Finite Group Databasebeta

← #13321665

Subgroups — #13321665

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742871C211yesmaximal, characteristic
2C7×#2356C211yesmaximal, characteristic
3#11739881C211yesmaximal, characteristic
4#11727320C214yesmaximal
5#11727300C212yesmaximal
6#11727225C212yesmaximal
7#11727187C214yesmaximal
8C2×C4×C7×Q8C2211yescharacteristic
9#90503—21no—
10#90261C414yescomplemented
11#90260—24no—
12Q8×Dic14C414yescomplemented
13#90191—21no—
14#90191—22no—
15#90161C2214yes—
16#90161C2212yes—
17C7×#391—21no—
18C7×#387—22no—
19C7×#383C2212yes—
20C7×#382C2211yescharacteristic
21C4×C7×C4⋊C4—21no—
22C4×C7×C4⋊C4C2211yescharacteristic
23C7×#376C2212yes—
24#89796—24no—
25#89698C2214yes—
26#89697C2212yes—
27#89247C2214yes—
28#89247C2218yes—
29#89247C2214yes—
30#17632—71nomaximal, Sylow
31C22×C7×Q8D411yescharacteristic
32C4×C7×Q8C2×C414yes—
33C4×C7×Q8D414yes—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D412yes—
38C2×C42×C7—22no—
39C2×C42×C7—21no—
40C2×C42×C7C2311yescharacteristic
41C2×C42×C7C2312yes—
42Q8×Dic7—24no—
43Q8×Dic7—24no—
44#6489—24no—
45#6489—22no—
46#6487—24no—
47#6487—22no—
48#6487C2314yes—
49#6453—28no—
50#6453—24no—
51#6451—28no—
52#6451—24no—
53#6433—22no—
54#6433—24no—
55#6432C2×C414yes—
56#6432C2×C412yes—
57#6432C2×C414yes—
58#6432C2×C412yes—
59C4×Dic14—22no—
60C4×Dic14—24no—
61C4×Dic14C2×C418yes—
62C4×Dic14C2×C414yes—
63C7×#95—22no—
64C7×#95—22no—
65C7×#95—21no—
66C7×#95—22no—
67C7×#95C2311yescharacteristic
68C7×#95C2312yes—
69C7×#95C2311yescharacteristic
70C22⋊(C7⋊C8)C2314yes—
71C4⋊(C7⋊C8)—22no—
72C4⋊(C7⋊C8)—24no—
73#6377—24no—
74C4×C7⋊C8—22no—
75#3502—71no—
76#2983—71no—
77#2356D711yescharacteristic
78#1518—72no—
79#1515—74no—
80#1506—74no—
81#1503—72no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8C22⋊C414yes—
85C22×C4×C7—21no—
86C22×C4×C7—22no—
87C22×C4×C7—21no—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2411yescharacteristic
93C2×Dic14—22no—
94C2×Dic14—24no—
95C2×Dic14C22×C414yes—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4C22⋊C418yes—
99C7×C4⋊C4C22⋊C414yes—
100C42×C7—44no—
101C42×C7—42no—
102C42×C7C22×C414yes—
103C42×C7C22×C412yes—
104C42×C7C2×D412yes—
105C42×C7C2×D414yes—
106C22⋊Dic7—24no—
107C4⋊Dic7—22no—
108C4⋊Dic7—24no—
109C4⋊Dic7—24no—
110C4⋊Dic7—22no—
111C4⋊Dic7—24no—
112C4⋊Dic7—22no—
113C4⋊Dic7C22×C414yes—
114Dic7⋊C4—24no—
115Dic7⋊C4—28no—
116Dic7⋊C4—24no—
117C4×Dic7—28no—
118C4×Dic7—24no—
119C2×C7⋊C8—24no—
120Q82—74no—
121#557—74no—
122#557—144no—
123#544—141no—
124#544—142no—
125#533—72no—
126#533—74no—
127#519—141no—
128C2×C4×Q8C2×D711yescharacteristic
129#446—144no—
130#432—74no—
131#431—72no—
132#391—21no—
133#387—22no—
134#383C2×D712yes—
135#382C2×D711yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×D711yescharacteristic
138#376C2×D712yes—
139#328—74no—
140#328—74no—
141#328—78no—
142C23×C7C22×D411yescharacteristic
143C7×Q8—28no—
144C2×C4×C7—22no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7C2×C22⋊C414yes—
159C2×C4×C7C2×C22⋊C412yes—
160C2×C4×C7C2×C22⋊C414yes—
161C2×C4×C7C2×C22⋊C412yes—
162C2×C4×C7#13712yes—
163C2×C4×C7#13712yes—
164C2×C4×C7C23×C411yescharacteristic
165C2×C4×C7C22×D411yescharacteristic
166C2×Dic7—24no—
167C2×Dic7—24no—
168Dic14—44no—
169Dic14—84no—
170Dic14—82no—
171C7⋊C8—44no—
172C22×Q8C22⋊D711yescharacteristic
173C4⋊Q8—72no—
174C4⋊Q8—74no—
175C4⋊Q8—72no—
176C4⋊Q8—74no—
177C4⋊Q8—148no—
178C4⋊Q8—144no—
179#125—148no—
180#125—144no—
181#125—142no—
182#125—144no—
183C22⋊Q8—74no—
184C22⋊Q8—144no—
185C22⋊Q8—142no—
186C4×Q8—74no—
187C4×Q8—78no—
188C4×Q8—144no—
189C4×Q8—144no—
190C4×Q8—142no—
191C4×Q8—144no—
192C4×Q8C4×D714yes—
193C4×Q8D2814yes—
194#117—144no—
195#117—142no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊D711yescharacteristic
199C2×C4⋊C4C22⋊D712yes—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C22×D712yes—
203C2×C42C22×D711yescharacteristic
204C4⋊C8—142no—
205C4⋊C8—144no—
206C22⋊C8—74no—
207#97—144no—
208C4×C8—142no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95C22×D711yescharacteristic
214#95C22×D712yes—
215#95C22×D711yescharacteristic
216C22×C7—21no—
217C22×C7—21no—
218C22×C7C22×C22⋊C411yescharacteristic
219C22×C7C2×#13711yescharacteristic
220C22×C7C2×#13711yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—24no—
224C4×C7—24no—
225C4×C7—22no—
226C4×C7—42no—
227C4×C7—81no—
228C4×C7—82no—
229C4×C7#41914yes—
230Dic7—44no—
231Dic7—44no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—74no—
235C2×Q8—142no—
236C2×Q8—144no—
237C2×Q8(C2×D7)⋊C414yes—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4C2×C22⋊D712yes—
242C22×C4C2×C22⋊D712yes—
243C22×C4C2×C22⋊D711yescharacteristic
244C22×C4C2×C22⋊D711yescharacteristic
245C22×C4C23×D711yescharacteristic
246C2×C8—144no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—74no—
250C4⋊C4—144no—
251C4⋊C4—148no—
252C4⋊C4—144no—
253C4⋊C4—142no—
254C4⋊C4—144no—
255C4⋊C4—142no—
256C4⋊C4—142no—
257C4⋊C4—144no—
258C4⋊C4—144no—
259C4⋊C4(C2×D7)⋊C418yes—
260C4⋊C4(C2×D7)⋊C414yes—
261C22⋊C4—144no—
262C42—44no—
263C42—42no—
264C42—148no—
265C42—144no—
266C42C2×C4×D714yes—
267C42C2×C4×D712yes—
268C42C2×D2812yes—
269C42C2×D2814yes—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7#288411yescharacteristic
273C2×C7C2×#41911yescharacteristic
274C2×C7#290111yescharacteristic
275C23C22×C22⋊D711yescharacteristic
276Q8—28no—
277Q8—284no—
278Q8—564no—
279Q8—562no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—144no—
295C2×C4—144no—
296C2×C4C2×(C2×D7)⋊C414yes—
297C2×C4C2×(C2×D7)⋊C412yes—
298C2×C4C2×(C2×D7)⋊C414yes—
299C2×C4C2×(C2×D7)⋊C412yes—
300C2×C4#651412yes—
301C2×C4#651412yes—
302C2×C4C22×C4×D711yescharacteristic
303C2×C4C22×D2811yescharacteristic
304C8—284no—
305C7#1763211yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22C22×(C2×D7)⋊C411yescharacteristic
309C22C2×#651411yescharacteristic
310C22C2×#651411yescharacteristic
311C4—24no—
312C4—24no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—82no—
319C4—284no—
320C4—284no—
321C4#8977614yes—
322C2—21no—
323C2—21no—
324C2C2×#8977611yescharacteristic
325C2#1173998111yescharacteristic
326C2#1174000111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2356C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#2356D7
hypocenterC7#17632
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
10. #90261C4semidirect
12. Q8×Dic14C4semidirect
305. C7#17632semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#23562
#117398812
#117273202
#117273002
#117272252
#117271872
#176327