Finite Group Databasebeta

← #13509032

Subgroups — #13509032

Order 1792.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C7×#3185C211yesmaximal, characteristic, complemented
2#11735986C211yesmaximal, characteristic
3#11729104C212yesmaximal, complemented
4#11729071C212yesmaximal, complemented
5#11728845C212yesmaximal
6#11728844C212yesmaximal, complemented
7#11728671C214yesmaximal, complemented
8#11727618C211yesmaximal, characteristic, complemented
9C7×#533—21no—
10#90439C2211yescharacteristic
11#90436—21no—
12C7×#498C2211yescharacteristic, complemented
13C7×#494C2211yescharacteristic, complemented
14C7×#487—21no—
15C7×#470—21no—
16C7×#470C2212yes—
17C7×#441—21no—
18C7×#441C2212yescomplemented
19C4×C7×M16C2211yescharacteristic
20#89951C2214yes—
21#89897C2211yescharacteristic
22C2×#6379—21no—
23#89489—21no—
24#89489C2212yes—
25#89472—21no—
26#89472C2212yes—
27#89360C2214yescomplemented
28#89348C2212yes—
29#89283C2214yes—
30#89282C2214yescomplemented
31#89261C2212yescomplemented
32#89258C2212yescomplemented
33#33689—71nomaximal, Sylow
34C2×#1157—21no—
35C2×C7×Q16—21no—
36C2×C7×Q16C2312yes—
37C2×C7×M16D411yescharacteristic
38C2×C7×M16C2311yescharacteristic
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×#125—41no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8—41no—
44C4×C7×Q8—22no—
45C4×C7×Q8—41no—
46C7×#117—41no—
47C2×C42×C7D411yescharacteristic
48C2×#1136—21no—
49#6472—22no—
50#6471—22no—
51#6470—42no—
52D7×M16—42no—
53#6467—21no—
54#6467C2312yes—
55#6464—22no—
56#6464—22no—
57C7×#106—21no—
58C7×#106C2312yes—
59C7×#103—22no—
60C7×#103—22no—
61C7×#97D411yescharacteristic
62C7×#97C2311yescharacteristic
63C4×C7×C8D411yescharacteristic
64C4×C7×C8C2311yescharacteristic
65#6403—24no—
66#6402—24no—
67#6393—21no—
68#6393C2312yes—
69#6386C2314yes—
70#6379—22no—
71#6379—22no—
72#3185D711yescharacteristic, complemented
73#2956—71no—
74#2239—72no—
75#2230—72no—
76#2214—72no—
77#2191—72no—
78#2180—74no—
79#2165—71no—
80C2×C7×Q8—22no—
81C2×C7×Q8—41no—
82C22×C4×C7—21no—
83C22×C4×C7C2×D411yescharacteristic
84C2×C22⋊D7—41no—
85#1157—22no—
86#1157—22no—
87#1157—41no—
88#1157—41no—
89C2×D28—21no—
90C2×C4×D7—41no—
91C2×Dic14—21no—
92C7×Q16—24no—
93C7×Q16—42no—
94C7×M16—22no—
95C7×M16—22no—
96C2×C7×C8C2×D411yescharacteristic
97C2×C7×C8C2×D411yescharacteristic
98C2×C7×C8C2×D411yescharacteristic
99C2×C7×C8C2411yescharacteristic
100C7×C4⋊C4—22no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—42no—
106C7×C22⋊C4—42no—
107C42×C7—21no—
108C42×C7—42no—
109C42×C7C2×D411yescharacteristic
110C42×C7C2×D411yescharacteristic
111#1136—22no—
112#1136—41no—
113C2×C7⋊C8—41no—
114Dic28—21no—
115Dic28#4112yes—
116D56—21no—
117D56#4112yes—
118#1132—22no—
119#1132—22no—
120#1131—42no—
121#1131—42no—
122C8×D7—42no—
123C8×D7—42no—
124C7⋊C16—24no—
125#575—71no—
126#568—141no—
127#533—21no—
128#510—74no—
129#498C2×D711yescharacteristic, complemented
130#494C2×D711yescharacteristic, complemented
131#487—21no—
132#470—21no—
133#470C2×D712yes—
134#444—72no—
135#444—141no—
136#443—72no—
137#443—141no—
138#441—21no—
139#441C2×D712yescomplemented
140#430—71no—
141C2×#104—141no—
142C4×M16C2×D711yescharacteristic
143#365—74no—
144#364—72no—
145#362—74no—
146#360—74no—
147#349—72no—
148#347—72no—
149C23×C7AGL(1,ℤ/8)11yescharacteristic
150C22×D7—41no—
151C7×Q8—42no—
152C7×Q8—42no—
153C7×Q8—82no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C7C22×D411yescharacteristic
163C22⋊D7—42no—
164C22⋊D7—42no—
165C2×Dic7—41no—
166D28—22no—
167D28—41no—
168C4×D7—42no—
169C4×D7—42no—
170Dic14—22no—
171Dic14—41no—
172C7×C8—21no—
173C7×C8—21no—
174C7×C8AGL(1,ℤ/8)12yes—
175C7×C8C2×#4112yes—
176C7⋊C8—42no—
177C2×#41—141no—
178#137—142no—
179AGL(1,ℤ/8)—142no—
180#135—72no—
181#135—141no—
182C2×Q16—21no—
183C2×Q16C22×D712yes—
184#131—142no—
185#131—142no—
186#131—282no—
187C2×M16—141no—
188C2×M16—282no—
189C2×M16C22⋊D711yescharacteristic
190C2×M16C22×D711yescharacteristic
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193#125—41no—
194C22⋊Q8—22no—
195C22⋊Q8—41no—
196C4×Q8—22no—
197C4×Q8—41no—
198#117—41no—
199C2×C42C22⋊D711yescharacteristic
200Q32—144no—
201SD32—144no—
202M32—74no—
203#108—72no—
204#108—141no—
205#106—21no—
206#106C22×D712yes—
207#104—142no—
208#104—142no—
209#103—22no—
210#103—22no—
211#97C22⋊D711yescharacteristic
212#97C22×D711yescharacteristic
213C4×C8C22⋊D711yescharacteristic
214C4×C8C22×D711yescharacteristic
215C22×C7—21no—
216C22×C7—41no—
217C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
218C2×D7—42no—
219C2×D7—161no—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—82no—
224C4×C7—82no—
225C4×C7#52511yescharacteristic
226C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
227Dic7—42no—
228#41—142no—
229#41—142no—
230#41—281no—
231#41—281no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—141no—
235C2×D4—141no—
236C2×D4—281no—
237C22×C4—21no—
238C22×C4—281no—
239C22×C4C2×C22⋊D711yescharacteristic
240Q16—24no—
241Q16—42no—
242Q16—72no—
243Q16—141no—
244SD16—142no—
245SD16—142no—
246D8—72no—
247D8—141no—
248M16—22no—
249M16—22no—
250M16—142no—
251M16—282no—
252M16—281no—
253M16—282no—
254C2×C8—282no—
255C2×C8—281no—
256C2×C8—282no—
257C2×C8C2×C22⋊D711yescharacteristic
258C2×C8C2×C22⋊D711yescharacteristic
259C2×C8C2×C22⋊D711yescharacteristic
260C2×C8C23×D711yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C22⋊C4—42no—
268C42—21no—
269C42—42no—
270C42C2×C22⋊D711yescharacteristic
271C42C2×C22⋊D711yescharacteristic
272C16—144no—
273C2×C7—21no—
274C2×C7—41no—
275C2×C7#310911yescharacteristic
276D7—82no—
277C23—281no—
278C23#649511yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—82no—
282Q8—142no—
283Q8—281no—
284D4—142no—
285D4—281no—
286D4—282no—
287D4—282no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—282no—
296C2×C4—281no—
297C2×C4—282no—
298C2×C4#649511yescharacteristic
299C2×C4C22×C22⋊D711yescharacteristic
300C8—21no—
301C8—21no—
302C8—282no—
303C8#651212yes—
304C8D7×#4112yes—
305C7#3368911yescharacteristic, complemented, Sylow
306C22—21no—
307C22—41no—
308C22—282no—
309C22—1121no—
310C22C2×#649511yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—282no—
317C4C2×#651211yescharacteristic
318C4#9051911yescharacteristic
319C2—21no—
320C2—41no—
321C2—562no—
322C2#1174074511yescharacteristic
subgroupG/H
centerC2#11740745
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3185C2
socleC2×C7#3109
prime radical#90439C22
hypercenter#3185D7
hypocenterC7#33689
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
1. C7×#3185C2semidirect
3. #11729104C2semidirect
4. #11729071C2semidirect
6. #11728844C2semidirect
7. #11728671C2semidirect
8. #11727618C2semidirect
12. C7×#498C22semidirect
13. C7×#494C22semidirect
18. C7×#441C22semidirect
27. #89360C22semidirect
30. #89282C22semidirect
31. #89261C22semidirect
32. #89258C22semidirect
72. #3185D7semidirect
129. #498C2×D7semidirect
130. #494C2×D7semidirect
139. #441C2×D7semidirect
305. C7#33689semidirect

Maximal subgroups

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

subgroupindex
C7×#31852
#117359862
#117291042
#117290712
#117288452
#117288442
#117286712
#117276182
#336897