Finite Group Databasebeta

← #13242301

Subgroups — #13242301

Order 1792.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11733636C211yesmaximal, characteristic
2#11726351C211yesmaximal, characteristic, complemented
3#11726350C211yesmaximal, characteristic, complemented
4C7×#1271C211yesmaximal, characteristic, complemented
5#11725444C211yesmaximal, characteristic, complemented
6#11725285C211yesmaximal, characteristic
7#11725285C211yesmaximal, characteristic
8C2×#6461—21no—
9C7×#389C2211yescharacteristic
10C2×#6389—21no—
11C2×C8×Dic7—21no—
12#89762—21no—
13#89760C411yescharacteristic
14#89760C411yescharacteristic
15#89757C411yescharacteristic
16#89745C411yescharacteristic
17#89711C2211yescharacteristic
18C42×Dic7C2211yescharacteristic
19#89442—21no—
20#89441—21no—
21C7×#336C2211yescharacteristic
22C7×#336C2211yescharacteristic
23#89345—21no—
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7692—71nomaximal, Sylow
28C22×D28C2×C411yescharacteristic
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×(C2×D7)⋊C4—21no—
36C2×(C2×D7)⋊C4—21no—
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7C2×C411yescharacteristic
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7C2×C411yescharacteristic
45C2×C4×Dic7C2×C411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C2×C4×Dic7D411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C22×C7⋊C8—21no—
51C22×C7⋊C8—21no—
52#6461—42no—
53#6461—42no—
54C7×#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57#6389—24no—
58C8×Dic7—24no—
59#2860—71no—
60#1745—71no—
61#1744—71no—
62#1271—71no—
63#1271D711yescharacteristic, complemented
64#1270—71no—
65#1270—71no—
66C23×D7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×Dic7—21no—
73C22×Dic7C22⋊C411yescharacteristic
74C22×Dic7C22⋊C411yescharacteristic
75C22×Dic7C22⋊C411yescharacteristic
76C22×Dic7C22⋊C411yescharacteristic
77C2×D28—21no—
78C2×D28—21no—
79C2×D28—41no—
80C2×D28—41no—
81C2×C7×C8—22no—
82C2×C7×C8—22no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C2×C7×C8—41no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C42×C7—41no—
93(C2×D7)⋊C4—42no—
94(C2×D7)⋊C4—42no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—42no—
100(C2×D7)⋊C4—42no—
101C4⋊Dic7—42no—
102C4⋊Dic7—41no—
103Dic7⋊C4—42no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109C4×Dic7—21no—
110C4×Dic7—22no—
111C4×Dic7—22no—
112C4×Dic7—41no—
113C2×C7⋊C8—22no—
114C2×C7⋊C8—22no—
115C2×C7⋊C8—41no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119C2×#126—141no—
120C2×#97—141no—
121C2×C4×C8—141no—
122#390—71no—
123#390—71no—
124#390—71no—
125#389—71no—
126#389—71no—
127#389C2×D711yescharacteristic
128#388—141no—
129#388—141no—
130#388—141no—
131C43—71no—
132#336—71no—
133#336—71no—
134#336—141no—
135#336—141no—
136#336C2×D711yescharacteristic
137#336C2×D711yescharacteristic
138C23×C7C2×C22⋊C411yescharacteristic
139C22×D7—41no—
140C22×D7—41no—
141C22×D7—41no—
142C22×D7—41no—
143C22×D7—41no—
144C22×D7—41no—
145C22×D7—41no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7#13111yescharacteristic
159C2×C4×C7#13111yescharacteristic
160C2×C4×C7#13511yescharacteristic
161C2×C4×C7#13511yescharacteristic
162C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7#10412yes—
178C2×Dic7#10412yes—
179D28—81no—
180D28—81no—
181C7×C8—42no—
182C7×C8—42no—
183C7⋊C8—42no—
184C7⋊C8—42no—
185C22×D4—71no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—141no—
189C22×C8—141no—
190#126—282no—
191#126—282no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—141no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4C4×D711yescharacteristic
197C2×C4⋊C4C4×D711yescharacteristic
198C2×C22⋊C4—141no—
199C2×C22⋊C4—141no—
200C2×C22⋊C4—141no—
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4—141no—
204C2×C42—71no—
205C2×C42—71no—
206C2×C42—71no—
207C2×C42—71no—
208C2×C42—71no—
209C2×C42—71no—
210C2×C42C22×D711yescharacteristic
211#97—144no—
212C4×C8—144no—
213#95—21no—
214#95—141no—
215#95—141no—
216C22×C7#40811yescharacteristic
217C22×C7#41011yescharacteristic
218C22×C7#41311yescharacteristic
219C22×C7#41611yescharacteristic
220C22×C7#41811yescharacteristic
221C22×C7C2×#10411yescharacteristic
222C22×C7C2×#10411yescharacteristic
223C2×D7—81no—
224C2×D7—81no—
225C2×D7—81no—
226C2×D7—81no—
227C2×D7—81no—
228C2×D7—81no—
229C2×D7—81no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—81no—
237C4×C7—81no—
238Dic7—22no—
239Dic7—22no—
240Dic7—41no—
241Dic7—41no—
242Dic7—81no—
243C24—141no—
244C2×D4—141no—
245C2×D4—141no—
246C2×D4—281no—
247C2×D4—281no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—141no—
255C22×C4C2×C4×D711yescharacteristic
256C22×C4D4×D711yescharacteristic
257C22×C4D4×D711yescharacteristic
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—142no—
265C2×C8—142no—
266C2×C8—281no—
267C2×C8—281no—
268C2×C8—281no—
269C2×C8—281no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—281no—
276C4⋊C4—282no—
277C4⋊C4—282no—
278C22⋊C4—282no—
279C22⋊C4—282no—
280C22⋊C4—282no—
281C22⋊C4—282no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C22⋊C4—281no—
285C22⋊C4—281no—
286C42—41no—
287C42—142no—
288C42—142no—
289C42—141no—
290C42—141no—
291C42—141no—
292C42—141no—
293C42—141no—
294C42—141no—
295C42—281no—
296C2×C7#147311yescharacteristic
297C2×C7#148411yescharacteristic
298C2×C7#149411yescharacteristic
299C2×C7#150011yescharacteristic
300C2×C7#150511yescharacteristic
301C2×C7#152411yescharacteristic
302C2×C7#155311yescharacteristic
303D7—161no—
304C23—281no—
305C23—281no—
306C23—281no—
307C23—281no—
308C23—281no—
309C23—281no—
310C23—281no—
311C23D7×C22⋊C411yescharacteristic
312D4—561no—
313D4—561no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—72no—
327C2×C4—72no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4#646411yescharacteristic
342C2×C4#647011yescharacteristic
343C2×C4#647411yescharacteristic
344C2×C4#647711yescharacteristic
345C2×C4#647911yescharacteristic
346C2×C4#648211yescharacteristic
347C8—42no—
348C8—42no—
349C8—282no—
350C8—282no—
351C7#769211yescharacteristic, complemented, Sylow
352C22—561no—
353C22—561no—
354C22—561no—
355C22—561no—
356C22—561no—
357C22—561no—
358C22—561no—
359C22#8949911yescharacteristic
360C22#8951211yescharacteristic
361C22#8952311yescharacteristic
362C22#8954211yescharacteristic
363C22#8957611yescharacteristic
364C22D7×#10411yescharacteristic
365C22D7×#10411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—81no—
373C4—81no—
374C4—142no—
375C4—142no—
376C4—281no—
377C4—281no—
378C4—561no—
379C2—1121no—
380C2#1172588311yescharacteristic
381C2#1172592511yescharacteristic
382C2#1172599511yescharacteristic
383C2#1172603211yescharacteristic
384C2#1172606211yescharacteristic
385C2#1172611511yescharacteristic
386C2#1172618011yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socleC23×C7C2×C22⋊C4
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1271D7
hypocenterC7#7692
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
2. #11726351C2semidirect
3. #11726350C2semidirect
4. C7×#1271C2semidirect
5. #11725444C2semidirect
63. #1271D7semidirect
351. C7#7692semidirect

Maximal subgroups

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

subgroupindex
#117336362
#117263512
#117263502
C7×#12712
#117254442
#117252852
#76927