no—
185C42D4×D711yescharacteristic
186C42D4×D712yes—
187C42#115911yescharacteristic
188C2×C7#166012yes—
189C2×C7#166312yes—
190C2×C7#200512yes—
191C2×C7#200611yescharacteristic
192C23#651611yescharacteristic
193Q8—284no—
194Q8—562no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—282no—
206C2×C4—561no—
207C2×C4—561no—
208C2×C4#650012yes—
209C2×C4#650111yescharacteristic
210C2×C4#650212yes—
211C2×C4#650311yescharacteristic
212C8—148no—
213C8—282no—
214C7#1008711yescharacteristic, complemented, Sylow
215C22#8999511yescharacteristic
216C22#9001311yescharacteristic
217C22#9001711yescharacteristic
218C22#9001814yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—562no—
227C2#1172962212yes—
228C2#1172963311yescharacteristic
229C2#1172971012yes—
230C2#1172971512yes—
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2822C2
socle, prime radicalC23×C7#120
hypercenter#2822D7
hypocenterC7#10087
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
214. C7#10087semidirect

Maximal subgroups

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

subgroupindex
C7×#28222
C2×#898372
C2×#893382
#117273062
#117254362
#100877