Finite Group Databasebeta

← #354×D101 #27172791

Subgroups — #354×D101 #27172791

Order 12928.

284 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 57 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#124×D101C211yesmaximal, characteristic
2#99×D101C212yesmaximal
3#20264972C211yesmaximal, characteristic, complemented
4#20264971C212yesmaximal, complemented
5#354×C101C211yesmaximal, characteristic, complemented
6C2×Q8×D101C411yescharacteristic, complemented
7C2×D4×D101C2211yescharacteristic
8#15692244C411yescharacteristic, complemented
9#15692243C411yescharacteristic, complemented
10#15692242C2211yescharacteristic
11#15692241—22no—
12C22⋊C4×D101—22no—
13C22⋊C4×D101—22no—
14C42×D101C411yescharacteristic, complemented
15#15692170C2212yes—
16#15692169—22no—
17#124×C101C2211yescharacteristic
18#99×C101C2212yes—
19C23×D101D412yes—
20C2×Dic202—21no—
21C2×Dic202C2×C411yescharacteristic, complemented
22Q8×D101—41no—
23Q8×D101—41no—
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101D412yes—
26D4×D101—41no—
27D4×D101—41no—
28C2×C4×D101—21no—
29C2×C4×D101—21no—
30C2×C4×D101—42no—
31C2×C4×D101C2×C411yescharacteristic
32C22⋊Dic101—22no—
33C22⋊Dic101—22no—
34#13108051—22no—
35#13108051—22no—
36#13108051—42no—
37C4×Dic101—21no—
38C4×Dic101C2×C411yescharacteristic, complemented
39C2×Q8×C101C2×C411yescharacteristic, complemented
40C2×D4×C101C2311yescharacteristic
41C22⋊C4×C101—22no—
42C22⋊C4×C101—22no—
43C42×C101C2×C411yescharacteristic, complemented
44C23×C101C2×D412yes—
45C22×D101—22no—
46C22×D101—22no—
47C22×D101—44no—
48C22×D101C22⋊C411yescharacteristic
49C22×D101C22⋊C412yes—
50Q8×C101—41no—
51D4×C101—41no—
52C2×C4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—42no—
55C2×C4×C101C22×C411yescharacteristic
56C22⋊D101—42no—
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—42no—
60C2×Dic101C22⋊C411yescharacteristic
61D404—41no—
62C4×D101—21no—
63C4×D101—21no—
64C4×D101—42no—
65C4×D101—41no—
66C4×D101—41no—
67C4×D101—84no—
68Dic202—42no—
69Dic202—41no—
70C22×C101—22no—
71C22×C101—42no—
72C22×C101C2×C22⋊C411yescharacteristic
73C2×D101—22no—
74C2×D101—22no—
75C2×D101—44no—
76C2×D101—42no—
77C2×D101—82no—
78C2×D101#9911yescharacteristic
79C2×D101#9911yescharacteristic
80C4×C101—21no—
81C4×C101—41no—
82C4×C101—41no—
83C4×C101—82no—
84Dic101—21no—
85Dic101—41no—
86Dic101—41no—
87Dic101—82no—
88C2×C101—21no—
89C2×C101—42no—
90C2×C101C2×#9911yescharacteristic
91D101—21no—
92D101—42no—
93D101#35412yesdirect product factor
94C2×#354—1011nomaximal, Sylow
95C101C2×#35411yescharacteristic, complemented, Sylow
96C2×#124—1011no—
97C2×#99—1012no—
98#354—1011no—
99#354—1012no—
100#354D10111yescharacteristic, direct product factor
101C22×Q8—1011no—
102C22×D4—1011no—
103#124—1011no—
104#124—1011no—
105#124—1011no—
106#124—2022no—
107#124C2×D10111yescharacteristic
108C2×C22⋊C4—2022no—
109C2×C22⋊C4—2022no—
110C2×C42—1011no—
111#99—1012no—
112#99—2022no—
113#99C2×D10112yes—
114C24—1012no—
115C2×Q8—1011no—
116C2×Q8—4041no—
117C2×Q8—4041no—
118C2×Q8—2021no—
119C2×Q8C4×D10111yescharacteristic, complemented
120C2×D4—1012no—
121C2×D4—1011no—
122C2×D4—4041no—
123C2×D4—4041no—
124C2×D4C22×D10111yescharacteristic
125C22×C4—1011no—
126C22×C4—4042no—
127C22×C4—2021no—
128C22×C4—2021no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—4042no—
132C22⋊C4—2022no—
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C42—1011no—
137C42—2021no—
138C42C4×D10111yescharacteristic, complemented
139C23—1011no—
140C23—1012no—
141C23—4044no—
142C23—2022no—
143C23—2022no—
144C23D4×D10112yes—
145Q8—41no—
146Q8—4041no—
147Q8—4042no—
148D4—41no—
149D4—4041no—
150D4—4042no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—8084no—
155C2×C4—1011no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—4042no—
159C2×C4—4042no—
160C2×C4—2021no—
161C2×C4—2021no—
162C2×C4—2021no—
163C2×C4—2021no—
164C2×C4C2×C4×D10111yescharacteristic
165C22—22no—
166C22—42no—
167C22—8082no—
168C22—1011no—
169C22—1011no—
170C22—4042no—
171C22—4044no—
172C22—2022no—
173C22—2022no—
174C22C22⋊C4×D10111yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—8082no—
180C4—4041no—
181C4—4041no—
182C4—2021no—
183C2—21no—
184C2—42no—
185C2—1012no—
186C2—4042no—
187C2—2021no—
188C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#354×C101C2
socleC2×C101C2×#99
prime radicalC2×D4×D101C22
cosocleC2×D4×C101C23
hypercenter#354D101
hypocenterC101C2×#354
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #20264972C2semidirect
4. #20264971C2semidirect
5. #354×C101C2semidirect
6. C2×Q8×D101C4semidirect
8. #15692244C4semidirect
9. #15692243C4semidirect
14. C42×D101C4semidirect
21. C2×Dic202C2×C4semidirect
38. C4×Dic101C2×C4semidirect
39. C2×Q8×C101C2×C4semidirect
43. C42×C101C2×C4semidirect
93. D101#354direct (N × H)
95. C101C2×#354semidirect
100. #354D101direct (N × H)
119. C2×Q8C4×D101semidirect
138. C42C4×D101semidirect

Maximal subgroups

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

subgroupindex
#124×D1012
#99×D1012
#202649722
#202649712
#354×C1012
C2×#354101