Finite Group Databasebeta

← #13251640

Subgroups — #13251640

Order 1792.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2982C211yesmaximal, characteristic
2#11739926C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic
4#11728524C212yesmaximal
5#11725544C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×Dic14C2211yescharacteristic
8C7×C8×Q8—24no—
9C7×#431—22no—
10C7×#431C2212yes—
11#90002—21no—
12C2×#6410—21no—
13#89758C412yes—
14#89751C412yes—
15#89745—21no—
16#89708C2211yescharacteristic
17#89705—22no—
18#89696C2212yes—
19#89325—21no—
20#89251—24no—
21#9484—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×Dic14C2×C411yescharacteristic
24C4×C7×Q8—22no—
25C4×C7×Q8—22no—
26C2×C7×C4⋊C4C2×C412yes—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C42×C7D412yes—
29C2×C42×C7C2311yescharacteristic
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7C2×C411yescharacteristic
32C2×Dic7⋊C4—22no—
33C2×Dic7⋊C4—21no—
34C2×C4×Dic7—21no—
35C2×#1136—22no—
36C4×Dic14—22no—
37C4×Dic14—22no—
38C7×C4⋊C8—24no—
39C7×C4⋊C8—22no—
40C7×C22⋊C8—22no—
41C4×C7×C8—24no—
42C4×C7×C8D414yes—
43#6410—42no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C7⋊#95—22no—
47C7⋊#95—21no—
48C7⋊#95C2×C412yes—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51#2982D711yescharacteristic
52#2581—71no—
53#1801—71no—
54#1487—72no—
55#1286—72no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8SD1612yes—
59C2×C7×Q8Q1612yes—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×Dic7—21no—
66C22×Dic7—22no—
67C22×Dic7—21no—
68C2×Dic14—22no—
69C2×Dic14—42no—
70C2×C7×C8—24no—
71C7×C4⋊C4—24no—
72C7×C4⋊C4—22no—
73C42×C7—22no—
74C42×C7—22no—
75C42×C7C2×D412yes—
76C42×C7#4112yes—
77C4⋊Dic7—22no—
78C4⋊Dic7—42no—
79Dic7⋊C4—44no—
80Dic7⋊C4—41no—
81Dic7⋊C4—41no—
82C4×Dic7—42no—
83#1136—42no—
84#1136—42no—
85C2×C7⋊C8—42no—
86C2×C4×Q8—71no—
87C2×C4×Q8C2×D711yescharacteristic
88C8×Q8—24no—
89#431—22no—
90#431C2×D712yes—
91#423—72no—
92C2×#101—141no—
93#400—72no—
94#395—72no—
95#391—141no—
96#389—141no—
97#387—142no—
98#382—71no—
99#343—141no—
100Q8⋊C8—144no—
101C23×C7C2×C22⋊C411yescharacteristic
102C7×Q8—24no—
103C7×Q8—24no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7#10314yes—
111C2×C4×C7C4×D412yes—
112C2×C4×C7#12011yescharacteristic
113C2×C4×C7#12112yes—
114C2×C4×C7#12311yescharacteristic
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7C2×Q1611yescharacteristic
117C2×C4×C7#13512yes—
118C2×Dic7—22no—
119C2×Dic7—41no—
120C2×Dic7—41no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124C2×Dic7—41no—
125C2×Dic7—42no—
126Dic14—24no—
127Dic14—42no—
128C7×C8—44no—
129C7⋊C8—82no—
130C22×Q8—71no—
131C22×Q8D2811yescharacteristic
132C2×M16—142no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—142no—
136C4×Q8—142no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—142no—
139C2×C4⋊C4—141no—
140C2×C4⋊C4—141no—
141C2×C4⋊C4C4×D712yes—
142C2×C4⋊C4D2811yescharacteristic
143C2×C42—141no—
144C2×C42C22⋊D712yes—
145C2×C42C22×D711yescharacteristic
146C4⋊C8—24no—
147C4⋊C8—22no—
148C4⋊C8—142no—
149C4⋊C8—142no—
150#101—282no—
151C22⋊C8—22no—
152C4×C8—24no—
153C4×C8C22⋊D714yes—
154#95—72no—
155#95—142no—
156#95—142no—
157#95—142no—
158#95—141no—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7#38611yescharacteristic
162C22×C7C2×#10311yescharacteristic
163C22×C7#41611yescharacteristic
164C4×C7—22no—
165C4×C7—24no—
166C4×C7—21no—
167C4×C7—21no—
168C4×C7—42no—
169C4×C7#46512yes—
170C4×C7#46712yes—
171Dic7—42no—
172Dic7—82no—
173Dic7—81no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—142no—
177C2×Q8—282no—
178C2×Q8#113212yes—
179C2×Q8Dic2812yes—
180C22×C4—141no—
181C22×C4—142no—
182C22×C4—141no—
183C22×C4(C2×D7)⋊C412yes—
184C22×C4(C2×D7)⋊C412yes—
185C22×C4C2×C4×D711yescharacteristic
186C22×C4C2×D2811yescharacteristic
187C22×C4C2×C22⋊D711yescharacteristic
188M16—282no—
189M16—282no—
190C2×C8—24no—
191C2×C8—282no—
192C4⋊C4—24no—
193C4⋊C4—22no—
194C4⋊C4—142no—
195C4⋊C4—284no—
196C4⋊C4—282no—
197C4⋊C4—281no—
198C4⋊C4—281no—
199C42—22no—
200C42—22no—
201C42—282no—
202C42#115712yes—
203C42C2×C22⋊D712yes—
204C2×C7—21no—
205C2×C7—21no—
206C2×C7#189111yescharacteristic
207C2×C7#189211yescharacteristic
208C2×C7#191211yescharacteristic
209C23C2×(C2×D7)⋊C411yescharacteristic
210Q8—24no—
211Q8—24no—
212Q8—144no—
213Q8—282no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—142no—
221C2×C4—281no—
222C2×C4—281no—
223C2×C4—282no—
224C2×C4—282no—
225C2×C4—282no—
226C2×C4—282no—
227C2×C4—281no—
228C2×C4#639014yes—
229C2×C4C2×#113211yescharacteristic
230C2×C4#646712yes—
231C2×C4C2×Dic2811yescharacteristic
232C2×C4C4×C22⋊D712yes—
233C2×C4#649211yescharacteristic
234C2×C4#649312yes—
235C2×C4#651611yescharacteristic
236C8—44no—
237C8—562no—
238C7#948411yescharacteristic, complemented, Sylow
239C22—21no—
240C22—21no—
241C22C2×#639011yescharacteristic
242C22#8988511yescharacteristic
243C22#8998511yescharacteristic
244C4—21no—
245C4—21no—
246C4—24no—
247C4—22no—
248C4—42no—
249C4—282no—
250C4—562no—
251C4—561no—
252C4#8988612yes—
253C4#8988812yes—
254C2—21no—
255C2—21no—
256C2#1172941711yescharacteristic
257C2#1172972611yescharacteristic
258C2#1172978311yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2982C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2982D7
hypocenterC7#9484
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
238. C7#9484semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117399262
#117293912
#117285242
#117255442
#94847