Finite Group Databasebeta

← #501×D101 #27181647

Subgroups — #501×D101 #27181647

Order 12928.

384 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D101C211yesmaximal, characteristic
2#125×D101C211yesmaximal, characteristic
3#20265587C211yesmaximal, characteristic, complemented
4#20265586C211yesmaximal, characteristic, complemented
5#20265585C212yesmaximal, complemented
6#20265584C212yesmaximal, complemented
7#20265583C211yesmaximal, characteristic, complemented
8#107×D101C212yesmaximal
9#106×D101C212yesmaximal
10#97×D101C211yesmaximal, characteristic
11#501×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101—21no—
13C2×C8×D101C2212yes—
14#15692255C2211yescharacteristic
15#15692254—21no—
16#15692254C2212yes—
17#15692253—21no—
18#15692247C2211yescharacteristic
19#15692246C2211yescharacteristic
20#15692246C2211yescharacteristic
21#15692245—21no—
22#15692245—21no—
23C4⋊C4×D101—21no—
24C4⋊C4×D101—21no—
25C4⋊C4×D101—21no—
26C4⋊C4×D101C2211yescharacteristic
27C4⋊C4×D101C2211yescharacteristic
28C4⋊C4×D101C2212yes—
29C42×D101C2211yescharacteristic
30#15692189C2212yes—
31#15692188C2212yes—
32#15692188C2212yes—
33#15692187C2212yes—
34#15692186C2212yes—
35#15692186C2212yes—
36#15692166C2211yescharacteristic
37#15692165C2212yes—
38C4⋊Q8×C101C2211yescharacteristic
39#125×C101C2211yescharacteristic
40#107×C101C2212yes—
41#106×C101C2212yes—
42#97×C101C2211yescharacteristic
43C2×Dic202—21no—
44C2×Dic202—22no—
45Q8×D101—44no—
46C2×C4×D101—22no—
47C2×C4×D101—21no—
48C2×C4×D101—21no—
49C2×C4×D101D411yescharacteristic
50C2×C4×D101D411yescharacteristic
51C2×C4×D101C2311yescharacteristic
52C2×#11717579C2312yes—
53C8×D101Q818yescomplemented
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101—21no—
57C4⋊Dic101C2311yescharacteristic
58C4⋊Dic101C2312yes—
59C4⋊Dic101C2311yescharacteristic
60#13108053—21no—
61#13108053—21no—
62#13108053—21no—
63#13108053—21no—
64#13108053—22no—
65#13108053—21no—
66#13108053—21no—
67#13108053—22no—
68C4×Dic101D411yescharacteristic
69C4×Dic101D411yescharacteristic
70C4×Dic101C2311yescharacteristic
71C2×Q8×C101—21no—
72C2×C8×C101C2312yes—
73C4⋊C4×C101—21no—
74C4⋊C4×C101—21no—
75C4⋊C4×C101—21no—
76C4⋊C4×C101C2311yescharacteristic
77C4⋊C4×C101C2312yes—
78C4⋊C4×C101C2311yescharacteristic
79C42×C101C2311yescharacteristic
80C22×D101C2×D411yescharacteristic
81Q8×C101—42no—
82C2×C4×C101—21no—
83C2×C4×C101—21no—
84C2×C4×C101—22no—
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2411yescharacteristic
88C2×Dic101—21no—
89C2×Dic101—21no—
90C2×Dic101—22no—
91C2×Dic101C2×D411yescharacteristic
92C2×Dic101C2×D411yescharacteristic
93C2×Dic101C2×D411yescharacteristic
94C4×D101—22no—
95C4×D101—22no—
96C4×D101—42no—
97C4×D101—44no—
98C4×D101—42no—
99C4×D101C2×Q814yes—
100Dic202—44no—
101Dic202—42no—
102C8×C101C2×Q814yescomplemented
103#11717579C2×Q814yescomplemented
104C22×C101C22×D411yescharacteristic
105C2×D101C4⋊Q812yes—
106C2×D101AGL(1,ℤ/8)12yes—
107C2×D101#13712yes—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101—42no—
111C4×C101—41no—
112C4×C101—41no—
113C4×C101C22×Q812yes—
114Dic101—21no—
115Dic101—21no—
116Dic101—42no—
117Dic101—41no—
118Dic101—41no—
119Dic101C4⋊Q812yes—
120C2×C101C2×C4⋊Q811yescharacteristic
121C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C101C2×#13711yescharacteristic
123D101#50114yesdirect product factor
124C2×#501—1011nomaximal, Sylow
125C101C2×#50111yescharacteristic, complemented, Sylow
126C2×C4⋊Q8—1011no—
127C2×#125—1011no—
128#501—1011no—
129#501—1011no—
130#501—1012no—
131#501—1012no—
132#501—1011no—
133#501D10111yescharacteristic, direct product factor
134C2×#107—1012no—
135C2×#106—1012no—
136C2×#97—1011no—
137C22×Q8—2021no—
138C22×C8—1012no—
139C4⋊Q8—1012no—
140C4⋊Q8—1011no—
141C4⋊Q8—2021no—
142C4⋊Q8—2021no—
143C4⋊Q8C2×D10111yescharacteristic
144#125—1011no—
145#125—1011no—
146#125—1011no—
147#125—2021no—
148#125—2021no—
149#125C2×D10111yescharacteristic
150C2×C4⋊C4—1011no—
151C2×C4⋊C4—1011no—
152C2×C4⋊C4—1012no—
153C2×C4⋊C4—2021no—
154C2×C4⋊C4—2021no—
155C2×C4⋊C4—2021no—
156C2×C42—1011no—
157#107—1012no—
158#107—1012no—
159#107—1012no—
160#107C2×D10112yes—
161#106—1012no—
162#106—1012no—
163#106—1012no—
164#106C2×D10112yes—
165#97—1012no—
166#97—1011no—
167#97C2×D10111yescharacteristic
168C2×Q8—21no—
169C2×Q8—4044no—
170C2×Q8—2022no—
171C2×Q8—2021no—
172C22×C4—1011no—
173C22×C4—1011no—
174C22×C4—1011no—
175C22×C4—2021no—
176C22×C4—2021no—
177C22×C4—2022no—
178C2×C8—1012no—
179C2×C8—1018no—
180C2×C8C22×D10112yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—1011no—
185C4⋊C4—1012no—
186C4⋊C4—1011no—
187C4⋊C4—2021no—
188C4⋊C4—2021no—
189C4⋊C4—2022no—
190C4⋊C4—2021no—
191C4⋊C4—2021no—
192C4⋊C4—2022no—
193C4⋊C4—2021no—
194C4⋊C4—2021no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4—2021no—
198C4⋊C4C22×D10111yescharacteristic
199C4⋊C4C22×D10112yes—
200C4⋊C4C22×D10111yescharacteristic
201C42—1011no—
202C42—1011no—
203C42—1011no—
204C42C22×D10111yescharacteristic
205C23—1011no—
206Q8—42no—
207Q8—4042no—
208Q8—4044no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—1011no—
213C2×C4—1014no—
214C2×C4—1011no—
215C2×C4—1011no—
216C2×C4—4044no—
217C2×C4—4042no—
218C2×C4—4042no—
219C2×C4—2022no—
220C2×C4—2022no—
221C2×C4—2021no—
222C2×C4—2021no—
223C2×C4—2022no—
224C2×C4D4×D10111yescharacteristic
225C2×C4D4×D10111yescharacteristic
226C2×C4C23×D10111yescharacteristic
227C8—1014no—
228C8Q8×D10114yescomplemented
229C22—1012no—
230C22—1012no—
231C22—1012no—
232C22C2×D4×D10111yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—1012no—
239C4—4042no—
240C4—4041no—
241C4—4041no—
242C4—2021no—
243C4—2021no—
244C4C2×Q8×D10112yes—
245C2—1014no—
246C2C4⋊Q8×D10111yescharacteristic
247C2AGL(1,ℤ/8)×D10111yescharacteristic
248C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#501×C101C2
socleC22×C101C22×D4
prime radicalC22×D101C2×D4
hypercenter#501D101
hypocenterC101C2×#501
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #20265587C2semidirect
4. #20265586C2semidirect
5. #20265585C2semidirect
6. #20265584C2semidirect
7. #20265583C2semidirect
11. #501×C101C2semidirect
53. C8×D101Q8semidirect
102. C8×C101C2×Q8semidirect
103. #11717579C2×Q8semidirect
123. D101#501direct (N × H)
125. C101C2×#501semidirect
133. #501D101direct (N × H)
228. C8Q8×D101semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1012
#125×D1012
#202655872
#202655862
#202655852
#202655842
#202655832
#107×D1012
#106×D1012
#97×D1012
#501×C1012
C2×#501101