Finite Group Databasebeta

← #13247379

Subgroups — #13247379

Order 1792.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#11739908C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3#11729378C211yesmaximal, characteristic
4#11725521C212yesmaximal
5#11725519C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C7×#423C2212yes—
8C2×C7×#101—21no—
9#90002—21no—
10#89753—22no—
11C4⋊C4×Dic7—21no—
12#89709C412yes—
13#89708C2211yescharacteristic
14C4×C4⋊Dic7C2211yescharacteristic
15#89704C412yes—
16#89697C2212yes—
17C7×#343—21no—
18#89323—22no—
19#8768—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C2×C7×M16—22no—
22C4×C7×Q8—22no—
23C4×C7×Q8—22no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C42×C7C2×C412yes—
27C2×C42×C7C2311yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7—21no—
30C2×C4×Dic7—22no—
31C2×C4×Dic7—21no—
32C7×C4⋊C8—22no—
33C7×C4⋊C8—22no—
34C7×#101—42no—
35C7⋊#95—22no—
36C7⋊#95—22no—
37C7⋊#95—22no—
38C7⋊#95—21no—
39C7⋊#95C2×C411yescharacteristic
40C7⋊#95C2×C412yes—
41C7⋊#95C2×C411yescharacteristic
42C22⋊(C7⋊C8)—22no—
43C4⋊(C7⋊C8)—22no—
44C4×C7⋊C8—22no—
45#2356—71no—
46#1801D711yescharacteristic
47#1461—71no—
48#1303—72no—
49#1286—72no—
50C2×C7×Q8—22no—
51C2×C7×Q8—22no—
52C2×C7×Q8—21no—
53C2×C7×Q8—21no—
54C22×C4×C7C22⋊C412yes—
55C22×C4×C7C22⋊C412yes—
56C22×C4×C7C22×C411yescharacteristic
57C22×C4×C7C2×D411yescharacteristic
58C22×C4×C7C2×D411yescharacteristic
59C22×Dic7—21no—
60C22×Dic7—22no—
61C22×Dic7—21no—
62C7×M16—42no—
63C7×M16—42no—
64C2×C7×C8—42no—
65C7×C4⋊C4—24no—
66C7×C4⋊C4—22no—
67C42×C7—22no—
68C42×C7—22no—
69C42×C7#4112yes—
70C42×C7#4112yes—
71C4⋊Dic7—42no—
72C4⋊Dic7—42no—
73C4×Dic7—44no—
74C4×Dic7—41no—
75C4×Dic7—41no—
76C2×C7⋊C8—42no—
77C2×C4×Q8C2×D711yescharacteristic
78#431—72no—
79#423C2×D712yes—
80C2×#101—21no—
81#391—141no—
82#387—142no—
83#383—72no—
84#382—71no—
85C4×C4⋊C4—71no—
86C4×C4⋊C4—141no—
87#376—72no—
88#343—21no—
89#324—142no—
90C23×C7C2×C22⋊C411yescharacteristic
91C7×Q8—42no—
92C7×Q8—42no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—41no—
98C2×C4×C7—41no—
99C2×C4×C7—42no—
100C2×C4×C7#11711yescharacteristic
101C2×C4×C7#11711yescharacteristic
102C2×C4×C7#12312yes—
103C2×C4×C7#12312yes—
104C2×C4×C7#13512yes—
105C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
106C2×C4×C7#13711yescharacteristic
107C2×Dic7—22no—
108C2×Dic7—41no—
109C2×Dic7—41no—
110C2×Dic7—42no—
111C2×Dic7—41no—
112C2×Dic7—41no—
113C2×Dic7—42no—
114C2×Dic7—42no—
115C2×Dic7—41no—
116C7×C8—82no—
117C7⋊C8—82no—
118C22×Q8C22⋊D711yescharacteristic
119C2×M16—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C2×C4⋊C4—141no—
123C2×C4⋊C4—141no—
124C2×C4⋊C4D2812yes—
125C2×C4⋊C4C22⋊D711yescharacteristic
126C2×C42—142no—
127C2×C42—141no—
128C2×C42C4×D712yes—
129C2×C42C22×D711yescharacteristic
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—142no—
133#101—42no—
134C22⋊C8—142no—
135C4×C8—142no—
136#95—71no—
137#95—72no—
138#95—71no—
139#95—142no—
140#95—142no—
141#95—142no—
142#95—141no—
143C22×C7—21no—
144C22×C7—21no—
145C22×C7#38111yescharacteristic
146C22×C7#41611yescharacteristic
147C22×C7#41711yescharacteristic
148C4×C7—22no—
149C4×C7—21no—
150C4×C7—21no—
151C4×C7—42no—
152C4×C7—42no—
153C4×C7#48212yes—
154C4×C7#48512yes—
155Dic7—42no—
156Dic7—81no—
157Dic7—82no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×Q8—21no—
161C2×Q8—22no—
162C22×C4—141no—
163C22×C4—142no—
164C22×C4—141no—
165C22×C4(C2×D7)⋊C412yes—
166C22×C4(C2×D7)⋊C412yes—
167C22×C4C2×C4×D711yescharacteristic
168C22×C4C2×D2811yescharacteristic
169C22×C4C2×C22⋊D711yescharacteristic
170M16—42no—
171M16—42no—
172C2×C8—42no—
173C2×C8—282no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C42—22no—
179C42—22no—
180C42—284no—
181C42—281no—
182C42—281no—
183C42#115912yes—
184C42#115912yes—
185C2×C7—21no—
186C2×C7—21no—
187C2×C7#182511yescharacteristic
188C2×C7#182711yescharacteristic
189C2×C7#183011yescharacteristic
190C23C2×(C2×D7)⋊C411yescharacteristic
191Q8—42no—
192Q8—42no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—142no—
201C2×C4—281no—
202C2×C4—281no—
203C2×C4—282no—
204C2×C4—281no—
205C2×C4—281no—
206C2×C4—282no—
207C2×C4—281no—
208C2×C4—282no—
209C2×C4#644011yescharacteristic
210C2×C4#644011yescharacteristic
211C2×C4#644912yes—
212C2×C4#649511yescharacteristic
213C2×C4#649812yes—
214C2×C4#650511yescharacteristic
215C2×C4#651312yes—
216C8—82no—
217C8—562no—
218C7#876811yescharacteristic, complemented, Sylow
219C22—21no—
220C22—21no—
221C22#8971511yescharacteristic
222C22#8973811yescharacteristic
223C22#8977411yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—282no—
230C4—561no—
231C4—562no—
232C4#8980712yes—
233C4#8981712yes—
234C2—21no—
235C2—21no—
236C2#1172725711yescharacteristic
237C2#1172785511yescharacteristic
238C2#1172788511yescharacteristic
subgroupG/H
centerC22#89715
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1801C2
socleC22×C7#381
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1801D7
hypocenterC7#8768
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
218. C7#8768semidirect

Maximal subgroups

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

subgroupindex
#117399082
C7×#18012
#117293782
#117255212
#117255192
#87687