Finite Group Databasebeta

← #13272946

Subgroups — #13272946

Order 1792.

540 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 75 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#11734047C211yesmaximal, characteristic, complemented
2C7×#1793C211yesmaximal, characteristic
3#11729897C212yesmaximal
4#11725505C212yesmaximal, complemented
5#11725450C211yesmaximal, characteristic, complemented
6C2×C7×2−1+4C2211yescharacteristic
7C2×C7×#104—21no—
8C7×#417—21no—
9C2×C7×#101C2211yescharacteristic
10C7×#391C2211yescharacteristic
11C2×#6411—22no—
12#90004—22no—
13#90002C2212yes—
14C2×#6410C2212yes—
15#89758—21no—
16#89700—21no—
17#89433—22no—
18#89427—22no—
19#13379—71nomaximal, Sylow
20C7×2−1+4—21no—
21C7×2−1+4—22no—
22C7×2−1+4—42no—
23C7×2−1+4Q812yes—
24C2×C7×#41—22no—
25C2×C7×#41—21no—
26C2×C7×#41—41no—
27C22×C7×Q8—41no—
28C22×C7×Q8C2311yescharacteristic
29C2×C7×M16—21no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7D412yes—
36C2×Dic7⋊C4—22no—
37C2×C4×Dic7D412yes—
38C2×#1136—22no—
39C7×#104—42no—
40C7×#103—41no—
41C7×#102—41no—
42C7×#101—22no—
43C7×#95—22no—
44#6411—44no—
45#6410—22no—
46#6410—22no—
47#6409—42no—
48#6406—42no—
49C7⋊#95—21no—
50C7⋊#95—22no—
51C7⋊#95—22no—
52C7⋊#95—22no—
53#2809—71no—
54#1793—72no—
55#1793D711yescharacteristic
56#1394—72no—
57#1394—71no—
58C7×#41—24no—
59C7×#41—22no—
60C7×#41—24no—
61C7×#41—22no—
62C7×#41—42no—
63C7×#41—41no—
64C7×#41—42no—
65C7×#41—42no—
66C7×#41—44no—
67C7×#41—42no—
68C7×#41—41no—
69C2×C7×Q8—22no—
70C2×C7×Q8—21no—
71C2×C7×Q8—41no—
72C2×C7×Q8—41no—
73C2×C7×Q8—42no—
74C2×C7×Q8—42no—
75C2×C7×Q8—44no—
76C2×C7×Q8C2×Q811yescharacteristic
77C2×C7×Q8#4111yescharacteristic
78C2×C7×Q8#4112yes—
79C2×C7×D4—22no—
80C2×C7×D4—21no—
81C2×C7×D4—41no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—42no—
85C22×C4×C7—41no—
86C22×C4×C7C2×D412yes—
87C22×C4×C7C2×D411yescharacteristic
88C22×Dic7—22no—
89C22×Dic7—22no—
90C7×M16—41no—
91C7×M16—41no—
92C2×C7×C8—41no—
93C7×C4⋊C4—41no—
94C42×C7—41no—
95C4⋊Dic7—42no—
96C4⋊Dic7—42no—
97C4⋊Dic7—81no—
98C4⋊Dic7—81no—
99Dic7⋊C4—44no—
100C4×Dic7—42no—
101#1136—42no—
102#1136—42no—
103C2×C7⋊C8—42no—
104C2×2−1+4C2×D711yescharacteristic
105C2×#104—21no—
106C2×#104—142no—
107#417—21no—
108#417—142no—
109C2×#101—72no—
110C2×#101C2×D711yescharacteristic
111#400—141no—
112#398—142no—
113#391—72no—
114#391C2×D711yescharacteristic
115#389—141no—
116#387—142no—
117C23×C7—41no—
118C23×C7#12011yescharacteristic
119C7×Q8—24no—
120C7×Q8—22no—
121C7×Q8—41no—
122C7×Q8—41no—
123C7×Q8—42no—
124C7×Q8—42no—
125C7×Q8—41no—
126C7×D4—24no—
127C7×D4—22no—
128C7×D4—42no—
129C7×D4—41no—
130C7×D4—42no—
131C7×D4—42no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—22no—
137C2×C4×C7—44no—
138C2×C4×C7—42no—
139C2×C4×C7—44no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7C22⋊Q812yes—
147C2×C4×C7C22⋊Q811yescharacteristic
148C2×C4×C7#12412yes—
149C2×C4×C7#12411yescharacteristic
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C7×C8—81no—
157C7⋊C8—82no—
1582−1+4—21no—
1592−1+4—22no—
1602−1+4—42no—
1612−1+4Dic1412yes—
162C2×#41—22no—
163C2×#41—21no—
164C2×#41—41no—
165C22×Q8—41no—
166C22×Q8C22×D711yescharacteristic
167C2×M16—21no—
168C2×M16—142no—
169C2×C4⋊C4—72no—
170C2×C4⋊C4—142no—
171C2×C4⋊C4—141no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C4⋊C4C22⋊D711yescharacteristic
175C2×C42—72no—
176C2×C42C22⋊D711yescharacteristic
177#104—42no—
178#104—284no—
179#103—41no—
180#103—282no—
181#102—41no—
182#102—282no—
183#101—22no—
184#101—142no—
185#101—142no—
186#95—22no—
187#95—141no—
188#95—142no—
189#95—142no—
190#95—142no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—42no—
194C22×C7—81no—
195C22×C7—81no—
196C22×C7#39311yescharacteristic
197C22×C7B(4,𝔽2)11yescharacteristic
198C22×C7B(4,𝔽2)11yescharacteristic
199C4×C7—22no—
200C4×C7—21no—
201C4×C7—22no—
202C4×C7—21no—
203C4×C7—42no—
204C4×C7—42no—
205C4×C7—81no—
206Dic7—82no—
207Dic7—82no—
208#41—22no—
209#41—24no—
210#41—24no—
211#41—22no—
212#41—41no—
213#41—42no—
214#41—42no—
215#41—42no—
216#41—44no—
217#41—42no—
218#41—41no—
219C2×Q8—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8C2×Dic1411yescharacteristic
227C2×Q8#115711yescharacteristic
228C2×Q8#115912yes—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—142no—
237C22×C4—142no—
238C22×C4D4×D712yes—
239C22×C4C2×C22⋊D711yescharacteristic
240M16—41no—
241M16—41no—
242M16—282no—
243M16—282no—
244C2×C8—41no—
245C2×C8—282no—
246C4⋊C4—41no—
247C4⋊C4—282no—
248C4⋊C4—284no—
249C4⋊C4—282no—
250C4⋊C4—561no—
251C4⋊C4—561no—
252C42—41no—
253C42—282no—
254C2×C7—21no—
255C2×C7—21no—
256C2×C7—82no—
257C2×C7#202911yescharacteristic
258C2×C7#220311yescharacteristic
259C2×C7#220411yescharacteristic
260C23—41no—
261C23#650011yescharacteristic
262Q8—24no—
263Q8—22no—
264Q8—41no—
265Q8—41no—
266Q8—42no—
267Q8—41no—
268Q8—42no—
269D4—24no—
270D4—22no—
271D4—41no—
272D4—42no—
273D4—42no—
274D4—42no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—282no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—282no—
293C2×C4—282no—
294C2×C4—282no—
295C2×C4#644112yes—
296C2×C4#644812yes—
297C2×C4#648711yescharacteristic
298C2×C4#649911yescharacteristic
299C8—81no—
300C8—562no—
301C7#1337911yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—42no—
305C22—81no—
306C22—81no—
307C22#8971911yescharacteristic
308C22#8980411yescharacteristic
309C22#8980411yescharacteristic
310C4—21no—
311C4—22no—
312C4—22no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—562no—
318C4—562no—
319C2—21no—
320C2—21no—
321C2—82no—
322C2#1172725111yescharacteristic
323C2#1172874311yescharacteristic
324C2#1172874811yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC22×C7×Q8C23
FrattiniC22×Q8C22×D7
FittingC7×#1793C2
socleC22×C7B(4,𝔽2)
prime radicalC2×C7×2−1+4C22
hypercenter#1793D7
hypocenterC7#13379
LCSG ▷ C22×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1793

Complements

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

NH ≅ G/Ntype
1. #11734047C2semidirect
4. #11725505C2semidirect
5. #11725450C2semidirect
301. C7#13379semidirect

Maximal subgroups

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

subgroupindex
#117340472
C7×#17932
#117298972
#117255052
#117254502
#133797