Finite Group Databasebeta

← #13251204

Subgroups — #13251204

Order 1792.

486 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 115 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#11739920C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic
3C7×#1801C211yesmaximal, characteristic
4#11728260C212yesmaximal
5#11725521C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6507—21no—
8C2×#6432C2211yescharacteristic
9C7×#423C2212yes—
10C2×C7×#101—21no—
11#89796—24no—
12#89755C414yes—
13#89746—21no—
14C4×C4⋊Dic7C2211yescharacteristic
15#89700—22no—
16#89698—22no—
17#89697C2212yes—
18C7×#343—21no—
19#89277—24no—
20#9402—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×Dic14—21no—
23C2×C7×M16—22no—
24C4×C7×Q8—22no—
25C4×C7×Q8—22no—
26C2×C7×C4⋊C4C2×C412yes—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C42×C7D412yes—
29C2×C42×C7C2311yescharacteristic
30#6507—44no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7C2×C412yes—
34C2×Dic7⋊C4—24no—
35C2×C4×Dic7—21no—
36#6432—22no—
37#6432—22no—
38C7×C4⋊C8—22no—
39C7×C4⋊C8—22no—
40C7×#101—42no—
41C7⋊#95—24no—
42C7⋊#95C2×C412yes—
43C22⋊(C7⋊C8)—22no—
44C4⋊(C7⋊C8)—24no—
45C4⋊(C7⋊C8)D414yes—
46#6377—24no—
47C4×C7⋊C8—22no—
48#2983—71no—
49#2546—71no—
50#1801D711yescharacteristic
51#1546—72no—
52#1303—72no—
53C2×C7×Q8—22no—
54C2×C7×Q8—21no—
55C2×C7×Q8—22no—
56C2×C7×Q8—21no—
57C22×C4×C7C22⋊C412yes—
58C22×C4×C7C22⋊C412yes—
59C22×C4×C7C22×C411yescharacteristic
60C22×C4×C7C2×D411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—22no—
63C22×Dic7—22no—
64C2×Dic14—42no—
65C2×Dic14—42no—
66C2×Dic14—41no—
67C2×Dic14—41no—
68C7×M16—42no—
69C7×M16—42no—
70C2×C7×C8—42no—
71C7×C4⋊C4—24no—
72C7×C4⋊C4—22no—
73C42×C7—22no—
74C42×C7—22no—
75C42×C7C2×D412yes—
76C42×C7#4112yes—
77C4⋊Dic7—24no—
78C4⋊Dic7—24no—
79C4⋊Dic7—41no—
80C4⋊Dic7—41no—
81Dic7⋊C4—48no—
82C4×Dic7—42no—
83C2×C7⋊C8—24no—
84C2×C4⋊Q8—71no—
85C2×C4⋊Q8—141no—
86C2×C4×Q8C2×D711yescharacteristic
87#446—144no—
88#432—142no—
89#431—72no—
90#423C2×D712yes—
91C2×#101—21no—
92#398—74no—
93#391—141no—
94#389—142no—
95C4×C4⋊C4—71no—
96#343—21no—
97#332—144no—
98C23×C7C2×C22⋊C411yescharacteristic
99C7×Q8—42no—
100C7×Q8—42no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—41no—
107C2×C4×C7—42no—
108C2×C4×C7C4×D412yes—
109C2×C4×C7#12011yescharacteristic
110C2×C4×C7#12112yes—
111C2×C4×C7#12311yescharacteristic
112C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C7AGL(1,ℤ/8)12yes—
114C2×C4×C7#13711yescharacteristic
115C2×Dic7—24no—
116C2×Dic7—42no—
117C2×Dic7—42no—
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—42no—
121Dic14—82no—
122Dic14—82no—
123C7×C8—82no—
124C7⋊C8—44no—
125C22×Q8—141no—
126C22×Q8C22⋊D711yescharacteristic
127C2×M16—22no—
128C4⋊Q8—142no—
129C4⋊Q8—142no—
130C4⋊Q8—284no—
131C4×Q8—22no—
132C4×Q8—22no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4—144no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4C4×D712yes—
138C2×C4⋊C4C22⋊D711yescharacteristic
139C2×C42—141no—
140C2×C42D2812yes—
141C2×C42C22×D711yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—74no—
145C4⋊C8—144no—
146#101—42no—
147C22⋊C8—142no—
148#97—144no—
149C4×C8—142no—
150#95—72no—
151#95—144no—
152C22×C7—21no—
153C22×C7—21no—
154C22×C7#38611yescharacteristic
155C22×C7#41711yescharacteristic
156C22×C7#41811yescharacteristic
157C4×C7—21no—
158C4×C7—22no—
159C4×C7—42no—
160C4×C7—42no—
161C4×C7#45612yes—
162C4×C7#46812yes—
163C4×C7#46912yes—
164Dic7—44no—
165Dic7—82no—
166C2×Q8—21no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—281no—
171C2×Q8—282no—
172C2×Q8—281no—
173C2×Q8—282no—
174C22×C4—142no—
175C22×C4—142no—
176C22×C4(C2×D7)⋊C412yes—
177C22×C4(C2×D7)⋊C412yes—
178C22×C4C2×C4×D711yescharacteristic
179C22×C4C2×D2811yescharacteristic
180C22×C4C2×C22⋊D711yescharacteristic
181M16—42no—
182M16—42no—
183C2×C8—42no—
184C2×C8—144no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—144no—
188C4⋊C4—144no—
189C4⋊C4—288no—
190C4⋊C4—281no—
191C4⋊C4—281no—
192C42—22no—
193C42—22no—
194C42—282no—
195C42D4×D712yes—
196C42#115912yes—
197C2×C7—21no—
198C2×C7—21no—
199C2×C7#188611yescharacteristic
200C2×C7#189511yescharacteristic
201C2×C7#191511yescharacteristic
202C23C2×(C2×D7)⋊C411yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—562no—
206Q8—562no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—144no—
215C2×C4—282no—
216C2×C4—282no—
217C2×C4—282no—
218C2×C4—282no—
219C2×C4—282no—
220C2×C4(C22⋊D7)⋊C412yes—
221C2×C4#644511yescharacteristic
222C2×C4#644911yescharacteristic
223C2×C4#649511yescharacteristic
224C2×C4#650212yes—
225C2×C4#650511yescharacteristic
226C2×C4#651212yes—
227C8—82no—
228C8—284no—
229C7#940211yescharacteristic, complemented, Sylow
230C22—21no—
231C22—21no—
232C22#8973011yescharacteristic
233C22#8973811yescharacteristic
234C22#8977311yescharacteristic
235C4—22no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—284no—
240C4—562no—
241C4#8952412yes—
242C4#8981112yes—
243C4#8982112yes—
244C2—21no—
245C2—21no—
246C2#1172727111yescharacteristic
247C2#1172785711yescharacteristic
248C2#1172789211yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1801C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1801D7
hypocenterC7#9402
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
229. C7#9402semidirect

Maximal subgroups

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

subgroupindex
#117399202
#117398812
C7×#18012
#117282602
#117255212
#94027