Finite Group Databasebeta

← #13242377

Subgroups — #13242377

Order 1792.

372 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 103 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#11730754C211yesmaximal, characteristic
2C7×#1831C211yesmaximal, characteristic
3#11727143C211yesmaximal, characteristic
4#11725286C212yesmaximal
5#11725284C212yesmaximal
6C2×C7×C4⋊C8—21no—
7C43×C7C2211yescharacteristic
8#89709—21no—
9#89705—21no—
10#89705—21no—
11#89705—21no—
12#89701C411yescharacteristic
13#89701C411yescharacteristic
14#89701C2211yescharacteristic
15#89700C411yescharacteristic
16#89700C411yescharacteristic
17#89700C2211yescharacteristic
18C2×#6377—21no—
19C7×#336—21no—
20C7×#336C2212yes—
21#89319—21no—
22#89319C2212yes—
23#7709—71nomaximal, Sylow
24C22×Dic14C2×C411yescharacteristic
25C22×C7×C8—22no—
26C2×C42×C7C2×C411yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7C2×C411yescharacteristic
34C2×C4⋊Dic7C2×C411yescharacteristic
35C2×C4⋊Dic7C2×C411yescharacteristic
36C2×Dic7⋊C4—21no—
37C2×Dic7⋊C4—21no—
38C2×Dic7⋊C4—21no—
39C22×C7⋊C8—22no—
40C7×C4⋊C8—42no—
41C7⋊#95—21no—
42C7⋊#95—21no—
43C7⋊#95—21no—
44C7⋊#95—21no—
45C7⋊#95—21no—
46C7⋊#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50#6377—24no—
51#2863—71no—
52#1831D711yescharacteristic
53#1745—71no—
54#1272—72no—
55#1271—72no—
56C22×C4×C7C22⋊C411yescharacteristic
57C22×C4×C7C22⋊C411yescharacteristic
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×Dic7—21no—
64C22×Dic7—21no—
65C22×Dic7—21no—
66C22×Dic7—21no—
67C2×Dic14—22no—
68C2×Dic14—41no—
69C2×Dic14—41no—
70C2×C7×C8—42no—
71C2×C7×C8—42no—
72C2×C7×C8—42no—
73C42×C7—22no—
74C42×C7—22no—
75C42×C7—21no—
76C42×C7—22no—
77C42×C7—21no—
78C42×C7—41no—
79C42×C7—41no—
80C42×C7—41no—
81C4⋊Dic7—22no—
82C4⋊Dic7—41no—
83C4⋊Dic7—41no—
84Dic7⋊C4—42no—
85Dic7⋊C4—42no—
86Dic7⋊C4—41no—
87Dic7⋊C4—41no—
88C2×C7⋊C8—24no—
89C2×C7⋊C8—42no—
90C2×C7⋊C8—42no—
91C2×C4⋊C8—21no—
92C2×#97—141no—
93#391—71no—
94#391—71no—
95#391—71no—
96#389—71no—
97#389—71no—
98#389—71no—
99#387—141no—
100#387—141no—
101#387—141no—
102#383—141no—
103C43C2×D711yescharacteristic
104#336—21no—
105#336—72no—
106#336—141no—
107#336C2×D712yes—
108C23×C7C2×C22⋊C411yescharacteristic
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7#10412yes—
127C2×C4×C7#13112yes—
128C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C7#13712yes—
130C2×C4×C7#13711yescharacteristic
131C2×Dic7—41no—
132C2×Dic7—41no—
133C2×Dic7—41no—
134C2×Dic7—41no—
135C2×Dic7—41no—
136C2×Dic7—41no—
137C2×Dic7—41no—
138C2×Dic7—41no—
139C2×Dic7—41no—
140C2×Dic7—41no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143Dic14—82no—
144C7×C8—82no—
145C7⋊C8—44no—
146C22×Q8—71no—
147C22×C8—22no—
148C22×C8—142no—
149C2×C4⋊C4—71no—
150C2×C4⋊C4—71no—
151C2×C4⋊C4—71no—
152C2×C4⋊C4—141no—
153C2×C4⋊C4—141no—
154C2×C4⋊C4—141no—
155C2×C42C4×D711yescharacteristic
156C2×C42C4×D711yescharacteristic
157C2×C42D2811yescharacteristic
158C2×C42D2811yescharacteristic
159C2×C42C22⋊D711yescharacteristic
160C2×C42C22⋊D711yescharacteristic
161C2×C42C22×D711yescharacteristic
162C4⋊C8—42no—
163#97—144no—
164#95—141no—
165#95—141no—
166#95—141no—
167#95—141no—
168#95—141no—
169#95—141no—
170#95—141no—
171#95—141no—
172#95—141no—
173C22×C7#40811yescharacteristic
174C22×C7#41011yescharacteristic
175C22×C7#41311yescharacteristic
176C22×C7#41711yescharacteristic
177C22×C7#41911yescharacteristic
178C22×C7C2×#10411yescharacteristic
179C22×C7#42111yescharacteristic
180C4×C7—22no—
181C4×C7—22no—
182C4×C7—22no—
183C4×C7—21no—
184C4×C7—21no—
185C4×C7—41no—
186C4×C7—41no—
187C4×C7—41no—
188Dic7—81no—
189Dic7—81no—
190Dic7—81no—
191Dic7—81no—
192C2×Q8—142no—
193C2×Q8—281no—
194C2×Q8—281no—
195C22×C4—141no—
196C22×C4—141no—
197C22×C4—141no—
198C22×C4—141no—
199C22×C4(C2×D7)⋊C411yescharacteristic
200C22×C4(C2×D7)⋊C411yescharacteristic
201C22×C4(C2×D7)⋊C411yescharacteristic
202C22×C4(C2×D7)⋊C411yescharacteristic
203C22×C4C2×C4×D711yescharacteristic
204C22×C4C2×D2811yescharacteristic
205C22×C4C2×C22⋊D711yescharacteristic
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—144no—
210C2×C8—282no—
211C2×C8—282no—
212C4⋊C4—142no—
213C4⋊C4—281no—
214C4⋊C4—281no—
215C4⋊C4—282no—
216C4⋊C4—281no—
217C4⋊C4—281no—
218C4⋊C4—282no—
219C42—22no—
220C42—22no—
221C42—21no—
222C42—21no—
223C42—22no—
224C42—41no—
225C42—41no—
226C42—41no—
227C2×C7#147311yescharacteristic
228C2×C7#148311yescharacteristic
229C2×C7#149711yescharacteristic
230C2×C7#150611yescharacteristic
231C2×C7#151211yescharacteristic
232C2×C7#152911yescharacteristic
233C2×C7#155911yescharacteristic
234C23C2×(C2×D7)⋊C411yescharacteristic
235Q8—562no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—281no—
254C2×C4—281no—
255C2×C4—281no—
256C2×C4—281no—
257C2×C4—281no—
258C2×C4—281no—
259C2×C4—281no—
260C2×C4—281no—
261C2×C4—281no—
262C2×C4—281no—
263C2×C4—281no—
264C2×C4—281no—
265C2×C4#639912yes—
266C2×C4#646412yes—
267C2×C4#647212yes—
268C2×C4#649511yescharacteristic
269C2×C4#650511yescharacteristic
270C8—82no—
271C8—284no—
272C7#770911yescharacteristic, complemented, Sylow
273C22#8973811yescharacteristic
274C22#8977711yescharacteristic
275C22#8977811yescharacteristic
276C22#8988211yescharacteristic
277C22#8989211yescharacteristic
278C22#8990411yescharacteristic
279C22C2×#639911yescharacteristic
280C4—22no—
281C4—22no—
282C4—21no—
283C4—21no—
284C4—22no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—561no—
289C4—561no—
290C4—561no—
291C4—561no—
292C2#1172729811yescharacteristic
293C2#1172818111yescharacteristic
294C2#1172826411yescharacteristic
295C2#1172852511yescharacteristic
296C2#1172853911yescharacteristic
297C2#1172855911yescharacteristic
298C2#1172859011yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1831C2
socle, prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1831D7
hypocenterC7#7709
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
272. C7#7709semidirect

Maximal subgroups

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

subgroupindex
#117307542
C7×#18312
#117271432
#117252862
#117252842
#77097