Finite Group Databasebeta

← #13250208

Subgroups — #13250208

Order 1792.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×M16C211yesmaximal, characteristic, complemented
2C2×#89799C211yesmaximal, characteristic
3#11729376C211yesmaximal, characteristic, complemented
4#11728256C211yesmaximal, characteristic
5#11727630C211yesmaximal, characteristic
6#11725517C211yesmaximal, characteristic, complemented
7#11725474C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C22×#1143C412yes—
10C2×#6437C2211yescharacteristic
11C7×#446—21no—
12C7×#446—22no—
13C7×C8×Q8—21no—
14C7×#423—21no—
15C7×#423C2211yescharacteristic
16C4×C7×M16—21no—
17C4×C7×M16C2211yescharacteristic
18C2×#6397—21no—
19C2×#6397—21no—
20#89799—21no—
21#89799—21no—
22#89799—21no—
23#89799—21no—
24C2×#6383C412yes—
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27C2×C4⋊(C7⋊C8)C2211yescharacteristic
28Q8⋊(C7⋊C8)—21no—
29Q8⋊(C7⋊C8)—21no—
30#89280—22no—
31#9254—71nomaximal, Sylow
32C22×C7×Q8C2×C411yescharacteristic
33C22×D28—21no—
34C22×D28—21no—
35C22×D28C2×C412yes—
36C2×C7×M16—21no—
37C2×C7×M16D411yescharacteristic
38C2×C7×M16D411yescharacteristic
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8D411yescharacteristic
42C4×C7×Q8D411yescharacteristic
43C4×C7×Q8D411yescharacteristic
44C4×C7×Q8D411yescharacteristic
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4C2×C411yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7C2311yescharacteristic
49C2×#1143—22no—
50C2×#1143—22no—
51C2×#1143—44no—
52C2×#1136—21no—
53C22×C7⋊C8C2×C412yes—
54#6437—21no—
55#6437—21no—
56#6437—21no—
57#6437—21no—
58C7×C4⋊C8—22no—
59C7×C4⋊C8—22no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8D412yes—
62C7×#97—22no—
63C7×#97—21no—
64C4×C7×C8—22no—
65C4×C7×C8—21no—
66C22⋊(C7⋊C8)—21no—
67#6397—42no—
68#6397—42no—
69#6383—22no—
70#6383—22no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)—21no—
73C4⋊(C7⋊C8)—21no—
74C4⋊(C7⋊C8)—21no—
75C4⋊(C7⋊C8)—21no—
76C4×C7⋊C8—21no—
77C2×#460—71no—
78Q8×M16D711yescharacteristic, complemented
79#1965—71no—
80#1545—71no—
81#1487—71no—
82#1292—71no—
83#1283—71no—
84C23×D7—22no—
85C2×C7×Q8—21no—
86C2×C7×Q8—22no—
87C2×C7×Q8—41no—
88C2×C7×Q8C22⋊C411yescharacteristic
89C2×C7×Q8C22⋊C411yescharacteristic
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7C22×C411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C2×D28—22no—
96C2×D28—22no—
97C2×D28—42no—
98C2×D28—42no—
99C2×D28—41no—
100C2×D28—41no—
101C2×D28—41no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×D28—41no—
106C2×D28—41no—
107C2×D28—82no—
108C7×M16—21no—
109C7×M16—21no—
110C7×M16—21no—
111C7×M16—41no—
112C7×M16—41no—
113C7×M16SD1612yescomplemented
114C2×C7×C8—21no—
115C2×C7×C8—21no—
116C2×C7×C8—22no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4C22⋊C411yescharacteristic
123C7×C4⋊C4C22⋊C411yescharacteristic
124C42×C7—21no—
125C42×C7—21no—
126C42×C7—21no—
127C42×C7—21no—
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7C2×D411yescharacteristic
131C42×C7#4111yescharacteristic
132#1143—44no—
133#1143—44no—
134#1136—41no—
135#1136—41no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—42no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C22×SD16—72no—
141C2×#127—71no—
142C2×C4×Q8C2×D711yescharacteristic
143#460—141no—
144#460—141no—
145#460—141no—
146#460—141no—
147#446—21no—
148#446—22no—
149C8×Q8—21no—
150#431—71no—
151#423—21no—
152#423—71no—
153#423C2×D711yescharacteristic
154C2×C4⋊C8—71no—
155C2×#102—72no—
156C2×#100—141no—
157C2×#100—141no—
158C4×M16—21no—
159C4×M16C2×D711yescharacteristic
160#331—142no—
161Q8⋊C8—141no—
162Q8⋊C8—141no—
163C23×C7C4×D411yescharacteristic
164C22×D7—42no—
165C22×D7—42no—
166C22×D7—42no—
167C22×D7—82no—
168C22×D7—82no—
169C7×Q8—21no—
170C7×Q8—21no—
171C7×Q8—21no—
172C7×Q8—42no—
173C7×Q8#10012yescomplemented
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—81no—
183C2×C4×C7C2×C22⋊C411yescharacteristic
184C2×C4×C7C4×D411yescharacteristic
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12311yescharacteristic
189C2×C4×C7C2×SD1611yescharacteristic
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C7#13711yescharacteristic
193D28—42no—
194D28—42no—
195D28—82no—
196D28—82no—
197D28—82no—
198D28—81no—
199D28—81no—
200D28—81no—
201D28—81no—
202C7×C8—22no—
203C7×C8—41no—
204C7×C8—42no—
205C7⋊C8—42no—
206C7⋊C8—81no—
207C7⋊C8—81no—
208C22×Q8C4×D711yescharacteristic
209C22×D4—72no—
210C22×D4—141no—
211C22×D4—141no—
212C2×SD16—142no—
213C2×SD16—142no—
214C2×SD16—284no—
215C2×M16—21no—
216C2×M16—141no—
217C2×M16C22⋊D711yescharacteristic
218C2×M16C22⋊D711yescharacteristic
219C22×C8—72no—
220#127—141no—
221#127—141no—
222#127—141no—
223#127—141no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8D2811yescharacteristic
227C4×Q8D2811yescharacteristic
228C4×Q8C22⋊D711yescharacteristic
229C4×Q8C22⋊D711yescharacteristic
230C2×C4⋊C4—21no—
231C2×C4⋊C4C4×D711yescharacteristic
232C2×C42—21no—
233C2×C42C22×D711yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—141no—
238C4⋊C8—141no—
239C4⋊C8—141no—
240C4⋊C8—141no—
241C4⋊C8—141no—
242C4⋊C8C22⋊D712yes—
243#102—142no—
244#102—142no—
245#100—282no—
246#100—282no—
247C22⋊C8—141no—
248#97—22no—
249#97—21no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—141no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7#38611yescharacteristic
256C22×C7C4×SD1611yescharacteristic
257C22×C7#44011yescharacteristic
258C2×D7—82no—
259C2×D7—82no—
260C2×D7—82no—
261C2×D7—162no—
262C2×D7—162no—
263C4×C7—21no—
264C4×C7—22no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—41no—
268C4×C7—41no—
269C4×C7—41no—
270C4×C7C2×#10011yescharacteristic
271C4×C7#41311yescharacteristic
272C4×C7#44811yescharacteristic
273C4×C7#44911yescharacteristic
274C4×C7#46011yescharacteristic
275C4×C7#46411yescharacteristic
276C24—142no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8(C2×D7)⋊C411yescharacteristic
281C2×Q8(C2×D7)⋊C411yescharacteristic
282C2×D4—142no—
283C2×D4—142no—
284C2×D4—282no—
285C2×D4—282no—
286C2×D4—281no—
287C2×D4—281no—
288C2×D4—281no—
289C2×D4—281no—
290C2×D4—281no—
291C2×D4—281no—
292C2×D4—281no—
293C2×D4—281no—
294C2×D4—562no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×C4×D711yescharacteristic
298C22×C4#115711yescharacteristic
299C22×C4C2×C22⋊D711yescharacteristic
300SD16—284no—
301SD16—284no—
302M16—21no—
303M16—21no—
304M16—21no—
305M16—41no—
306M16—41no—
307M16—281no—
308M16—281no—
309M16#114312yescomplemented
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—142no—
314C2×C8—282no—
315C2×C8—281no—
316C2×C8—281no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4(C2×D7)⋊C411yescharacteristic
323C4⋊C4(C2×D7)⋊C411yescharacteristic
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42C2×D2811yescharacteristic
329C42#115711yescharacteristic
330C42C2×C22⋊D711yescharacteristic
331C42C2×C22⋊D711yescharacteristic
332C2×C7—21no—
333C2×C7—21no—
334C2×C7#188011yescharacteristic
335C2×C7#190011yescharacteristic
336C2×C7#190311yescharacteristic
337D7—162no—
338C23—282no—
339C23—282no—
340C23—282no—
341C23—562no—
342C23—562no—
343C23C4×C22⋊D711yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—21no—
347Q8—42no—
348Q8#639712yescomplemented
349D4—282no—
350D4—282no—
351D4—562no—
352D4—562no—
353D4—562no—
354D4—561no—
355D4—561no—
356D4—561no—
357D4—561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4C2×(C2×D7)⋊C411yescharacteristic
368C2×C4C4×C22⋊D711yescharacteristic
369C2×C4#649211yescharacteristic
370C2×C4#649311yescharacteristic
371C2×C4#649311yescharacteristic
372C2×C4C2×#114311yescharacteristic
373C2×C4#650511yescharacteristic
374C2×C4#651211yescharacteristic
375C2×C4#651311yescharacteristic
376C2×C4#651611yescharacteristic
377C8—22no—
378C8—41no—
379C8—42no—
380C8—282no—
381C8—561no—
382C8—561no—
383C7#925411yescharacteristic, complemented, Sylow
384C22—21no—
385C22—21no—
386C22—562no—
387C22—562no—
388C22—562no—
389C22—1122no—
390C22—1122no—
391C22C4×#114311yescharacteristic
392C22#8979811yescharacteristic
393C22#8998511yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4#8979911yescharacteristic
402C4#8980011yescharacteristic
403C4C2×#639711yescharacteristic
404C4#8990411yescharacteristic
405C4#8999911yescharacteristic
406C4#9001011yescharacteristic
407C2—21no—
408C2—21no—
409C2—1122no—
410C2#1172938711yescharacteristic
411C2#1172975611yescharacteristic
412C2#1172977711yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×Q8×M16C2
socleC22×C7#386
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenterQ8×M16D7
hypocenterC7#9254
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
1. C7×Q8×M16C2semidirect
3. #11729376C2semidirect
6. #11725517C2semidirect
7. #11725474C2semidirect
78. Q8×M16D7semidirect
113. C7×M16SD16semidirect
173. C7×Q8#100semidirect
309. M16#1143semidirect
348. Q8#6397semidirect
383. C7#9254semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×M162
C2×#897992
#117293762
#117282562
#117276302
#117255172
#117254742
#92547