Finite Group Databasebeta

← #13251670

Subgroups — #13251670

Order 1792.

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

normal subgroups only

This group has 522 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
1C7×#2983C211yesmaximal, characteristic, complemented
2#11739944C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic, complemented
4#11728516C212yesmaximal
5#11725544C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6509—21no—
8C2×C4×D28C2211yescharacteristic
9C7×#446—24no—
10C7×#432—22no—
11C7×#431C2212yes—
12C2×#6410—21no—
13#89763C412yes—
14#89759—21no—
15#89755C412yes—
16#89712—22no—
17#89708C2211yescharacteristic
18#89696C2212yes—
19#89325—21no—
20#89254—24no—
21#9490—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×D28C2×C411yescharacteristic
24C22×C4×D7—21no—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C2×C7×C4⋊C4C2×C412yes—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C42×C7D412yes—
30C2×C42×C7C2311yescharacteristic
31#6509—44no—
32C2×(C2×D7)⋊C4—22no—
33C2×(C2×D7)⋊C4—22no—
34C2×(C2×D7)⋊C4—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—22no—
38C2×#1136—22no—
39C4×D28—22no—
40C4×D28—22no—
41C7×C4⋊C8—24no—
42C7×C4⋊C8—22no—
43C7×C22⋊C8—22no—
44C7×#97—24no—
45C4×C7×C8D414yes—
46#6410—42no—
47C7⋊#95—22no—
48C7⋊#95—21no—
49C7⋊#95C2×C412yes—
50C4⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52#2983D711yescharacteristic, complemented
53#2536—71no—
54#1801—71no—
55#1486—72no—
56#1286—72no—
57C23×D7—21no—
58C2×C7×Q8—22no—
59C2×C7×Q8—21no—
60C2×C7×Q8—22no—
61C2×C7×Q8—21no—
62C22×C4×C7C22⋊C412yes—
63C22×C4×C7C22⋊C412yes—
64C22×C4×C7C22×C411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×Dic7—22no—
68C22×Dic7—21no—
69C2×D28—22no—
70C2×D28—42no—
71C2×C4×D7—42no—
72C2×C4×D7—42no—
73C2×C4×D7—41no—
74C2×C4×D7—41no—
75C2×C7×C8—24no—
76C7×C4⋊C4—24no—
77C7×C4⋊C4—22no—
78C42×C7—22no—
79C42×C7—22no—
80C42×C7C2×D412yes—
81C42×C7#4112yes—
82(C2×D7)⋊C4—44no—
83(C2×D7)⋊C4—44no—
84(C2×D7)⋊C4—41no—
85(C2×D7)⋊C4—41no—
86C4⋊Dic7—22no—
87C4⋊Dic7—42no—
88Dic7⋊C4—44no—
89#1136—42no—
90#1136—42no—
91C2×C7⋊C8—42no—
92C2×C22⋊Q8—141no—
93C2×C4×Q8C2×D711yescharacteristic
94C2×C4×D4—71no—
95#446—24no—
96#432—22no—
97#431C2×D712yes—
98#423—72no—
99C2×#101—141no—
100#398—72no—
101#394—72no—
102#388—142no—
103#382—71no—
104#380—141no—
105#343—141no—
106D4⋊C8—144no—
107C23×C7C2×C22⋊C411yescharacteristic
108C22×D7—22no—
109C22×D7—41no—
110C22×D7—41no—
111C22×D7—41no—
112C22×D7—41no—
113C22×D7—41no—
114C22×D7—41no—
115C7×Q8—42no—
116C7×Q8—42no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7C4×D412yes—
125C2×C4×C7#12011yescharacteristic
126C2×C4×C7#12112yes—
127C2×C4×C7#12311yescharacteristic
128C2×C4×C7AGL(1,ℤ/8)12yes—
129C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C7#13711yescharacteristic
131C2×Dic7—42no—
132C2×Dic7—42no—
133C2×Dic7—41no—
134C2×Dic7—41no—
135C2×Dic7—41no—
136C2×Dic7—42no—
137D28—24no—
138D28—42no—
139C4×D7—82no—
140C4×D7—82no—
141C7×C8—44no—
142C7⋊C8—82no—
143C22×Q8D2811yescharacteristic
144C22×D4—71no—
145C23×C4—141no—
146C2×M16—142no—
147C22⋊Q8—284no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×D4—142no—
151C4×D4—142no—
152C2×C4⋊C4—71no—
153C2×C4⋊C4—142no—
154C2×C4⋊C4—141no—
155C2×C4⋊C4C4×D712yes—
156C2×C4⋊C4D2811yescharacteristic
157C2×C22⋊C4—142no—
158C2×C22⋊C4—142no—
159C2×C22⋊C4—141no—
160C2×C42C22⋊D712yes—
161C2×C42C22×D711yescharacteristic
162C4⋊C8—24no—
163C4⋊C8—22no—
164C4⋊C8—142no—
165C4⋊C8—142no—
166#101—282no—
167C22⋊C8—22no—
168#97—24no—
169C4×C8C22⋊D714yes—
170#95—72no—
171#95—142no—
172#95—141no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7#38611yescharacteristic
176C22×C7#41711yescharacteristic
177C22×C7#41811yescharacteristic
178C2×D7—42no—
179C2×D7—42no—
180C2×D7—42no—
181C2×D7—81no—
182C2×D7—81no—
183C2×D7—81no—
184C2×D7—81no—
185C4×C7—22no—
186C4×C7—21no—
187C4×C7—21no—
188C4×C7—42no—
189C4×C7—42no—
190C4×C7#46812yes—
191C4×C7#46912yes—
192Dic7—82no—
193Dic7—81no—
194C24—141no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×D4—142no—
200C2×D4—282no—
201C22×C4—142no—
202C22×C4—141no—
203C22×C4—282no—
204C22×C4—281no—
205C22×C4—281no—
206C22×C4—282no—
207C22×C4(C2×D7)⋊C412yes—
208C22×C4(C2×D7)⋊C412yes—
209C22×C4C2×C4×D711yescharacteristic
210C22×C4C2×D2811yescharacteristic
211C22×C4C2×C22⋊D711yescharacteristic
212M16—282no—
213M16—282no—
214C2×C8—24no—
215C2×C8—282no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—142no—
219C4⋊C4—284no—
220C4⋊C4—282no—
221C22⋊C4—284no—
222C22⋊C4—284no—
223C22⋊C4—281no—
224C22⋊C4—281no—
225C42—22no—
226C42—22no—
227C42#115712yes—
228C42C2×C22⋊D712yes—
229C2×C7—21no—
230C2×C7—21no—
231C2×C7#189111yescharacteristic
232C2×C7#189511yescharacteristic
233C2×C7#191211yescharacteristic
234D7—82no—
235C23—142no—
236C23—281no—
237C23—281no—
238C23—281no—
239C23—281no—
240C23—281no—
241C23—281no—
242C23C2×(C2×D7)⋊C411yescharacteristic
243Q8—42no—
244Q8—42no—
245D4—144no—
246D4—282no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—282no—
255C2×C4—282no—
256C2×C4—282no—
257C2×C4—281no—
258C2×C4—281no—
259C2×C4—281no—
260C2×C4—562no—
261C2×C4—562no—
262C2×C4#647112yes—
263C2×C4#647111yescharacteristic
264C2×C4#647211yescharacteristic
265C2×C4C4×C22⋊D712yes—
266C2×C4#649211yescharacteristic
267C2×C4#649312yes—
268C2×C4#651611yescharacteristic
269C8—44no—
270C8—562no—
271C7#949011yescharacteristic, complemented, Sylow
272C22—21no—
273C22—21no—
274C22—282no—
275C22—282no—
276C22—282no—
277C22—561no—
278C22—561no—
279C22—561no—
280C22—561no—
281C22#8990311yescharacteristic
282C22#8990511yescharacteristic
283C22#8998511yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—562no—
290C4—561no—
291C4#8990612yes—
292C4#8990712yes—
293C2—21no—
294C2—21no—
295C2—562no—
296C2#1172941711yescharacteristic
297C2#1172974411yescharacteristic
298C2#1172978311yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2983C2
socleC22×C7#386
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#2983D7
hypocenterC7#9490
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
1. C7×#2983C2semidirect
3. #11729391C2semidirect
5. #11725544C2semidirect
52. #2983D7semidirect
271. C7#9490semidirect

Maximal subgroups

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

subgroupindex
C7×#29832
#117399442
#117293912
#117285162
#117255442
#94907