Finite Group Databasebeta

← #13271998

Subgroups — #13271998

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#11741179C211yesmaximal, characteristic
2C7×#2112C211yesmaximal, characteristic, complemented
3#11729717C211yesmaximal, characteristic, complemented
4#11729192C211yesmaximal, characteristic, complemented
5#11729191C211yesmaximal, characteristic, complemented
6#11729179C211yesmaximal, characteristic, complemented
7#11729177C211yesmaximal, characteristic, complemented
8C23×C7×C8C2211yescharacteristic
9#90482C411yescharacteristic
10C2×#6464C2211yescharacteristic
11C2×C7×M32C2211yescharacteristic
12C2×C7×M32C2211yescharacteristic
13#89979C411yescharacteristic
14#89883—22no—
15#89882C411yescharacteristic
16C8×C22⋊D7—22no—
17#89874C411yescharacteristic
18#89873C2211yescharacteristic
19C2×#6386C2211yescharacteristic
20C2×#6386C2211yescharacteristic
21C7×C22⋊C16—22no—
22C22⋊(C7⋊C16)—22no—
23#89310—22no—
24#89307—22no—
25#89306—22no—
26#89303—22no—
27#13200—71nomaximal, Sylow
28C23×C4×C7C2×C411yescharacteristic
29C2×#1157C2×C411yescharacteristic
30C22×C7×C8—21no—
31C22×C7×C8—21no—
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C22×C7×C8C2×C411yescharacteristic
37C22×C7×C8C2311yescharacteristic
38#6516—21no—
39#6493—21no—
40#6492—21no—
41C4×C22⋊D7—22no—
42#6489C2×C411yescharacteristic
43#6487—21no—
44C2×#1136C2×C411yescharacteristic
45C2×#1136C2×C411yescharacteristic
46#6464—21no—
47#6464—21no—
48#6464—21no—
49#6464—21no—
50C2×#1131—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#2117—71no—
73#2112—71no—
74#2112D711yescharacteristic, complemented
75#2110—71no—
76C24×C7M1611yescharacteristic
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7M1611yescharacteristic
84C22×C4×C7C22×C411yescharacteristic
85C2×C22⋊D7—22no—
86C2×C22⋊D7—21no—
87#1157—21no—
88#1157—21no—
89#1157—21no—
90#1157—21no—
91C2×D28M1611yescharacteristic
92C2×C4×D7—21no—
93C2×Dic14M1611yescharacteristic
94C2×C7×C8—21no—
95C2×C7×C8—21no—
96C2×C7×C8—21no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
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—
115#1136—21no—
116#1136—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8C22⋊C412yes—
119#1131—41no—
120#1131—41no—
121C8×D7—41no—
122C8×D7—41no—
123C7×C16—22no—
124C7×C16—41no—
125C7⋊C16—22no—
126C7⋊C16—41no—
127C2×#131—71no—
128C23×C8C2×D711yescharacteristic
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—142no—
142#350—142no—
143C22⋊C16—22no—
144C22⋊C16—142no—
145#347—142no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—42no—
150C23×C7C2×M1611yescharacteristic
151C22×D7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7C2×C22⋊C411yescharacteristic
160C2×C4×C7C4×D411yescharacteristic
161C2×C4×C7C4×D411yescharacteristic
162C2×C4×C7C2×M1611yescharacteristic
163C2×C4×C7#13111yescharacteristic
164C2×C4×C7#13111yescharacteristic
165C22⋊D7—44no—
166C22⋊D7—41no—
167C22⋊D7—41no—
168C2×Dic7—22no—
169C2×Dic7—21no—
170D28—21no—
171D28—21no—
172C4×D7—41no—
173C4×D7—41no—
174Dic14—21no—
175Dic14—21no—
176C7×C8—41no—
177C7×C8#12111yescharacteristic
178C7×C8#12111yescharacteristic
179C7×C8#12411yescharacteristic
180C7×C8#12711yescharacteristic
181C7⋊C8—22no—
182C7⋊C8—41no—
183C2×#41—71no—
184C23×C4C4×D711yescharacteristic
185#131—141no—
186#131—141no—
187#131—141no—
188#131—141no—
189C2×M16—71no—
190C2×M16—71no—
191C2×M16—141no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—141no—
199C22×C8C4×D711yescharacteristic
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—22no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7#40711yescharacteristic
233C22×C7#43511yescharacteristic
234C22×C7#43511yescharacteristic
235C2×D7—42no—
236C2×D7—41no—
237C4×C7—41no—
238C4×C7#39011yescharacteristic
239C4×C7#40811yescharacteristic
240C4×C7#43511yescharacteristic
241C4×C7#43511yescharacteristic
242Dic7—42no—
243Dic7—41no—
244C24#113111yescharacteristic
245#41—141no—
246#41—141no—
247#41—141no—
248#41—141no—
249C2×Q8—71no—
250C2×D4—71no—
251C2×D4—142no—
252C2×D4—141no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—141no—
260C22×C4#113111yescharacteristic
261C22×C4C2×C4×D711yescharacteristic
262M16—141no—
263M16—141no—
264M16—141no—
265M16—141no—
266M16—281no—
267M16—281no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—72no—
276C2×C8—141no—
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—42no—
299C2×C7#195511yescharacteristic
300D7—81no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—42no—
305C23—141no—
306C23C2×#113111yescharacteristic
307Q8—141no—
308Q8—141no—
309D4—141no—
310D4—141no—
311D4—284no—
312D4—281no—
313D4—281no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—142no—
322C2×C4—141no—
323C2×C4—281no—
324C2×C4—281no—
325C2×C4D7×C22⋊C411yescharacteristic
326C2×C4(C22⋊D7)⋊C411yescharacteristic
327C2×C4#646411yescharacteristic
328C2×C4D7×M1611yescharacteristic
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—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—282no—
348C22—281no—
349C22#8949811yescharacteristic
350C22#8950211yescharacteristic
351C22#8988311yescharacteristic
352C4—41no—
353C4—282no—
354C4—281no—
355C4#8949911yescharacteristic
356C4#8950011yescharacteristic
357C4#8972911yescharacteristic
358C4#8989911yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—561no—
364C2#1172767111yescharacteristic
subgroupG/H
centerC8#6503
commutatorC22×C4×C7C22×C4
FrattiniC22×C8C22×D7
FittingC7×#2112C2
socleC2×C7#1955
prime radical#90482C4
cosocleC22×C7×C8C23
hypercenter#2112D7
hypocenterC7#13200
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C7×#2112C2semidirect
3. #11729717C2semidirect
4. #11729192C2semidirect
5. #11729191C2semidirect
6. #11729179C2semidirect
7. #11729177C2semidirect
57. C2×C7×C16D4semidirect
68. C2×C7⋊C16D4semidirect
74. #2112D7semidirect
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
#117411792
C7×#21122
#117297172
#117291922
#117291912
#117291792
#117291772
#132007