Finite Group Databasebeta

← #13272000

Subgroups — #13272000

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741239C211yesmaximal, characteristic
2C7×#2117C211yesmaximal, characteristic, complemented
3#11729760C211yesmaximal, characteristic, complemented
4#11729185C211yesmaximal, characteristic, complemented
5#11729185C211yesmaximal, characteristic, complemented
6#11729179C211yesmaximal, characteristic, complemented
7#11729177C211yesmaximal, characteristic, complemented
8C2×C7×#131C2211yescharacteristic
9#90519C411yescharacteristic
10C22×C8×D7C2211yescharacteristic
11C2×C7×M32C2211yescharacteristic
12C2×C7×M32C2211yescharacteristic
13#90006C411yescharacteristic
14#89899—22no—
15#89898C411yescharacteristic
16#89891C411yescharacteristic
17C8×C22⋊D7—22no—
18#89873C2211yescharacteristic
19C2×#6386C2211yescharacteristic
20C2×#6386C2211yescharacteristic
21C7×#350—22no—
22#89356—22no—
23#89307—22no—
24#89306—22no—
25(C2×D7)⋊C16—22no—
26(C2×D7)⋊C16—22no—
27#13200—71nomaximal, Sylow
28C2×C7×#41C2×C411yescharacteristic
29C22×C4×D7C2×C411yescharacteristic
30C7×#131—21no—
31C7×#131—21no—
32C7×#131—21no—
33C7×#131—21no—
34C2×C7×M16—21no—
35C2×C7×M16C2×C411yescharacteristic
36C22×C7×C8—21no—
37C22×C7×C8C2311yescharacteristic
38#6509—21no—
39#6501—21no—
40#6500—21no—
41#6498—21no—
42C4×C22⋊D7—22no—
43#6489C2×C411yescharacteristic
44C2×#1136C2×C411yescharacteristic
45C22×C7⋊C8C2×C411yescharacteristic
46C2×C8×D7—21no—
47C2×C8×D7—21no—
48C2×C8×D7—21no—
49C2×C8×D7—21no—
50C2×C8×D7—21no—
51C2×C8×D7—21no—
52C7×M32—21no—
53C7×M32—21no—
54C7×M32—21no—
55C7×M32—21no—
56C2×C7×C16—21no—
57C2×C7×C16D412yescomplemented
58C22⋊(C7⋊C8)—22no—
59(C2×D7)⋊C8—22no—
60#6389D412yes—
61Dic7⋊C8—22no—
62C8×Dic7D412yes—
63#6386—21no—
64#6386—21no—
65#6386—21no—
66#6386—21no—
67C2×C7⋊C16—21no—
68C2×C7⋊C16D412yescomplemented
69#2929—71no—
70#2153—71no—
71#2117—71no—
72#2117D711yescharacteristic, complemented
73#2112—71no—
74#2112—71no—
75#2110—71no—
76C23×D7M1611yescharacteristic
77C7×#41—21no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41—21no—
81C2×C7×Q8M1611yescharacteristic
82C2×C7×D4—21no—
83C2×C7×D4M1611yescharacteristic
84C22×C4×C7—21no—
85C22×C4×C7C22×C411yescharacteristic
86C2×C22⋊D7—22no—
87C22×Dic7M1611yescharacteristic
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×C4×D7—21no—
92C2×C4×D7—21no—
93C2×C4×D7—21no—
94C7×M16—21no—
95C7×M16—21no—
96C7×M16—41no—
97C7×M16—41no—
98C2×C7×C8—21no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8C2×D411yescharacteristic
102C2×C7×C8C2×D411yescharacteristic
103C2×C7×C8C2×D411yescharacteristic
104C2×C7×C8C2×D411yescharacteristic
105C2×C7×C8#4111yescharacteristic
106C2×C7×C8#4111yescharacteristic
107C22⋊Dic7—22no—
108C22⋊Dic7—21no—
109(C2×D7)⋊C4—22no—
110C4⋊Dic7—21no—
111Dic7⋊C4—22no—
112C4×Dic7C22⋊C412yes—
113#1136—21no—
114#1136—21no—
115C2×C7⋊C8—21no—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8C22⋊C412yes—
119C8×D7—41no—
120C8×D7—41no—
121C8×D7—41no—
122C8×D7—41no—
123C7×C16—22no—
124C7×C16—41no—
125C7⋊C16—22no—
126C7⋊C16—41no—
127C2×#131C2×D711yescharacteristic
128C23×C8—71no—
129#525—71no—
130C2×M32—71no—
131C2×M32—71no—
132C2×M32C2×D711yescharacteristic
133C2×M32C2×D711yescharacteristic
134#435—142no—
135C8×D4—142no—
136#424—71no—
137#408—71no—
138#407—71no—
139#405—71no—
140#364—142no—
141#350—22no—
142#350—142no—
143C22⋊C16—142no—
144C22⋊C16—142no—
145#347—142no—
146C23×C7—21no—
147C23×C7C2×M1611yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—42no—
152C7×Q8—21no—
153C7×Q8—21no—
154C7×D4—21no—
155C7×D4—21no—
156C7×D4—41no—
157C7×D4—41no—
158C2×C4×C7—21no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7C2×C22⋊C411yescharacteristic
162C2×C4×C7C4×D411yescharacteristic
163C2×C4×C7C4×D411yescharacteristic
164C2×C4×C7C2×M1611yescharacteristic
165C2×C4×C7#13111yescharacteristic
166C2×C4×C7#13111yescharacteristic
167C22⋊D7—44no—
168C2×Dic7—22no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C4×D7—41no—
173C4×D7—41no—
174C4×D7—41no—
175C4×D7—41no—
176C7×C8—41no—
177C7×C8#12111yescharacteristic
178C7×C8#12111yescharacteristic
179C7×C8#12411yescharacteristic
180C7×C8#12711yescharacteristic
181C7⋊C8—22no—
182C7⋊C8—41no—
183C2×#41C4×D711yescharacteristic
184C23×C4—71no—
185#131—21no—
186#131—21no—
187#131—21no—
188#131—21no—
189C2×M16—21no—
190C2×M16—71no—
191C2×M16C4×D711yescharacteristic
192C22×C8—21no—
193C22×C8—71no—
194C22×C8—141no—
195C22×C8—141no—
196C22×C8—141no—
197C22×C8—141no—
198C22×C8—141no—
199C22×C8—141no—
200C22×C8C22×D711yescharacteristic
201#123—141no—
202C22⋊Q8—141no—
203#121—141no—
204#120—141no—
205C4×D4—142no—
206#117—71no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—141no—
212M32—141no—
213M32—141no—
214M32—141no—
215C2×C16—21no—
216C2×C16—72no—
217C2×C16—141no—
218C2×C16C22⋊D712yescomplemented
219C4⋊C8—142no—
220C22⋊C8—142no—
221C22⋊C8—142no—
222#97—72no—
223C4×C8—72no—
224C22×C7—41no—
225C22×C7—42no—
226C22×C7—41no—
227C22×C7#40711yescharacteristic
228C22×C7#43511yescharacteristic
229C22×C7#43511yescharacteristic
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C4×C7—41no—
238C4×C7#39011yescharacteristic
239C4×C7#40811yescharacteristic
240C4×C7#43511yescharacteristic
241C4×C7#43511yescharacteristic
242Dic7—42no—
243Dic7—41no—
244C24—71no—
245#41—21no—
246#41—21no—
247#41—21no—
248#41—21no—
249C2×Q8#113111yescharacteristic
250C2×D4—21no—
251C2×D4—142no—
252C2×D4#113111yescharacteristic
253C22×C4—21no—
254C22×C4—71no—
255C22×C4—141no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C2×C4×D711yescharacteristic
262M16—21no—
263M16—21no—
264M16—41no—
265M16—41no—
266M16—141no—
267M16—141no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—72no—
272C2×C8—141no—
273C2×C8—141no—
274C2×C8—141no—
275C2×C8—281no—
276C2×C8—281no—
277C2×C8—281no—
278C2×C8—281no—
279C2×C8#115711yescharacteristic
280C2×C8D4×D711yescharacteristic
281C2×C8D4×D711yescharacteristic
282C2×C8D4×D711yescharacteristic
283C2×C8#115911yescharacteristic
284C2×C8C2×C22⋊D711yescharacteristic
285C4⋊C4—142no—
286C4⋊C4—141no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—141no—
290C42—72no—
291C16—22no—
292C16—41no—
293C16—142no—
294C16—281no—
295C2×C7—21no—
296C2×C7—21no—
297C2×C7—21no—
298C2×C7—81no—
299C2×C7#195511yescharacteristic
300D7—42no—
301C23—21no—
302C23—141no—
303C23—141no—
304C23—141no—
305C23—282no—
306C23D7×M1611yescharacteristic
307Q8—21no—
308Q8—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—41no—
313D4—284no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—141no—
318C2×C4—142no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—281no—
322C2×C4—281no—
323C2×C4—281no—
324C2×C4—281no—
325C2×C4D7×C22⋊C411yescharacteristic
326C2×C4(C22⋊D7)⋊C411yescharacteristic
327C2×C4C2×#113111yescharacteristic
328C2×C4#646411yescharacteristic
329C2×C4#647011yescharacteristic
330C2×C4C4×C22⋊D711yescharacteristic
331C8—41no—
332C8—142no—
333C8—281no—
334C8#644711yescharacteristic
335C8#644811yescharacteristic
336C8#650211yescharacteristic
337C8#650311yescharacteristic
338C7#1320011yescharacteristic, complemented, Sylow
339C22—41no—
340C22—42no—
341C22—41no—
342C22—142no—
343C22—142no—
344C22—142no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22#8949811yescharacteristic
350C22#8950011yescharacteristic
351C22#8989911yescharacteristic
352C4—41no—
353C4—282no—
354C4—281no—
355C4#8949911yescharacteristic
356C4#8950211yescharacteristic
357C4#8972911yescharacteristic
358C4#8988311yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—282no—
364C2#1172767111yescharacteristic
subgroupG/H
centerC8#6503
commutatorC22×C4×C7C22×C4
FrattiniC22×C8C22×D7
FittingC7×#2117C2
socleC2×C7#1955
prime radical#90519C4
cosocleC22×C7×C8C23
hypercenter#2117D7
hypocenterC7#13200
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. C7×#2117C2semidirect
3. #11729760C2semidirect
4. #11729185C2semidirect
5. #11729185C2semidirect
6. #11729179C2semidirect
7. #11729177C2semidirect
57. C2×C7×C16D4semidirect
68. C2×C7⋊C16D4semidirect
72. #2117D7semidirect
218. C2×C16C22⋊D7semidirect
338. C7#13200semidirect

Maximal subgroups

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

subgroupindex
#117412392
C7×#21172
#117297602
#117291852
#117291792
#117291772
#132007