Finite Group Databasebeta

← #13271219

Subgroups — #13271219

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3067C211yesmaximal, characteristic
2#11741225C211yesmaximal, characteristic, complemented
3#11729834C211yesmaximal, characteristic, complemented
4C2×#89361C211yesmaximal, characteristic, complemented
5#11729769C211yesmaximal, characteristic, complemented
6#11725539C211yesmaximal, characteristic, complemented
7#11725536C211yesmaximal, characteristic, complemented
8C7×#575—21no—
9C2×C7×#135C2211yescharacteristic
10C2×C7×#131C2211yescharacteristic
11C7×#473C412yescomplemented
12C7×#472C412yescomplemented
13C7×#471—22no—
14C2×C7×#108C2211yescharacteristic
15C7×#413—21no—
16#90004—21no—
17#89912—22no—
18#89912—22no—
19C2×#6393C2211yescharacteristic
20C2×#6391C2211yescharacteristic
21C2×#6386—22no—
22C2×#6386C2211yescharacteristic
23C2×#6386C2211yescharacteristic
24#89362—22no—
25#89361—22no—
26#89349—21no—
27#13070—71nomaximal, Sylow
28C2×C7×#41—21no—
29C2×C7×#41D411yescharacteristic
30C7×#137—22no—
31C7×#137—41no—
32C7×AGL(1,ℤ/8)—22no—
33C7×AGL(1,ℤ/8)—41no—
34C7×#135—21no—
35C7×#135—21no—
36C7×#135—21no—
37C7×#135—21no—
38C7×#135—42no—
39C2×C7×Q16C2×C411yescharacteristic
40C2×C7×SD16—21no—
41C2×C7×D8C2×C411yescharacteristic
42C7×#131—21no—
43C7×#131—21no—
44C7×#131C2×C412yes—
45C7×#131D412yes—
46C2×C7×M16—21no—
47C2×C7×M16—21no—
48C2×C7×M16D411yescharacteristic
49C22×C7×C8—21no—
50C22×C7×C8C2311yescharacteristic
51C2×C4⋊Dic7—21no—
52C2×#1136—21no—
53C7×#108—21no—
54C7×#108C2×C412yes—
55C7×#101—22no—
56C7×#100—22no—
57#6409—41no—
58#6406—41no—
59#6393—21no—
60#6393—21no—
61#6391—21no—
62#6391—21no—
63#6386—24no—
64#6386—21no—
65#6386—21no—
66#6386—21no—
67#6386D412yes—
68C2×C7⋊C16—24no—
69C2×C7⋊C16—21no—
70C2×C7⋊C16D412yes—
71#3407—71no—
72#3067D711yescharacteristic
73C2×#360—71no—
74#2143—71no—
75#1815—71no—
76#1388—71no—
77#1383—71no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41—41no—
81C7×#41—41no—
82C7×#41—41no—
83C7×#41—41no—
84C7×#41C22⋊C412yes—
85C7×#41SD1612yes—
86C2×C7×Q8—21no—
87C2×C7×Q8D811yescharacteristic
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—41no—
91C2×C7×D4D811yescharacteristic
92C22×C4×C7—21no—
93C22×C4×C7—41no—
94C22×C4×C7C2×D411yescharacteristic
95C22×Dic7—41no—
96C7×Q16—21no—
97C7×Q16—21no—
98C7×Q16—42no—
99C7×SD16—42no—
100C7×SD16—42no—
101C7×SD16—41no—
102C7×SD16—41no—
103C7×D8—21no—
104C7×D8—21no—
105C7×D8—42no—
106C7×M16—21no—
107C7×M16—22no—
108C7×M16—22no—
109C7×M16—22no—
110C7×M16—41no—
111C7×M16C22⋊C412yes—
112C2×C7×C8—21no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—41no—
116C2×C7×C8C22×C411yescharacteristic
117C2×C7×C8C2×D411yescharacteristic
118C2×C7×C8C2×D411yescharacteristic
119C2×C7×C8C2×D411yescharacteristic
120C2×C7×C8#4111yescharacteristic
121C2×C7×C8#4111yescharacteristic
122C4⋊Dic7—41no—
123C4⋊Dic7—41no—
124#1136—41no—
125#1136—41no—
126C2×C7⋊C8—41no—
127C7⋊C16—22no—
128C7⋊C16—22no—
129#575—21no—
130C2×#135C2×D711yescharacteristic
131C2×#131C2×D711yescharacteristic
132#504—142no—
133#504—142no—
134C2×M32—71no—
135C2×M32—71no—
136C2×M32—142no—
137#473Dic712yescomplemented
138#472Dic712yescomplemented
139#471—22no—
140C2×#108—71no—
141C2×#108C2×D711yescharacteristic
142C2×#106—71no—
143#417—141no—
144#413—21no—
145#360—142no—
146#359—142no—
147#343—141no—
148C23×C7—21no—
149C23×C7—41no—
150C23×C7#13511yescharacteristic
151C7×Q8—21no—
152C7×Q8—41no—
153C7×Q8—41no—
154C7×Q8#10212yes—
155C7×D4—22no—
156C7×D4—22no—
157C7×D4—21no—
158C7×D4—41no—
159C7×D4—41no—
160C7×D4—81no—
161C7×D4—81no—
162C7×D4#10212yes—
163C2×C4×C7—21no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7C2×C22⋊C411yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7C2×SD1611yescharacteristic
174C2×C4×C7#13511yescharacteristic
175C2×Dic7—81no—
176C2×Dic7—161no—
177C7×C8—22no—
178C7×C8—42no—
179C7×C8C4×D411yescharacteristic
180C7×C8C4×D411yescharacteristic
181C7×C8#12011yescharacteristic
182C7×C8#12311yescharacteristic
183C7⋊C8—81no—
184C2×#41—21no—
185C2×#41C22⋊D711yescharacteristic
186#137—22no—
187#137—41no—
188AGL(1,ℤ/8)—22no—
189AGL(1,ℤ/8)—41no—
190#135—21no—
191#135—21no—
192#135—21no—
193#135—21no—
194#135—42no—
195C2×Q16C2×Dic711yescharacteristic
196C2×SD16—21no—
197C2×D8C2×Dic711yescharacteristic
198#131—21no—
199#131—21no—
200#131C2×Dic712yes—
201#131C22⋊D712yes—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—141no—
205C2×M16C22⋊D711yescharacteristic
206C22×C8—21no—
207C22×C8C22×D711yescharacteristic
208C2×C4⋊C4—141no—
209M32—72no—
210M32—144no—
211M32—141no—
212M32—141no—
213M32—141no—
214C2×C16—72no—
215C2×C16—144no—
216C2×C16—141no—
217#108—21no—
218#108—141no—
219#108—141no—
220#108C2×Dic712yes—
221#106—141no—
222#106—141no—
223#103—281no—
224#102—281no—
225#101—22no—
226#100—22no—
227C22×C7—22no—
228C22×C7—41no—
229C22×C7—81no—
230C22×C7—81no—
231C22×C7—161no—
232C22×C7#41611yescharacteristic
233C22×C7#46511yescharacteristic
234C22×C7#46611yescharacteristic
235C4×C7—22no—
236C4×C7—81no—
237C4×C7#38611yescharacteristic
238C4×C7C2×#10211yescharacteristic
239C4×C7#46511yescharacteristic
240C4×C7#46611yescharacteristic
241Dic7—161no—
242#41—21no—
243#41—21no—
244#41—41no—
245#41—41no—
246#41—41no—
247#41—41no—
248#41#114212yes—
249#41C22⋊Dic712yes—
250C2×Q8—21no—
251C2×Q8#114111yescharacteristic
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4#114111yescharacteristic
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—281no—
259C22×C4C2×C22⋊D711yescharacteristic
260Q16—21no—
261Q16—21no—
262Q16—42no—
263SD16—42no—
264SD16—42no—
265SD16—41no—
266SD16—41no—
267D8—21no—
268D8—21no—
269D8—42no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—281no—
276M16—281no—
277M16C22⋊Dic712yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—281no—
283C2×C8D4×D711yescharacteristic
284C2×C8D4×D711yescharacteristic
285C2×C8#115911yescharacteristic
286C2×C8#115911yescharacteristic
287C2×C8C22×Dic711yescharacteristic
288C2×C8C2×C22⋊D711yescharacteristic
289C4⋊C4—281no—
290C4⋊C4—281no—
291C16—142no—
292C16—142no—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7—21no—
296C2×C7—42no—
297C2×C7—161no—
298C2×C7#189311yescharacteristic
299C23—21no—
300C23—41no—
301C23#651311yescharacteristic
302Q8—21no—
303Q8—41no—
304Q8—41no—
305Q8#640612yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—41no—
310D4—41no—
311D4—81no—
312D4—81no—
313D4#640612yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—561no—
321C2×C4—1121no—
322C2×C4C2×#114111yescharacteristic
323C2×C4C2×#114211yescharacteristic
324C2×C4#650211yescharacteristic
325C2×C4#650211yescharacteristic
326C2×C4#651311yescharacteristic
327C2×C4C2×C22⋊Dic711yescharacteristic
328C8—22no—
329C8—42no—
330C8—561no—
331C8D4×Dic711yescharacteristic
332C8D4×Dic711yescharacteristic
333C8#649811yescharacteristic
334C8#650011yescharacteristic
335C7#1307011yescharacteristic, complemented, Sylow
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22#8981011yescharacteristic
342C22#8981311yescharacteristic
343C22#9000511yescharacteristic
344C4—22no—
345C4—81no—
346C4—1121no—
347C4#8981011yescharacteristic
348C4#8981311yescharacteristic
349C4C2×#640611yescharacteristic
350C4#8999211yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#1172943511yescharacteristic
subgroupG/H
centerC2#11729435
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
Fitting, prime radicalC7×#3067C2
socleC2×C7#1893
cosocleC22×C7×C8C23
hypercenter#3067D7
hypocenterC7#13070
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
2. #11741225C2semidirect
3. #11729834C2semidirect
4. C2×#89361C2semidirect
5. #11729769C2semidirect
6. #11725539C2semidirect
7. #11725536C2semidirect
11. C7×#473C4semidirect
12. C7×#472C4semidirect
137. #473Dic7semidirect
138. #472Dic7semidirect
335. C7#13070semidirect

Maximal subgroups

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

subgroupindex
C7×#30672
#117412252
#117298342
C2×#893612
#117297692
#117255392
#117255362
#130707