Finite Group Databasebeta

← #28378406

Subgroups — #28378406

Order 13952.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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#20295962C211yesmaximal, characteristic, complemented
3#20295879C212yesmaximal, complemented
4#20295878C212yesmaximal
5#20295561C211yesmaximal, characteristic
6C2×C4×#89137—21no—
7#16812766—21no—
8#16812756C2212yes—
9C4×#11724786—21no—
10(C2×#89137)⋊C4—21no—
11(C2×#89137)⋊C4C2212yes—
12#16812694—21no—
13#16812687C411yescharacteristic, complemented
14#16812684—21no—
15#16812679C2211yescharacteristic
16#16812674C2211yescharacteristic
17#16812664C411yescharacteristic, complemented
18#16812661—21no—
19#16812661C411yescharacteristic, complemented
20#16812657—21no—
21C42×D109C2211yescharacteristic
22#125×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24C2×#11724786—22no—
25C4×#89137—24no—
26C2×Dic218C2×C411yescharacteristic
27C2×D436—21no—
28C2×D436C2×C411yescharacteristic
29C2×C4×D109—21no—
30C2×C4×D109D411yescharacteristic
31C2×C4×D109D411yescharacteristic
32C2×C4×D109C2311yescharacteristic
33C4⋊Dic109—21no—
34#13182008—21no—
35#13182006—21no—
36#13182006—21no—
37#13182006—21no—
38#13182006—21no—
39#13182006—21no—
40C4×Dic109—21no—
41C4×Dic109C2×C411yescharacteristic
42C4×Dic109D411yescharacteristic
43C4×Dic109D411yescharacteristic
44C4⋊C4×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109C2×C411yescharacteristic
47C4⋊C4×C109C2×C411yescharacteristic
48C42×C109C2×C411yescharacteristic
49C22×D109—21no—
50C22×D109C2×D411yescharacteristic
51C2×#89137—24no—
52C2×#89137—42no—
53C2×#89137—42no—
54C2×C4×C109—21no—
55C2×C4×C109—21no—
56C2×C4×C109C22⋊C411yescharacteristic
57C2×C4×C109C22⋊C411yescharacteristic
58C2×C4×C109C22×C411yescharacteristic
59C2×Dic109—21no—
60C2×Dic109C22⋊C411yescharacteristic
61C2×Dic109C22⋊C411yescharacteristic
62C2×Dic109C2×D411yescharacteristic
63D436—41no—
64D436—42no—
65C4×D109—22no—
66C4×D109—22no—
67C4×D109—41no—
68C4×D109—42no—
69Dic218—41no—
70#11724786—42no—
71C22×C109C2×C22⋊C411yescharacteristic
72C2×D109—21no—
73C2×D109—21no—
74C2×D109—41no—
75C2×D109—41no—
76C2×D109—41no—
77C2×D109#13512yes—
78#89137—44no—
79C4×C109—21no—
80C4×C109—21no—
81C4×C109—21no—
82C4×C109—41no—
83C4×C109—41no—
84Dic109—21no—
85Dic109—21no—
86Dic109—41no—
87Dic109#10412yes—
88C2×C109#41011yescharacteristic
89C2×C109#41611yescharacteristic
90C2×C109C2×#10411yescharacteristic
91D109—22no—
92D109—81no—
93#1500—1091nomaximal, Sylow
94C109#150011yescharacteristic, complemented, Sylow
95#529—1091no—
96#431—1091no—
97#376—1091no—
98#328—1092no—
99#327—1092no—
100#126—2181no—
101#125#8913711yescharacteristic, complemented
102#124—1091no—
103#123—2181no—
104C22⋊Q8—1091no—
105#121—1091no—
106C4×Q8—1091no—
107C4×D4—1091no—
108C4×D4—2181no—
109#117—2181no—
110C2×C42—1091no—
111C2×C42—2181no—
112C4⋊C8—2181no—
113C22⋊C8—1092no—
114C4×C8—2181no—
115#95—1092no—
116#95—2181no—
117C2×Q8—1091no—
118C2×D4—1091no—
119C2×D4—2181no—
120C22×C4—1091no—
121C22×C4—1091no—
122C22×C4—1091no—
123C22×C4—2181no—
124C22×C4—2182no—
125C2×C8—2182no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—2181no—
129C4⋊C4—2181no—
130C4⋊C4C2×#8913711yescharacteristic
131C4⋊C4C2×#8913711yescharacteristic
132C22⋊C4—2181no—
133C22⋊C4—2181no—
134C22⋊C4—2181no—
135C22⋊C4—2181no—
136C22⋊C4—2181no—
137C42—1091no—
138C42—1091no—
139C42—1091no—
140C42—2181no—
141C42—2184no—
142C42C2×#8913711yescharacteristic
143C23—1091no—
144C23—2181no—
145Q8—4361no—
146D4—4361no—
147D4—4362no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—1091no—
151C2×C4—1091no—
152C2×C4—1091no—
153C2×C4—4361no—
154C2×C4—4362no—
155C2×C4—4362no—
156C2×C4—4362no—
157C2×C4—2182no—
158C2×C4—2182no—
159C2×C4—2184no—
160C2×C4—2181no—
161C2×C4C22⋊#8913711yescharacteristic
162C2×C4C22⋊#8913711yescharacteristic
163C2×C4C22×#8913711yescharacteristic
164C8—4362no—
165C22—1092no—
166C22—4361no—
167C22—4361no—
168C22—4361no—
169C22—2181no—
170C22—2181no—
171C22C2×C22⋊#8913711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—1092no—
178C4—4364no—
179C4—4361no—
180C4—2181no—
181C4—2181no—
182C2—8721no—
183C2—2182no—
184C2#2029600711yescharacteristic
185C2#2029601611yescharacteristic
186C2D109⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#125×C109C4
socleC22×C109C2×C22⋊C4
prime radical#16812674C22
cosocleC2×C4×D109C23
hypercenter#125#89137
hypocenterC109#1500
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #20296044C2semidirect
2. #20295962C2semidirect
3. #20295879C2semidirect
13. #16812687C4semidirect
17. #16812664C4semidirect
19. #16812661C4semidirect
22. #125×C109C4semidirect
94. C109#1500semidirect
101. #125#89137semidirect

Maximal subgroups

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

subgroupindex
#202960442
#202959622
#202958792
#202958782
#202955612
#1500109