Finite Group Databasebeta

← #355×D101 #27172808

Subgroups — #355×D101 #27172808

Order 12928.

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

normal subgroups only

This group has 260 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#101×D101C212yesmaximal, complemented
3#20264974C211yesmaximal, characteristic, complemented
4#20264973C212yesmaximal, complemented
5#355×C101C211yesmaximal, characteristic, complemented
6C2×Q8×D101C2211yescharacteristic
7C2×D4×D101C411yescharacteristic
8M16×D101—22no—
9#15692244C2211yescharacteristic
10#15692243C411yescharacteristic
11#15692242C411yescharacteristic
12#15692241—22no—
13C22⋊C4×D101—22no—
14C42×D101C411yescharacteristic
15#15692174C2212yescomplemented
16#15692173—22no—
17#124×C101C2211yescharacteristic
18#101×C101C2212yescomplemented
19C23×D101—21no—
20C2×Dic202C2×C411yescharacteristic
21C2×Dic202D412yes—
22Q8×D101—41no—
23Q8×D101—41no—
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101—21no—
26D4×D101—41no—
27D4×D101—41no—
28C2×C4×D101—21no—
29C2×C4×D101C2×C411yescharacteristic
30C2×C4×D101D412yes—
31#13108058—22no—
32#13108057—42no—
33C8×D101—42no—
34C22⋊Dic101—22no—
35#13108051—22no—
36#13108051—22no—
37C4×Dic101—21no—
38C4×Dic101C2×C411yescharacteristic
39C2×Q8×C101C2311yescharacteristic
40C2×D4×C101C2×C411yescharacteristic
41M16×C101—22no—
42C22⋊C4×C101—22no—
43C42×C101C2×C411yescharacteristic
44C23×C101—21no—
45C22×D101—21no—
46C22×D101—44no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101C22⋊C411yescharacteristic
50Q8×C101—41no—
51D4×C101—41no—
52C2×C4×C101—21no—
53C2×C4×C101C22×C411yescharacteristic
54C2×C4×C101C2×D412yes—
55C22⋊D101—42no—
56C2×Dic101—21no—
57C2×Dic101C22⋊C411yescharacteristic
58C2×Dic101C22⋊C412yes—
59D404—41no—
60C4×D101—22no—
61C4×D101—22no—
62C4×D101—21no—
63C4×D101—21no—
64C4×D101—42no—
65Dic202—41no—
66Dic202—42no—
67C8×C101—42no—
68#11717579—42no—
69C22×C101—42no—
70C22×C101—41no—
71C22×C101C2×C22⋊C411yescharacteristic
72C2×D101—22no—
73C2×D101—42no—
74C2×D101—41no—
75C2×D101—82no—
76C2×D101—82no—
77C2×D101#9911yescharacteristic
78C2×D101#9911yescharacteristic
79C4×C101—22no—
80C4×C101—21no—
81C4×C101—41no—
82Dic101—22no—
83Dic101—21no—
84Dic101—41no—
85C2×C101—21no—
86C2×C101—81no—
87C2×C101C2×#9911yescharacteristic
88D101—21no—
89D101—81no—
90D101#35512yesdirect product factor
91C2×#355—1011nomaximal, Sylow
92C101C2×#35511yescharacteristic, complemented, Sylow
93C2×#124—1011no—
94C2×#101—1012no—
95#355—1011no—
96#355—1012no—
97#355D10111yescharacteristic, direct product factor
98C22×Q8—1011no—
99C22×D4—1011no—
100C2×M16—2022no—
101#124—1011no—
102#124—1011no—
103#124—1011no—
104#124—2022no—
105#124C2×D10111yescharacteristic
106C2×C22⋊C4—2022no—
107C2×C42—1011no—
108#101—1012no—
109#101—2022no—
110#101C2×D10112yescomplemented
111C24—2021no—
112C2×Q8—1011no—
113C2×Q8—1012no—
114C2×Q8—4041no—
115C2×Q8—4041no—
116C2×Q8C22×D10111yescharacteristic
117C2×D4—1011no—
118C2×D4—4041no—
119C2×D4—4041no—
120C2×D4—2021no—
121C2×D4C4×D10111yescharacteristic
122C22×C4—1011no—
123C22×C4—1012no—
124C22×C4—2021no—
125M16—22no—
126M16—4042no—
127M16—2022no—
128C2×C8—4042no—
129C22⋊C4—22no—
130C22⋊C4—2022no—
131C22⋊C4—2022no—
132C22⋊C4—2022no—
133C42—1011no—
134C42—2021no—
135C42C4×D10111yescharacteristic
136C23—21no—
137C23—1011no—
138C23—4041no—
139C23—4041no—
140C23—4044no—
141C23—2021no—
142Q8—41no—
143Q8—4042no—
144Q8—4041no—
145D4—41no—
146D4—4041no—
147D4—4042no—
148C2×C4—21no—
149C2×C4—1011no—
150C2×C4—1012no—
151C2×C4—4042no—
152C2×C4—2022no—
153C2×C4—2022no—
154C2×C4—2021no—
155C2×C4—2021no—
156C2×C4—2021no—
157C2×C4C2×C4×D10111yescharacteristic
158C2×C4D4×D10112yes—
159C8—42no—
160C8—4042no—
161C22—42no—
162C22—41no—
163C22—8082no—
164C22—8082no—
165C22—1011no—
166C22—1011no—
167C22—4042no—
168C22—4041no—
169C22—2022no—
170C22C22⋊C4×D10111yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—4041no—
175C4—2021no—
176C4—2022no—
177C2—21no—
178C2—81no—
179C2—8081no—
180C2—1012no—
181C2—2021no—
182C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×Q8C22×D101
Fitting#355×C101C2
socleC2×C101C2×#99
prime radicalC2×D4×D101C4
cosocleC2×Q8×C101C23
hypercenter#355D101
hypocenterC101C2×#355
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D101C2semidirect
3. #20264974C2semidirect
4. #20264973C2semidirect
5. #355×C101C2semidirect
15. #15692174C22semidirect
18. #101×C101C22semidirect
90. D101#355direct (N × H)
92. C101C2×#355semidirect
97. #355D101direct (N × H)
110. #101C2×D101semidirect

Maximal subgroups

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

subgroupindex
#124×D1012
#101×D1012
#202649742
#202649732
#355×C1012
C2×#355101