Finite Group Databasebeta

← #13251690

Subgroups — #13251690

Order 1792.

514 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2921C211yesmaximal, characteristic, complemented
2#11739729C211yesmaximal, characteristic
3#11729326C211yesmaximal, characteristic, complemented
4#11727609C212yesmaximal, complemented
5#11725472C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C2×C4×D28C2211yescharacteristic
8C7×#433—24no—
9C7×#423—21no—
10C7×#407—21no—
11C4×C7×M16C2212yes—
12#89985—21no—
13#89763C411yescharacteristic
14#89761—21no—
15#89758C411yescharacteristic
16#89732C411yescharacteristic
17#89728—21no—
18#89721C411yescharacteristic
19#89712—21no—
20#89709C2211yescharacteristic
21#89698C2212yes—
22#89323—21no—
23#89320—21no—
24#89254—24no—
25#9494—71nomaximal, Sylow
26C23×C4×C7D411yescharacteristic
27C22×C22⋊D7—21no—
28C22×D28C2×C411yescharacteristic
29C22×C4×D7—21no—
30C2×C7×M16—22no—
31C7×#117—22no—
32C7×#117—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7C2311yescharacteristic
39C2×C22⋊Dic7—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×C4⋊Dic7C2×C411yescharacteristic
45C2×Dic7⋊C4—21no—
46C4×D28—22no—
47C4×D28—22no—
48C7×C4⋊C8—24no—
49C7×C22⋊C8—24no—
50C7×#97—22no—
51C4×C7×C8D414yescomplemented
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95C2×C411yescharacteristic
57C7⋊#95C2×C411yescharacteristic
58C22⋊(C7⋊C8)—22no—
59C4⋊(C7⋊C8)—22no—
60#6377—22no—
61#2921D711yescharacteristic, complemented
62#2598—71no—
63#1490—72no—
64#1464—71no—
65#1288—72no—
66C24×C7C22⋊C411yescharacteristic
67C23×D7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C2×C22⋊D7—42no—
79C2×C22⋊D7—42no—
80C2×C22⋊D7—81no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C22×Dic7—21no—
84C2×D28—22no—
85C2×D28—42no—
86C2×C4×D7—42no—
87C2×C4×D7—81no—
88C2×C4×D7—81no—
89C7×M16—42no—
90C7×M16—42no—
91C2×C7×C8—24no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C22⋊C4—22no—
95C7×C22⋊C4—22no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7C2×D412yes—
99C42×C7#4112yes—
100C22⋊Dic7—81no—
101(C2×D7)⋊C4—42no—
102(C2×D7)⋊C4—42no—
103(C2×D7)⋊C4—41no—
104(C2×D7)⋊C4—41no—
105(C2×D7)⋊C4—81no—
106C4⋊Dic7—22no—
107Dic7⋊C4—81no—
108C2×C7⋊C8—42no—
109C2×C4×D4—71no—
110C2×#117C2×D711yescharacteristic
111#433—24no—
112#432—72no—
113#423—21no—
114#407—21no—
115C4×M16C2×D712yes—
116#400—71no—
117#399—71no—
118#394—71no—
119#392—71no—
120#388—141no—
121#386—141no—
122#386—141no—
123#385—141no—
124#383—71no—
125D4⋊C8—144no—
126#324—141no—
127#323—141no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7C2×C22⋊C411yescharacteristic
134C22×D7—22no—
135C22×D7—41no—
136C22×D7—41no—
137C22×D7—41no—
138C22×D7—41no—
139C22×D7—41no—
140C22×D7—41no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7C4×D412yes—
155C2×C4×C7#12011yescharacteristic
156C2×C4×C7#12112yes—
157C2×C4×C7#12311yescharacteristic
158C22⋊D7—84no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168D28—24no—
169D28—42no—
170C4×D7—84no—
171C7×C8—44no—
172C7⋊C8—82no—
173C22×D4—71no—
174C22×D4—141no—
175C23×C4—141no—
176C23×C4C22⋊D711yescharacteristic
177C2×M16—22no—
178C4×D4—142no—
179C4×D4—142no—
180#117—22no—
181#117—22no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—141no—
184C2×C4⋊C4C4×D711yescharacteristic
185C2×C4⋊C4D2811yescharacteristic
186C2×C22⋊C4—141no—
187C2×C22⋊C4—141no—
188C2×C22⋊C4—141no—
189C2×C22⋊C4—141no—
190C2×C22⋊C4—141no—
191C2×C22⋊C4C4×D711yescharacteristic
192C2×C22⋊C4D2811yescharacteristic
193C2×C42C22⋊D711yescharacteristic
194C2×C42C22×D711yescharacteristic
195C4⋊C8—24no—
196C4⋊C8—142no—
197C22⋊C8—24no—
198C22⋊C8—142no—
199#97—22no—
200#97—142no—
201C4×C8C22⋊D714yescomplemented
202#95—71no—
203#95—71no—
204#95—141no—
205#95—141no—
206#95—141no—
207#95—141no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—81no—
215C22×C7#38611yescharacteristic
216C22×C7#42111yescharacteristic
217C22×C7#42111yescharacteristic
218C2×D7—42no—
219C2×D7—42no—
220C2×D7—42no—
221C2×D7—81no—
222C2×D7—81no—
223C2×D7—81no—
224C2×D7—81no—
225C4×C7—22no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—22no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232Dic7—81no—
233Dic7—81no—
234Dic7—81no—
235C24—141no—
236C24(C2×D7)⋊C411yescharacteristic
237C2×D4—142no—
238C2×D4—282no—
239C2×D4—282no—
240C2×D4—282no—
241C2×D4—561no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4—282no—
250C22×C4—561no—
251C22×C4—561no—
252C22×C4(C2×D7)⋊C411yescharacteristic
253C22×C4(C2×D7)⋊C411yescharacteristic
254C22×C4(C2×D7)⋊C411yescharacteristic
255C22×C4C2×C4×D711yescharacteristic
256C22×C4C2×D2811yescharacteristic
257C22×C4C2×C22⋊D711yescharacteristic
258M16—42no—
259M16—42no—
260C2×C8—24no—
261C2×C8—282no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—142no—
265C4⋊C4—561no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—282no—
269C22⋊C4—282no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C22⋊C4—561no—
273C22⋊C4—561no—
274C42—21no—
275C42—21no—
276C42#115712yes—
277C42C2×C22⋊D712yes—
278C2×C7—21no—
279C2×C7—21no—
280C2×C7—81no—
281C2×C7#189111yescharacteristic
282C2×C7#189711yescharacteristic
283C2×C7#191011yescharacteristic
284D7—82no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—142no—
291C23—281no—
292C23—281no—
293C23—281no—
294C23—281no—
295C23—281no—
296C23—281no—
297C23C2×(C2×D7)⋊C411yescharacteristic
298D4—144no—
299D4—282no—
300D4—564no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—281no—
317C2×C4—281no—
318C2×C4—281no—
319C2×C4—281no—
320C2×C4—281no—
321C2×C4—281no—
322C2×C4—281no—
323C2×C4—564no—
324C2×C4C4×C22⋊D712yes—
325C2×C4#649211yescharacteristic
326C2×C4#649312yes—
327C2×C4#651611yescharacteristic
328C8—44no—
329C8—562no—
330C7#949411yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—282no—
339C22—282no—
340C22—282no—
341C22—561no—
342C22—561no—
343C22—561no—
344C22—561no—
345C22#8977711yescharacteristic
346C22#8977711yescharacteristic
347C22#8998511yescharacteristic
348C4—22no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—561no—
356C4—561no—
357C4—561no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—562no—
362C2#1172934511yescharacteristic
363C2#1172977911yescharacteristic
364C2#1172978311yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2921C2
socleC22×C7#386
prime radical#89732C4
cosocleC2×C42×C7C23
hypercenter#2921D7
hypocenterC7#9494
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. C7×#2921C2semidirect
3. #11729326C2semidirect
4. #11727609C2semidirect
5. #11725472C2semidirect
51. C4×C7×C8D4semidirect
61. #2921D7semidirect
201. C4×C8C22⋊D7semidirect
330. C7#9494semidirect

Maximal subgroups

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

subgroupindex
C7×#29212
#117397292
#117293262
#117276092
#117254722
#94947