Finite Group Databasebeta

← #13248840

Subgroups — #13248840

Order 1792.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C2×#89721C211yesmaximal, characteristic
3C2×#89320C211yesmaximal, characteristic
4#11727256C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C22×C7×C22⋊C4C2211yescharacteristic
7C22×C22⋊Dic7—21no—
8C22×C4⋊Dic7C2211yescharacteristic
9C22×Dic7⋊C4—21no—
10C7×#388—24no—
11C7×#380C2212yes—
12C7×#379—21no—
13C4×C7×C22⋊C4—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2212yes—
16#89721—22no—
17#89721C414yescomplemented
18#89324—24no—
19#89320—22no—
20#9024—71nomaximal, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C2×C7×C4⋊C4D414yes—
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—24no—
36C2×C22⋊Dic7—44no—
37C2×C4⋊Dic7—22no—
38C2×C4⋊Dic7—22no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C412yes—
41C2×Dic7⋊C4—24no—
42C2×Dic7⋊C4—44no—
43C22×C7⋊C8—22no—
44C7×#95—22no—
45C7×#95—22no—
46C7⋊#95—21no—
47C7⋊#95C2×C412yes—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50C2×#399—71no—
51#2312D711yescharacteristic
52#1824—72no—
53C2×#323—71no—
54#1294—72no—
55C24×C7—21no—
56C24×C7—22no—
57C24×C7—21no—
58C24×C7—22no—
59C24×C7—42no—
60C24×C7—42no—
61C24×C7C22⋊C412yes—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—24no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7C22⋊C412yes—
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D412yes—
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7#4112yes—
73C22×Dic7—24no—
74C22×Dic7—22no—
75C22×Dic7—22no—
76C22×Dic7—42no—
77C22×Dic7—42no—
78C22×Dic7—42no—
79C22×Dic7—42no—
80C22×Dic7—41no—
81C22×Dic7—41no—
82C7×C4⋊C4—44no—
83C7×C22⋊C4—24no—
84C7×C22⋊C4—24no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—82no—
88C7×C22⋊C4—82no—
89C42×C7—82no—
90C22⋊Dic7—48no—
91C4⋊Dic7—24no—
92C4⋊Dic7—42no—
93Dic7⋊C4—48no—
94C2×C7⋊C8—44no—
95C2×C7⋊C8—42no—
96C22×C4⋊C4—71no—
97C22×C4⋊C4—141no—
98C22×C22⋊C4—141no—
99C22×C22⋊C4C2×D711yescharacteristic
100C2×C22⋊C8—72no—
101#399—74no—
102#399—142no—
103#388—24no—
104#380C2×D712yes—
105#379—21no—
106C4×C22⋊C4—21no—
107C2×#95—71no—
108#340—144no—
109#323—142no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—42no—
116C23×C7—42no—
117C23×C7—48no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—81no—
123C23×C7—81no—
124C23×C7—84no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C23×C7#13512yes—
129C23×C7#13712yes—
130C2×C4×C7—24no—
131C2×C4×C7—22no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—44no—
138C2×C4×C7—44no—
139C2×C4×C7—44no—
140C2×C4×C7—81no—
141C2×C4×C7C4×D412yes—
142C2×C4×C7#12112yes—
143C2×Dic7—24no—
144C2×Dic7—48no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—82no—
151C2×Dic7—82no—
152C7⋊C8—82no—
153C25C22⋊D711yescharacteristic
154C23×C4—141no—
155C23×C4—141no—
156C23×C4C22⋊D711yescharacteristic
157C23×C4C22×D711yescharacteristic
158C22×C8—142no—
159C2×C4⋊C4—72no—
160C2×C4⋊C4—142no—
161C2×C4⋊C4—142no—
162C2×C4⋊C4—144no—
163C2×C4⋊C4—141no—
164C2×C4⋊C4—284no—
165C2×C4⋊C4C22⋊D714yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—144no—
172C2×C22⋊C4—284no—
173C2×C22⋊C4C4×D712yes—
174C2×C22⋊C4D2812yes—
175C2×C42—22no—
176C22⋊C8—142no—
177C22⋊C8—142no—
178#95—22no—
179#95—22no—
180#95—72no—
181#95—141no—
182C22×C7—24no—
183C22×C7—24no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—44no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—42no—
192C22×C7—82no—
193C22×C7—84no—
194C22×C7—84no—
195C22×C7#38611yescharacteristic
196C22×C7#41611yescharacteristic
197C22×C7#41911yescharacteristic
198C22×C7#45212yes—
199C22×C7#45312yes—
200C22×C7#45812yes—
201C22×C7#48512yes—
202C4×C7—22no—
203C4×C7—22no—
204C4×C7—42no—
205C4×C7—84no—
206Dic7—44no—
207Dic7—82no—
208C24—22no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—42no—
213C24—42no—
214C24(C2×D7)⋊C412yes—
215C24C2×C22⋊D711yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—144no—
222C22×C4—142no—
223C22×C4—142no—
224C22×C4—282no—
225C22×C4—282no—
226C22×C4—282no—
227C22×C4—282no—
228C22×C4—281no—
229C22×C4—281no—
230C22×C4(C2×D7)⋊C412yes—
231C22×C4C2×C4×D711yescharacteristic
232C22×C4C2×D2811yescharacteristic
233C22×C4#115712yes—
234C22×C4C2×C22⋊D712yes—
235C2×C8—284no—
236C2×C8—282no—
237C4⋊C4—44no—
238C4⋊C4—144no—
239C4⋊C4—288no—
240C4⋊C4—282no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—288no—
248C42—82no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—22no—
252C2×C7—82no—
253C2×C7#161814yes—
254C2×C7#187311yescharacteristic
255C2×C7#188711yescharacteristic
256C2×C7#189011yescharacteristic
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—48no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—81no—
270C23—81no—
271C23—84no—
272C23C2×(C2×D7)⋊C411yescharacteristic
273C23#649211yescharacteristic
274C23#651312yes—
275C23#651412yes—
276C23#651611yescharacteristic
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—81no—
288C2×C4—144no—
289C2×C4—282no—
290C2×C4—288no—
291C2×C4—282no—
292C2×C4—282no—
293C2×C4—282no—
294C2×C4—282no—
295C2×C4—562no—
296C2×C4—562no—
297C2×C4C4×C22⋊D712yes—
298C2×C4#649312yes—
299C8—562no—
300C7#902411yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—24no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#8976812yes—
315C22#8977411yescharacteristic
316C22#8977611yescharacteristic
317C22#8998511yescharacteristic
318C22#9001112yes—
319C22#9001312yes—
320C22#9001512yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—84no—
325C4—284no—
326C4—562no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#1172933911yescharacteristic
332C2#1172940411yescharacteristic
333C2#1172940911yescharacteristic
334C2#1172963714yes—
subgroupG/H
centerC23#6516
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2312C2
socleC23×C7#120
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2312D7
hypocenterC7#9024
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
4. #11727256C2semidirect
5. #11725430C2semidirect
17. #89721C4semidirect
300. C7#9024semidirect

Maximal subgroups

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

subgroupindex
C7×#23122
C2×#897212
C2×#893202
#117272562
#117254302
#90247