Finite Group Databasebeta

← #13242829

Subgroups — #13242829

Order 1792.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#11733843C211yesmaximal, characteristic
2C7×#2071C211yesmaximal, characteristic
3#11727325C211yesmaximal, characteristic, complemented
4#11727323C211yesmaximal, characteristic, complemented
5#11725445C211yesmaximal, characteristic
6#11725284C211yesmaximal, characteristic, complemented
7#11725284C211yesmaximal, characteristic, complemented
8C2×#6498—21no—
9C2×C7×#102—21no—
10C7×#390C2211yescharacteristic
11C7×#384C2211yescharacteristic
12#89755—21no—
13C2×#6381—21no—
14#89725—21no—
15#89725—21no—
16#89722—21no—
17#89722—21no—
18#89700C2211yescharacteristic
19#89700C2211yescharacteristic
20C7×#340—21no—
21C7×#336C2211yescharacteristic
22#89324—21no—
23#89324—21no—
24#89324—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7788—71nomaximal, Sylow
28C22×C7×D4D411yescharacteristic
29C23×Dic7—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—21no—
37C2×C42×C7C2311yescharacteristic
38C2×C22⋊Dic7—21no—
39C2×C22⋊Dic7—21no—
40C2×C22⋊Dic7—21no—
41#6498—42no—
42#6498—42no—
43C2×C4⋊Dic7—21no—
44C2×C4⋊Dic7D411yescharacteristic
45C2×C4⋊Dic7D411yescharacteristic
46C2×C4⋊Dic7D411yescharacteristic
47C2×C4⋊Dic7D411yescharacteristic
48C2×Dic7⋊C4—21no—
49C2×Dic7⋊C4—21no—
50C22×C7⋊C8—21no—
51C22×C7⋊C8—21no—
52C7×#102—42no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58#6381—42no—
59#2724—71no—
60#2099—71no—
61#2083—71no—
62#2071D711yescharacteristic
63#1271—71no—
64#1271—71no—
65#1270—71no—
66C24×C7—21no—
67C2×C7×D4—21no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4D812yes—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7—21no—
80C22×Dic7—42no—
81C22×Dic7—81no—
82C22×Dic7—81no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C42×C7—41no—
97C22⋊Dic7—42no—
98C22⋊Dic7—42no—
99C22⋊Dic7—42no—
100C4⋊Dic7—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7—41no—
105C4⋊Dic7—41no—
106C4⋊Dic7—81no—
107Dic7⋊C4—41no—
108Dic7⋊C4—41no—
109Dic7⋊C4—41no—
110Dic7⋊C4—41no—
111C2×C7⋊C8—22no—
112C2×C7⋊C8—41no—
113C2×C7⋊C8—41no—
114C2×C7⋊C8—41no—
115C2×C7⋊C8—41no—
116C2×C7⋊C8—41no—
117C2×#123—141no—
118C2×#107—141no—
119C2×#102—21no—
120#398—141no—
121#397—141no—
122#397—141no—
123#390C2×D711yescharacteristic
124#389—71no—
125#389—71no—
126#385—141no—
127#385—141no—
128#384C2×D711yescharacteristic
129#340—21no—
130#340—141no—
131#340—141no—
132#340—141no—
133#336—71no—
134#336—71no—
135#336C2×D711yescharacteristic
136C23×C7—22no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7#12011yescharacteristic
144C7×D4—42no—
145C7×D4—81no—
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—81no—
156C2×C4×C7—81no—
157C2×C4×C7—81no—
158C2×C4×C7C2×D811yescharacteristic
159C2×C4×C7#13511yescharacteristic
160C2×C4×C7#13511yescharacteristic
161C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C7#13711yescharacteristic
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—84no—
177C7×C8—81no—
178C7⋊C8—42no—
179C7⋊C8—81no—
180C22×D4C22⋊D711yescharacteristic
181C23×C4—141no—
182C22×C8—21no—
183C22×C8—141no—
184C22×C8—141no—
185#123—282no—
186#123—282no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—141no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4C22⋊D711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—141no—
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C42C22×D711yescharacteristic
204#107—282no—
205#102—42no—
206#95—141no—
207#95—141no—
208#95—141no—
209#95—141no—
210#95—141no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7#44711yescharacteristic
219C22×C7#44911yescharacteristic
220C22×C7#45211yescharacteristic
221C22×C7#45411yescharacteristic
222C22×C7#45511yescharacteristic
223C22×C7#45611yescharacteristic
224C22×C7#45811yescharacteristic
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—81no—
232C4×C7—161no—
233Dic7—81no—
234Dic7—81no—
235Dic7—81no—
236Dic7—81no—
237C24—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4#114112yes—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—282no—
249C22×C4—561no—
250C22×C4—561no—
251C22×C4D4×D711yescharacteristic
252C22×C4D4×D711yescharacteristic
253C22×C4C2×C22⋊D711yescharacteristic
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—142no—
258C2×C8—281no—
259C2×C8—281no—
260C2×C8—281no—
261C2×C8—281no—
262C2×C8—281no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—281no—
274C4⋊C4—281no—
275C4⋊C4—281no—
276C4⋊C4—281no—
277C4⋊C4—281no—
278C4⋊C4—281no—
279C4⋊C4—561no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—282no—
285C22⋊C4—282no—
286C22⋊C4—282no—
287C42—41no—
288C2×C7—82no—
289C2×C7#160711yescharacteristic
290C2×C7#160911yescharacteristic
291C2×C7#161511yescharacteristic
292C2×C7#162511yescharacteristic
293C2×C7#164011yescharacteristic
294C2×C7#164711yescharacteristic
295C2×C7#166411yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#650011yescharacteristic
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—281no—
319C2×C4—281no—
320C2×C4—281no—
321C2×C4—281no—
322C2×C4—281no—
323C2×C4—281no—
324C2×C4—281no—
325C2×C4—281no—
326C2×C4—281no—
327C2×C4—281no—
328C2×C4—281no—
329C2×C4—281no—
330C2×C4—564no—
331C2×C4#647411yescharacteristic
332C2×C4#647511yescharacteristic
333C2×C4#647811yescharacteristic
334C2×C4#647911yescharacteristic
335C2×C4C2×#114111yescharacteristic
336C2×C4#651211yescharacteristic
337C8—81no—
338C8—282no—
339C8—561no—
340C7#778811yescharacteristic, complemented, Sylow
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#8980811yescharacteristic
349C22#8982511yescharacteristic
350C22#8983311yescharacteristic
351C22#8993011yescharacteristic
352C22#8994511yescharacteristic
353C22#8997311yescharacteristic
354C22#8997711yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—161no—
363C4—561no—
364C4—561no—
365C4—561no—
366C4—561no—
367C2—82no—
368C2#1172746211yescharacteristic
369C2#1172877211yescharacteristic
370C2#1172880211yescharacteristic
371C2#1172892011yescharacteristic
372C2#1172892411yescharacteristic
373C2#1172896011yescharacteristic
374C2#1172900111yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2071C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenter#2071D7
hypocenterC7#7788
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #11727325C2semidirect
4. #11727323C2semidirect
6. #11725284C2semidirect
7. #11725284C2semidirect
340. C7#7788semidirect

Maximal subgroups

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

subgroupindex
#117338432
C7×#20712
#117273252
#117273232
#117254452
#117252842
#77887