Finite Group Databasebeta

← #28378388

Subgroups — #28378388

Order 13952.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#20296044C211yesmaximal, characteristic, complemented
2#20296030C211yesmaximal, characteristic, complemented
3Q8⋊#11724786C212yesmaximal, complemented
4D4⋊#11724786C212yesmaximal, complemented
5#20295555C211yesmaximal, characteristic
6C2×#13182039—21no—
7#16812766—21no—
8C4⋊#11724786—21no—
9C4⋊#11724786C2212yes—
10#16812756—21no—
11C4×#11724786C2212yes—
12#16812692—21no—
13#16812691C411yescharacteristic
14#16812681—21no—
15#16812675C411yescharacteristic
16#16812670C411yescharacteristic
17Q8×Dic109C2211yescharacteristic
18D4×Dic109C2211yescharacteristic
19#16812659—21no—
20#16812655—21no—
21C42×D109C2211yescharacteristic
22#124×C109C411yescharacteristic
23#13182039—42no—
24C2×#11724786—22no—
25C2×#11724786—22no—
26C2×C22⋊D109—21no—
27C22×Dic109—21no—
28C2×C4×D109C2×C411yescharacteristic
29C2×C4×D109D411yescharacteristic
30C2×C4×D109D411yescharacteristic
31C4⋊Dic109—21no—
32C4⋊Dic109C2×C411yescharacteristic
33C4⋊Dic109C2×C411yescharacteristic
34#13182008—21no—
35#13182008—21no—
36#13182008—21no—
37C22⋊Dic109—21no—
38C22⋊Dic109—21no—
39#13182006—21no—
40C4×Dic109—21no—
41C4×Dic109D411yescharacteristic
42C4×Dic109D411yescharacteristic
43C4×Dic109C2311yescharacteristic
44C2×Q8×C109C2×C411yescharacteristic
45C2×D4×C109C2×C411yescharacteristic
46C22⋊C4×C109—21no—
47C22⋊C4×C109—21no—
48C42×C109C2×C411yescharacteristic
49C23×C109—21no—
50C22×D109C22⋊C411yescharacteristic
51Q8×C109—22no—
52D4×C109—22no—
53C2×C4×C109—21no—
54C2×C4×C109C22⋊C411yescharacteristic
55C2×C4×C109C22⋊C411yescharacteristic
56C2×C4×C109C22×C411yescharacteristic
57C22⋊D109—42no—
58C2×Dic109—21no—
59C2×Dic109—21no—
60C2×Dic109—41no—
61C2×Dic109—41no—
62C2×Dic109C22⋊C411yescharacteristic
63C2×Dic109C2×D411yescharacteristic
64C2×Dic109C2×D411yescharacteristic
65C4×D109—22no—
66C4×D109—21no—
67C4×D109—21no—
68C4×D109—41no—
69#11724786—42no—
70#11724786—42no—
71C22×C109—41no—
72C22×C109—41no—
73C22×C109—41no—
74C22×C109C2×C22⋊C411yescharacteristic
75C2×D109—21no—
76C2×D109—21no—
77C2×D109—21no—
78C4×C109—21no—
79C4×C109—21no—
80C4×C109—41no—
81C4×C109#13112yes—
82Dic109—21no—
83Dic109—21no—
84Dic109—41no—
85Dic109—41no—
86Dic109AGL(1,ℤ/8)11yescharacteristic
87Dic109#13711yescharacteristic
88C2×C109—81no—
89C2×C109#40811yescharacteristic
90C2×C109#41711yescharacteristic
91C2×C109#42111yescharacteristic
92D109—41no—
93#1495—1091nomaximal, Sylow
94C109#149511yescharacteristic, complemented, Sylow
95#529—1091no—
96#431—1091no—
97#423—1091no—
98Q8⋊C8—1092no—
99D4⋊C8—1092no—
100C2×M16—2181no—
101#126—2181no—
102#125—1091no—
103#124#8913711yescharacteristic
104#123—2181no—
105C22⋊Q8—1091no—
106#121—1091no—
107C4×Q8—1091no—
108C4×D4—1091no—
109C4×D4—2181no—
110#117—2181no—
111C2×C42—1091no—
112C4⋊C8—1092no—
113C4⋊C8—2181no—
114C4⋊C8—2181no—
115C22⋊C8—2181no—
116C4×C8—1092no—
117C2×Q8C2×#8913711yescharacteristic
118C2×D4—2181no—
119C2×D4C2×#8913711yescharacteristic
120C22×C4—1091no—
121C22×C4—1091no—
122C22×C4—1091no—
123C22×C4—2181no—
124M16—4362no—
125C2×C8—2182no—
126C2×C8—2182no—
127C4⋊C4—1091no—
128C4⋊C4—1091no—
129C4⋊C4—2181no—
130C4⋊C4—2181no—
131C4⋊C4—2181no—
132C4⋊C4—2181no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—2181no—
136C22⋊C4—2181no—
137C22⋊C4—2181no—
138C42—1091no—
139C42—1091no—
140C42—1091no—
141C42—2181no—
142C42C2×#8913711yescharacteristic
143C23—21no—
144C23—1091no—
145Q8—22no—
146D4—22no—
147D4—4362no—
148C2×C4—21no—
149C2×C4—1091no—
150C2×C4—1091no—
151C2×C4—1091no—
152C2×C4—4361no—
153C2×C4—4361no—
154C2×C4—4361no—
155C2×C4—2181no—
156C2×C4—2181no—
157C2×C4—2181no—
158C2×C4—2181no—
159C2×C4—2182no—
160C2×C4C22⋊#8913711yescharacteristic
161C2×C4C22⋊#8913711yescharacteristic
162C2×C4C22×#8913711yescharacteristic
163C8—4362no—
164C8—4362no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—2181no—
169C22—2181no—
170C22—2181no—
171C22C2×C22⋊#8913711yescharacteristic
172C4—21no—
173C4—21no—
174C4—41no—
175C4—1091no—
176C4—1091no—
177C4—4361no—
178C4—4361no—
179C4—2181no—
180C4—2181no—
181C4#1681278612yes—
182C2—81no—
183C2—4361no—
184C2#2029600011yescharacteristic
185C2#2029601711yescharacteristic
186C2#2029602311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#124×C109C4
socleC22×C109C2×C22⋊C4
prime radical#16812670C4
cosocleC4×Dic109C23
hypercenter#124#89137
hypocenterC109#1495
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20296044C2semidirect
2. #20296030C2semidirect
3. Q8⋊#11724786C2semidirect
4. D4⋊#11724786C2semidirect
94. C109#1495semidirect

Maximal subgroups

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

subgroupindex
#202960442
#202960302
Q8⋊#117247862
D4⋊#117247862
#202955552
#1495109