Finite Group Databasebeta

← #13251655

Subgroups — #13251655

Order 1792.

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

normal subgroups only

This group has 518 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×#2975C211yesmaximal, characteristic
2#11739833C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11728524C212yesmaximal, complemented
5#11725544C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C4×Dic14C2211yescharacteristic
8C7×#436—24no—
9C7×#431C2212yes—
10C7×#407—22no—
11#89993—21no—
12C2×#6407—21no—
13#89982—22no—
14#89745—21no—
15#89721C412yes—
16#89719C412yes—
17#89708C2211yescharacteristic
18#89696C2212yes—
19#89325—21no—
20#89251—24no—
21#9487—71nomaximal, Sylow
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D412yes—
24C22×Dic14C2×C411yescharacteristic
25C2×C7×M16—24no—
26C4×C7×D4—22no—
27C4×C7×D4—22no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C22⋊C4C2×C412yes—
30C2×C42×C7C2311yescharacteristic
31C2×C22⋊Dic7—22no—
32C2×C22⋊Dic7—22no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7C2×C411yescharacteristic
35C2×Dic7⋊C4—21no—
36C2×Dic7⋊C4—22no—
37C2×C4×Dic7—21no—
38C2×#1136—22no—
39C4×Dic14—22no—
40C4×Dic14—22no—
41C7×C4⋊C8—22no—
42C7×C22⋊C8—24no—
43C7×C22⋊C8—22no—
44C4×C7×C8D414yes—
45#6407—42no—
46C7⋊#95—22no—
47C7⋊#95—21no—
48C7⋊#95C2×C412yes—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51#2975D711yescharacteristic
52#2580—71no—
53#1800—71no—
54#1487—72no—
55#1286—72no—
56C24×C7C22⋊C412yes—
57C2×C7×D4—22no—
58C2×C7×D4—21no—
59C2×C7×D4—22no—
60C2×C7×D4—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—82no—
64C22×C4×C7C22⋊C412yes—
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—22no—
70C22×Dic7—21no—
71C2×Dic14—22no—
72C2×Dic14—42no—
73C7×M16—84no—
74C2×C7×C8—24no—
75C7×C4⋊C4—22no—
76C7×C22⋊C4—24no—
77C42×C7C2×D412yes—
78C42×C7#4112yes—
79C22⋊Dic7—44no—
80C22⋊Dic7—82no—
81C4⋊Dic7—22no—
82C4⋊Dic7—41no—
83C4⋊Dic7—41no—
84Dic7⋊C4—41no—
85Dic7⋊C4—41no—
86Dic7⋊C4—82no—
87C4×Dic7—42no—
88#1136—42no—
89#1136—42no—
90C2×C7⋊C8—42no—
91C2×C4×Q8—71no—
92C2×C4×D4C2×D711yescharacteristic
93#436—24no—
94#431C2×D712yes—
95#423—72no—
96C2×#100—141no—
97#407—22no—
98#399—72no—
99#393—72no—
100#390—141no—
101#389—141no—
102#385—142no—
103#382—71no—
104#343—141no—
105Q8⋊C8—144no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—22no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7C2×C22⋊C411yescharacteristic
112C7×D4—42no—
113C7×D4—42no—
114C2×C4×C7—22no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7—44no—
118C2×C4×C7—42no—
119C2×C4×C7—82no—
120C2×C4×C7C4×D412yes—
121C2×C4×C7#12011yescharacteristic
122C2×C4×C7#12112yes—
123C2×C4×C7#12311yescharacteristic
124C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C7#13712yes—
126C2×C4×C7#13711yescharacteristic
127C2×Dic7—22no—
128C2×Dic7—41no—
129C2×Dic7—41no—
130C2×Dic7—41no—
131C2×Dic7—42no—
132C2×Dic7—42no—
133C2×Dic7—41no—
134C2×Dic7—41no—
135C2×Dic7—42no—
136Dic14—24no—
137Dic14—42no—
138C7×C8—44no—
139C7⋊C8—82no—
140C22×Q8—71no—
141C22×D4D2811yescharacteristic
142C23×C4C22⋊D712yes—
143C2×M16—24no—
144C2×M16—142no—
145C4×Q8—142no—
146C4×Q8—142no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—71no—
150C2×C4⋊C4—141no—
151C2×C4⋊C4—141no—
152C2×C4⋊C4—142no—
153C2×C4⋊C4D2811yescharacteristic
154C2×C22⋊C4—142no—
155C2×C22⋊C4—142no—
156C2×C22⋊C4C4×D712yes—
157C2×C42—141no—
158C2×C42C22×D711yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—142no—
161C4⋊C8—142no—
162#100—282no—
163C22⋊C8—24no—
164C22⋊C8—22no—
165C4×C8C22⋊D714yes—
166#95—72no—
167#95—142no—
168#95—141no—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—44no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—82no—
176C22×C7#38611yescharacteristic
177C22×C7#41711yescharacteristic
178C22×C7#41911yescharacteristic
179C4×C7—22no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—42no—
183C4×C7#46812yes—
184C4×C7#47012yes—
185Dic7—42no—
186Dic7—82no—
187Dic7—81no—
188C24(C2×D7)⋊C412yes—
189C2×Q8—142no—
190C2×Q8—282no—
191C2×D4—21no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—82no—
198C22×C4—141no—
199C22×C4—142no—
200C22×C4—141no—
201C22×C4(C2×D7)⋊C412yes—
202C22×C4C2×C4×D711yescharacteristic
203C22×C4C2×D2811yescharacteristic
204C22×C4C2×C22⋊D711yescharacteristic
205M16—84no—
206M16—282no—
207M16—282no—
208C2×C8—24no—
209C2×C8—282no—
210C4⋊C4—22no—
211C4⋊C4—142no—
212C4⋊C4—281no—
213C4⋊C4—281no—
214C4⋊C4—281no—
215C4⋊C4—281no—
216C4⋊C4—562no—
217C22⋊C4—24no—
218C22⋊C4—284no—
219C22⋊C4—562no—
220C42—282no—
221C42#115712yes—
222C42C2×C22⋊D712yes—
223C2×C7—21no—
224C2×C7—21no—
225C2×C7—82no—
226C2×C7#189111yescharacteristic
227C2×C7#189411yescharacteristic
228C2×C7#191111yescharacteristic
229C23—22no—
230C23—22no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23C2×(C2×D7)⋊C411yescharacteristic
235Q8—144no—
236Q8—282no—
237D4—42no—
238D4—42no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—142no—
246C2×C4—281no—
247C2×C4—281no—
248C2×C4—282no—
249C2×C4—281no—
250C2×C4—282no—
251C2×C4—281no—
252C2×C4—282no—
253C2×C4—281no—
254C2×C4#647111yescharacteristic
255C2×C4#647212yes—
256C2×C4#647211yescharacteristic
257C2×C4C4×C22⋊D712yes—
258C2×C4#649211yescharacteristic
259C2×C4#649312yes—
260C2×C4#651611yescharacteristic
261C8—44no—
262C8—562no—
263C7#948711yescharacteristic, complemented, Sylow
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#8989211yescharacteristic
272C22#8990511yescharacteristic
273C22#8998511yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—282no—
279C4—562no—
280C4—561no—
281C4#8990612yes—
282C4#8990912yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#1172940611yescharacteristic
287C2#1172974311yescharacteristic
288C2#1172978311yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2975C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2975D7
hypocenterC7#9487
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
4. #11728524C2semidirect
5. #11725544C2semidirect
263. C7#9487semidirect

Maximal subgroups

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

subgroupindex
C7×#29752
#117398332
#117293712
#117285242
#117255442
#94877