Finite Group Databasebeta

← #13247828

Subgroups — #13247828

Order 1792.

374 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 105 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#11736484C211yesmaximal, characteristic
2#11728536C211yesmaximal, characteristic
3#11728503C211yesmaximal, characteristic, complemented
4#11728248C211yesmaximal, characteristic, complemented
5#11728230C211yesmaximal, characteristic
6C7×#1301C211yesmaximal, characteristic, complemented
7#11725513C211yesmaximal, characteristic, complemented
8C2×#6435C2211yescharacteristic
9C7×#431C2211yescharacteristic
10C7×#423C2211yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12#89898—21no—
13M16×Dic7—21no—
14C2×(C2×D7)⋊C8—21no—
15#89697C2211yescharacteristic
16#89696C2211yescharacteristic
17C2×C4⋊(C7⋊C8)C2211yescharacteristic
18#89612—22no—
19#89612—22no—
20#89345—21no—
21#89274—22no—
22#89264C412yes—
23#89263C412yes—
24#89252—22no—
25#8863—71nomaximal, Sylow
26C22×C4×D7D411yescharacteristic
27C2×C7×M16—21no—
28C22×C7×C8—21no—
29C2×C42×C7C2311yescharacteristic
30C2×(C2×D7)⋊C4C2×C411yescharacteristic
31C2×(C2×D7)⋊C4Q811yescharacteristic
32C2×Dic7⋊C4C2×C411yescharacteristic
33C2×Dic7⋊C4Q811yescharacteristic
34C2×C4×Dic7D411yescharacteristic
35C2×#1136—21no—
36C22×C7⋊C8—21no—
37#6435—21no—
38#6435—21no—
39#6435—21no—
40#6435—21no—
41C7×C4⋊C8—21no—
42C7×C4⋊C8—21no—
43C7×C4⋊C8—21no—
44C7×C4⋊C8—21no—
45C7×C4⋊C8D412yes—
46C7×C22⋊C8C2×C412yes—
47C4×C7×C8—21no—
48C22⋊(C7⋊C8)C2×C412yes—
49(C2×D7)⋊C8—24no—
50(C2×D7)⋊C8—42no—
51#6389—22no—
52C8×Dic7—22no—
53C4⋊(C7⋊C8)—21no—
54C4⋊(C7⋊C8)—21no—
55C4⋊(C7⋊C8)—21no—
56C4⋊(C7⋊C8)—21no—
57C4⋊(C7⋊C8)D412yes—
58C4×C7⋊C8—21no—
59#2921—71no—
60#1798—71no—
61#1571—71no—
62#1570—71no—
63#1458—71no—
64#1301—71no—
65#1301D711yescharacteristic, complemented
66C23×D7C4⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×Q811yescharacteristic
70C22×Dic7C4⋊C411yescharacteristic
71C22×Dic7C4⋊C411yescharacteristic
72C22×Dic7C4⋊C411yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C7×M16—41no—
80C7×M16—41no—
81C2×C7×C8—22no—
82C2×C7×C8—22no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C42×C7C2×D411yescharacteristic
87C42×C7C2×D411yescharacteristic
88C42×C7#4111yescharacteristic
89C42×C7#4111yescharacteristic
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100#1136—41no—
101#1136—41no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8—22no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106C2×C7⋊C8—41no—
107C2×#117—71no—
108#433—142no—
109#433—142no—
110#431—71no—
111#431C2×D711yescharacteristic
112#423—71no—
113#423C2×D711yescharacteristic
114C2×C4⋊C8—71no—
115C2×C4⋊C8C2×D711yescharacteristic
116#407—141no—
117C2×C22⋊C8—141no—
118C4×M16—141no—
119#336—141no—
120#335—142no—
121#334—142no—
122#324—72no—
123#323—72no—
124C23×C7C2×C4⋊C411yescharacteristic
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—41no—
129C22×D7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7C4×D411yescharacteristic
133C2×C4×C7C4×D411yescharacteristic
134C2×C4×C7#12011yescharacteristic
135C2×C4×C7C22⋊Q811yescharacteristic
136C2×C4×C7C22⋊Q811yescharacteristic
137C2×C4×C7#12311yescharacteristic
138C2×C4×C7#13511yescharacteristic
139C2×C4×C7#13511yescharacteristic
140C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C7#13711yescharacteristic
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C4×D7—41no—
150C4×D7—41no—
151C4×D7—41no—
152C4×D7—41no—
153C7×C8—42no—
154C7×C8—42no—
155C7×C8—81no—
156C7⋊C8—42no—
157C7⋊C8—42no—
158C7⋊C8—81no—
159C23×C4—71no—
160C2×M16—21no—
161C2×M16—141no—
162C22×C8—21no—
163C22×C8—141no—
164#117—141no—
165#117—141no—
166#117—141no—
167#117—141no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C22⋊C4—71no—
171C2×C22⋊C4—71no—
172C2×C42—71no—
173C2×C42C22×D711yescharacteristic
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—72no—
179C4⋊C8—141no—
180C4⋊C8—141no—
181C4⋊C8—141no—
182C4⋊C8—141no—
183C4⋊C8D2812yes—
184C22⋊C8—72no—
185C22⋊C8—144no—
186C22⋊C8—282no—
187C22⋊C8C4×D712yes—
188#97—142no—
189C4×C8—21no—
190C4×C8—142no—
191C4×C8—141no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7#38511yescharacteristic
195C22×C7#42711yescharacteristic
196C22×C7#42811yescharacteristic
197C2×D7—42no—
198C2×D7—41no—
199C2×D7—41no—
200C2×D7—41no—
201C2×D7—81no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7#44911yescharacteristic
207C4×C7#45211yescharacteristic
208C4×C7#48211yescharacteristic
209C4×C7#48511yescharacteristic
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C24—71no—
214C22×C4—71no—
215C22×C4—71no—
216C22×C4—71no—
217C22×C4—141no—
218C22×C4—141no—
219C22×C4—141no—
220C22×C4—141no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4C2×C4×D711yescharacteristic
224C22×C4D4×D711yescharacteristic
225C22×C4Q8×D711yescharacteristic
226M16—41no—
227M16—41no—
228M16—281no—
229M16—281no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—142no—
236C2×C8—142no—
237C2×C8—281no—
238C2×C8—281no—
239C2×C8—281no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C42—141no—
249C42—141no—
250C42C2×D2811yescharacteristic
251C42#115711yescharacteristic
252C42D4×D711yescharacteristic
253C42#115911yescharacteristic
254C2×C7—21no—
255C2×C7—21no—
256C2×C7#185411yescharacteristic
257C2×C7#185811yescharacteristic
258C2×C7#186811yescharacteristic
259D7—81no—
260C23—141no—
261C23—141no—
262C23—141no—
263C23—281no—
264C23—281no—
265C23D7×C4⋊C411yescharacteristic
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—141no—
271C2×C4—141no—
272C2×C4—141no—
273C2×C4—281no—
274C2×C4—281no—
275C2×C4—281no—
276C2×C4—281no—
277C2×C4—281no—
278C2×C4—281no—
279C2×C4C4×D2811yescharacteristic
280C2×C4(C22⋊D7)⋊C411yescharacteristic
281C2×C4#644511yescharacteristic
282C2×C4#644611yescharacteristic
283C2×C4#645911yescharacteristic
284C2×C4#646011yescharacteristic
285C2×C4#647411yescharacteristic
286C2×C4#647911yescharacteristic
287C2×C4#647911yescharacteristic
288C2×C4#648111yescharacteristic
289C8—42no—
290C8—42no—
291C8—81no—
292C8—282no—
293C8—282no—
294C8—561no—
295C7#886311yescharacteristic, complemented, Sylow
296C22—21no—
297C22—21no—
298C22—282no—
299C22—281no—
300C22—281no—
301C22—281no—
302C22—561no—
303C22#8944011yescharacteristic
304C22#8963211yescharacteristic
305C22#8965311yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—281no—
311C4—281no—
312C4—281no—
313C4#8955111yescharacteristic
314C4#8955311yescharacteristic
315C4#8958211yescharacteristic
316C4#8958711yescharacteristic
317C2—21no—
318C2—21no—
319C2—561no—
320C2#1172663011yescharacteristic
321C2#1172669911yescharacteristic
322C2#1172678811yescharacteristic
subgroupG/H
centerC2×C4#6445
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1301C2
socleC22×C7#385
prime radicalC23×D7C4⋊C4
cosocleC2×C42×C7C23
hypercenter#1301D7
hypocenterC7#8863
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #11728503C2semidirect
4. #11728248C2semidirect
6. C7×#1301C2semidirect
7. #11725513C2semidirect
65. #1301D7semidirect
295. C7#8863semidirect

Maximal subgroups

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

subgroupindex
#117364842
#117285362
#117285032
#117282482
#117282302
C7×#13012
#117255132
#88637