Finite Group Databasebeta

← #13514036

Subgroups — #13514036

Order 1792.

2350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×Q8C211yesmaximal, characteristic
2Q8×C22⋊Dic7C212yesmaximal
3#11739907C216yesmaximal, complemented
4Q8×C4⋊Dic7C212yesmaximal, complemented
5#11739899C216yesmaximal, complemented
6#11739895C212yesmaximal, complemented
7#11739604C216yesmaximal
8#11739507C213yesmaximal
9#11739505C213yesmaximal
10C23×C7×Q8C2211yescharacteristic
11C2×C4×C7×Q8C4116yescomplemented
12C2×C4×C7×Q8C2213yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×Q8C2213yes—
15C2×C4×C7×Q8C2211yescharacteristic
16C22×C7×C4⋊C4C2213yes—
17C22×C42×C7C2213yes—
18C2×Q8×Dic7C2214yes—
19C2×#6489C2216yes—
20#90002C22112yes—
21#89984C2216yes—
22#89984C2211yescharacteristic
23#89984C2213yes—
24#89983C2216yes—
25C4×C22⋊Dic7C2216yes—
26#89757C22112yes—
27#89753C22124yes—
28C4⋊C4×Dic7C22112yes—
29#89709C2216yes—
30#89708C2216yes—
31#89708C2216yes—
32#89707C2216yes—
33C4×C4⋊Dic7C2216yes—
34C4×C4⋊Dic7C2216yes—
35C4×C4⋊Dic7C2216yes—
36#89704C2216yes—
37#34196—71nomaximal, Sylow
38C22×C7×Q8—28no—
39C22×C7×Q8C2×C418yes—
40C22×C7×Q8C2313yes—
41C22×C7×Q8C2311yescharacteristic
42C23×C4×C7C2313yes—
43C23×C4×C7C2313yes—
44C23×C4×C7C2311yescharacteristic
45C4×C7×Q8C2×C4124yes—
46C4×C7×Q8C2×C4124yes—
47C4×C7×Q8C2×C418yes—
48C4×C7×Q8C2×C418yes—
49C2×C7×C4⋊C4C2×C4124yes—
50C2×C7×C4⋊C4C2313yes—
51C2×C7×C4⋊C4C2316yes—
52C2×C7×C4⋊C4C2313yes—
53C2×C42×C7C2×C4124yes—
54C2×C42×C7C2313yes—
55C2×C42×C7C2313yes—
56C2×C42×C7C2313yes—
57C2×C42×C7C2313yes—
58C2×C22⋊Dic7C2316yes—
59C2×C22⋊Dic7C2312yes—
60Q8×Dic7—216no—
61#6489—412no—
62C2×C4⋊Dic7C2316yes—
63C2×C4⋊Dic7C2312yes—
64C2×C4⋊Dic7C23112yes—
65C2×C4⋊Dic7C23112yes—
66C2×C4×Dic7C23112yes—
67C2×C4×Dic7C23112yes—
68C7⋊#95C2312yes—
69C7⋊#95C2316yes—
70C7⋊#95C23112yes—
71C7⋊#95C23112yes—
72C7⋊#95C23112yes—
73C7⋊#95C23112yes—
74C22×C4×Q8D711yescharacteristic
75#2355—76no—
76Q8×C4⋊C4—72no—
77#2344—76no—
78Q8×C22⋊C4—72no—
79#2320—76no—
80#2285—73no—
81#2283—73no—
82#2276—72no—
83C24×C7C2411yescharacteristic
84C2×C7×Q8—232no—
85C2×C7×Q8—212no—
86C2×C7×Q8C22×C4112yes—
87C2×C7×Q8C22×C414yes—
88C2×C7×Q8#4118yes—
89C22×C4×C7—26no—
90C22×C4×C7—26no—
91C22×C4×C7—212no—
92C22×C4×C7—24no—
93C22×C4×C7C22×C414yes—
94C22×C4×C7C22×C4112yes—
95C22×C4×C7C22×C4112yes—
96C22×C4×C7C2411yescharacteristic
97C22×C4×C7C2413yes—
98C22×C4×C7C2413yes—
99C22×C4×C7C2413yes—
100C22×C4×C7C2413yes—
101C22×C4×C7C2411yescharacteristic
102C22×Dic7C2414yes—
103C22×Dic7C24112yes—
104C7×C4⋊C4C22×C4124yes—
105C7×C4⋊C4C22×C4112yes—
106C7×C4⋊C4C22×C4112yes—
107C42×C7C22×C4112yes—
108C42×C7C22×C4112yes—
109C42×C7C22×C4112yes—
110C42×C7C22×C4112yes—
111C22⋊Dic7—212no—
112C22⋊Dic7—24no—
113C4⋊Dic7—224no—
114C4⋊Dic7—212no—
115C4⋊Dic7—24no—
116C4⋊Dic7—412no—
117C4×Dic7—224no—
118C4×Dic7—412no—
119C23×Q8C2×D711yescharacteristic
120C2×C4×Q8—74no—
121C2×C4×Q8Dic7116yescomplemented
122C2×C4×Q8C2×D713yes—
123C2×C4×Q8C2×D711yescharacteristic
124C2×C4×Q8C2×D713yes—
125C2×C4×Q8C2×D711yescharacteristic
126C2×#117—76no—
127C22×C4⋊C4C2×D713yes—
128C22×C42C2×D713yes—
129#391—712no—
130#389—712no—
131#387—724no—
132#384—76no—
133#383—76no—
134#382—76no—
135#382—76no—
136#381—76no—
137#380—76no—
138#380—71no—
139#380—73no—
140C4×C4⋊C4—712no—
141C4×C4⋊C4—76no—
142C4×C4⋊C4—76no—
143C4×C4⋊C4—76no—
144C4×C22⋊C4—76no—
145#376—76no—
146C23×C7—22no—
147C23×C7—22no—
148C23×C7C23×C412yes—
149C23×C7H(5,𝔽2)12yes—
150C23×C72−1+412yes—
151C23×C7C2511yescharacteristic
152C7×Q8—224no—
153C7×Q8#117116yescomplemented
154C2×C4×C7—224no—
155C2×C4×C7—26no—
156C2×C4×C7—23no—
157C2×C4×C7—23no—
158C2×C4×C7—22no—
159C2×C4×C7—23no—
160C2×C4×C7—23no—
161C2×C4×C7—224no—
162C2×C4×C7—28no—
163C2×C4×C7C23×C412yes—
164C2×C4×C7C23×C416yes—
165C2×C4×C7C23×C416yes—
166C2×C4×C7C23×C416yes—
167C2×C4×C7C23×C416yes—
168C2×C4×C7C23×C412yes—
169C2×C4×C7C2×#41112yes—
170C2×C4×C72−1+414yes—
171C2×Dic7—24no—
172C2×Dic7—212no—
173C2×Dic7—212no—
174C2×Dic7—212no—
175C2×Dic7—24no—
176C2×Dic7—24no—
177C22×Q8—28no—
178C22×Q8C2×Dic718yes—
179C22×Q8C22×D713yes—
180C22×Q8C22×D711yescharacteristic
181C23×C4C22×D713yes—
182C23×C4C22×D713yes—
183C23×C4C22×D711yescharacteristic
184C4×Q8—1416no—
185C4×Q8C2×Dic7124yes—
186C4×Q8C2×Dic7124yes—
187C4×Q8C2×Dic718yes—
188C4×Q8C2×Dic718yes—
189#117—2812no—
190C2×C4⋊C4—712no—
191C2×C4⋊C4—72no—
192C2×C4⋊C4—76no—
193C2×C4⋊C4—712no—
194C2×C4⋊C4C2×Dic7124yes—
195C2×C4⋊C4C22×D713yes—
196C2×C4⋊C4C22×D716yes—
197C2×C4⋊C4C22×D713yes—
198C2×C22⋊C4—76no—
199C2×C22⋊C4—72no—
200C2×C42—712no—
201C2×C42—712no—
202C2×C42C2×Dic7124yes—
203C2×C42C22×D713yes—
204C2×C42C22×D713yes—
205C2×C42C22×D713yes—
206C2×C42C22×D713yes—
207#95—72no—
208#95—76no—
209#95—712no—
210#95—712no—
211#95—712no—
212#95—712no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—24no—
216C22×C7—24no—
217C22×C7#52014yes—
218C22×C7C24×C411yescharacteristic
219C22×C7C22×#4112yes—
220C22×C7C2×H(5,𝔽2)11yescharacteristic
221C22×C7C2×2−1+412yes—
222C22×C7C2×2−1+411yescharacteristic
223C4×C7—26no—
224C4×C7—22no—
225C4×C7—26no—
226C4×C7—26no—
227C4×C7C2×#117112yes—
228C4×C7#51914yes—
229Dic7—412no—
230Dic7—44no—
231C24C23×D711yescharacteristic
232C2×Q8—232no—
233C2×Q8—212no—
234C2×Q8#115718yes—
235C2×Q8C22×Dic7112yes—
236C2×Q8C22×Dic714yes—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—212no—
240C22×C4—24no—
241C22×C4—74no—
242C22×C4—712no—
243C22×C4C22×Dic714yes—
244C22×C4C22×Dic7112yes—
245C22×C4C22×Dic7112yes—
246C22×C4C23×D711yescharacteristic
247C22×C4C23×D713yes—
248C22×C4C23×D713yes—
249C22×C4C23×D713yes—
250C22×C4C23×D713yes—
251C22×C4C23×D711yescharacteristic
252C4⋊C4—1424no—
253C4⋊C4—144no—
254C4⋊C4—1412no—
255C4⋊C4—2812no—
256C4⋊C4C22×Dic7124yes—
257C4⋊C4C22×Dic7112yes—
258C4⋊C4C22×Dic7112yes—
259C22⋊C4—1412no—
260C22⋊C4—144no—
261C42—1424no—
262C42—2812no—
263C42C22×Dic7112yes—
264C42C22×Dic7112yes—
265C42C22×Dic7112yes—
266C42C22×Dic7112yes—
267C2×C7—24no—
268C2×C7C22×#11711yescharacteristic
269C2×C7C2×#51911yescharacteristic
270C2×C7C2×#52011yescharacteristic
271C2×C7#352212yes—
272C2×C7#352312yes—
273C23—22no—
274C23—22no—
275C23#655312yes—
276C23#655412yes—
277C23C23×Dic712yes—
278C23C24×D711yescharacteristic
279Q8—224no—
280Q8#6489116yescomplemented
281C2×C4—224no—
282C2×C4—26no—
283C2×C4—22no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—224no—
289C2×C4—28no—
290C2×C4—144no—
291C2×C4—1412no—
292C2×C4—144no—
293C2×C4—1412no—
294C2×C4—1412no—
295C2×C4—144no—
296C2×C4C2×#1157112yes—
297C2×C4#655114yes—
298C2×C4C23×Dic712yes—
299C2×C4C23×Dic716yes—
300C2×C4C23×Dic716yes—
301C2×C4C23×Dic716yes—
302C2×C4C23×Dic716yes—
303C2×C4C23×Dic712yes—
304C7#3419611yescharacteristic, complemented, Sylow
305C22—22no—
306C22—22no—
307C22—24no—
308C22—24no—
309C22#9051814yes—
310C22C22×#115712yes—
311C22C2×#655112yes—
312C22C2×#655311yescharacteristic
313C22C2×#655411yescharacteristic
314C22C24×Dic711yescharacteristic
315C4—26no—
316C4—22no—
317C4—26no—
318C4—26no—
319C4—284no—
320C4—2812no—
321C4C2×#6489112yes—
322C4#9050314yes—
323C2—24no—
324C2#1174287212yes—
325C2#1174288912yes—
326C2C22×#648911yescharacteristic
327C2C2×#9050311yescharacteristic
328C2C2×#9051811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC22×C4×C7×Q8C2
socle, cosocleC23×C7C25
prime radicalC24×C7C24
hypercenterC22×C4×Q8D7
hypocenterC7#34196
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #11739907C2semidirect
4. Q8×C4⋊Dic7C2semidirect
5. #11739899C2semidirect
6. #11739895C2semidirect
11. C2×C4×C7×Q8C4semidirect
121. C2×C4×Q8Dic7semidirect
153. C7×Q8#117semidirect
280. Q8#6489semidirect
304. C7#34196semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×Q82
Q8×C22⋊Dic72
#117399072
Q8×C4⋊Dic72
#117398992
#117398952
#117396042
#117395072
#117395052
#341967