Finite Group Databasebeta

← #13249849

Subgroups — #13249849

Order 1792.

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

normal subgroups only

This group has 612 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×#2312C211yesmaximal, characteristic
2#11739562C211yesmaximal, characteristic
3C2×#89320C211yesmaximal, characteristic
4#11727264C212yesmaximal, complemented
5#11725431C212yesmaximal, complemented
6C22×C7×C22⋊C4C2211yescharacteristic
7C2×#6487C2211yescharacteristic
8C7×#388—24no—
9C7×#380C2212yes—
10C7×#379—21no—
11C4×C7×C22⋊C4—21no—
12#90019—21no—
13#89984C2211yescharacteristic
14#89982—21no—
15C2×C22⋊(C7⋊C8)C2212yes—
16#89721C412yescomplemented
17#89720—22no—
18#89719C412yescomplemented
19#89320—22no—
20#89247—24no—
21#9201—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×Dic14C2×C411yescharacteristic
26C2×C7×C4⋊C4—22no—
27C2×C7×C22⋊C4—22no—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4C2×C412yes—
33C2×C7×C22⋊C4D412yes—
34C2×C42×C7—22no—
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—22no—
38C2×C22⋊Dic7—22no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41#6487—22no—
42#6487—22no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—22no—
45C22×C7⋊C8—22no—
46C7×#95—22no—
47C7×#95D414yes—
48C7⋊#95—22no—
49C7⋊#95C2×C412yes—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#2626—71no—
53#2312D711yescharacteristic
54#1520—72no—
55C2×#323—71no—
56#1289—72no—
57C24×C7—21no—
58C24×C7—21no—
59C24×C7—22no—
60C24×C7—22no—
61C24×C7—81no—
62C24×C7—81no—
63C24×C7C22⋊C412yes—
64C24×C7C2×D411yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—24no—
68C22×C4×C7—81no—
69C22×C4×C7—81no—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D412yes—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7#4112yes—
75C22×Dic7—21no—
76C22×Dic7—22no—
77C22×Dic7—21no—
78C2×Dic14—22no—
79C2×Dic14—81no—
80C7×C4⋊C4—82no—
81C7×C22⋊C4—24no—
82C7×C22⋊C4—24no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—82no—
86C7×C22⋊C4—82no—
87C42×C7—82no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C22⋊Dic7—42no—
91C22⋊Dic7—82no—
92C22⋊Dic7—82no—
93C4⋊Dic7—22no—
94C4⋊Dic7—81no—
95Dic7⋊C4—42no—
96Dic7⋊C4—82no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8—82no—
99C2×C22⋊Q8—71no—
100C22×C22⋊C4C2×D711yescharacteristic
101C2×C22⋊C8—72no—
102#399—72no—
103#396—142no—
104#393—72no—
105#388—24no—
106#385—141no—
107#380—71no—
108#380C2×D712yes—
109#379—21no—
110#379—141no—
111C4×C22⋊C4—21no—
112#328—144no—
113#323—142no—
114C23×C7—22no—
115C23×C7—22no—
116C23×C7—22no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—81no—
126C23×C7—81no—
127C23×C7—84no—
128C23×C7—84no—
129C23×C7C2×C22⋊C411yescharacteristic
130C23×C7#12011yescharacteristic
131C23×C7#12311yescharacteristic
132C23×C7#13512yes—
133C23×C7#13712yes—
134C2×C4×C7—22no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—44no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—81no—
145C2×C4×C7C4×D412yes—
146C2×C4×C7#12112yes—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—42no—
154Dic14—82no—
155C7⋊C8—82no—
156C25C22⋊D711yescharacteristic
157C22×Q8—71no—
158C23×C4C22⋊D711yescharacteristic
159C23×C4C22×D711yescharacteristic
160C22×C8—142no—
161C22⋊Q8—142no—
162C22⋊Q8—142no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—141no—
166C2×C4⋊C4—142no—
167C2×C4⋊C4—141no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—141no—
174C2×C22⋊C4—141no—
175C2×C22⋊C4—142no—
176C2×C22⋊C4—142no—
177C2×C22⋊C4C4×D712yes—
178C2×C22⋊C4D2812yes—
179C2×C42—22no—
180C22⋊C8—142no—
181C22⋊C8—142no—
182#95—22no—
183#95—72no—
184#95—142no—
185#95C22⋊D714yes—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—44no—
195C22×C7—42no—
196C22×C7—42no—
197C22×C7—42no—
198C22×C7—82no—
199C22×C7—84no—
200C22×C7—84no—
201C22×C7#38611yescharacteristic
202C22×C7#41611yescharacteristic
203C22×C7#41911yescharacteristic
204C22×C7#45212yes—
205C22×C7#48512yes—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—42no—
209C4×C7—84no—
210Dic7—82no—
211Dic7—81no—
212Dic7—81no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—81no—
218C24—81no—
219C24(C2×D7)⋊C412yes—
220C24C2×C22⋊D711yescharacteristic
221C2×Q8—142no—
222C2×Q8—561no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—142no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4(C2×D7)⋊C412yes—
232C22×C4C2×C4×D711yescharacteristic
233C22×C4C2×D2811yescharacteristic
234C22×C4#115712yes—
235C22×C4C2×C22⋊D712yes—
236C2×C8—282no—
237C2×C8—562no—
238C4⋊C4—82no—
239C4⋊C4—142no—
240C4⋊C4—282no—
241C4⋊C4—562no—
242C4⋊C4—561no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—281no—
250C22⋊C4—282no—
251C22⋊C4—281no—
252C22⋊C4—562no—
253C22⋊C4—562no—
254C42—82no—
255C2×C7—21no—
256C2×C7—22no—
257C2×C7—21no—
258C2×C7—22no—
259C2×C7—82no—
260C2×C7#187311yescharacteristic
261C2×C7#191111yescharacteristic
262C2×C7#191311yescharacteristic
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—84no—
277C23—84no—
278C23C2×(C2×D7)⋊C411yescharacteristic
279C23#649211yescharacteristic
280C23#651312yes—
281C23#651412yes—
282C23#651611yescharacteristic
283Q8—562no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—81no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—282no—
302C2×C4C4×C22⋊D712yes—
303C2×C4#649312yes—
304C8—562no—
305C7#920111yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#8976812yes—
322C22#8977411yescharacteristic
323C22#8977611yescharacteristic
324C22#8998511yescharacteristic
325C22#9001112yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—562no—
331C4—561no—
332C4—561no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#1172933911yescharacteristic
339C2#1172940611yescharacteristic
340C2#1172941011yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2312C2
socleC22×C7#386
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2312D7
hypocenterC7#9201
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
4. #11727264C2semidirect
5. #11725431C2semidirect
16. #89721C4semidirect
18. #89719C4semidirect
305. C7#9201semidirect

Maximal subgroups

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

subgroupindex
C7×#23122
#117395622
C2×#893202
#117272642
#117254312
#92017