Finite Group Databasebeta

← #353×D109 #28364688

Subgroups — #353×D109 #28364688

Order 13952.

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

normal subgroups only

This group has 368 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#127×D109C211yesmaximal, characteristic
2#99×D109C212yesmaximal, complemented
3#20294819C211yesmaximal, characteristic, complemented
4#20294818C212yesmaximal, complemented
5#353×C109C211yesmaximal, characteristic, complemented
6C2×D4×D109—22no—
7C2×D4×D109C411yescharacteristic, complemented
8C2×D4×D109C2211yescharacteristic
9#16812696C411yescharacteristic, complemented
10#16812695—22no—
11#16812695C411yescharacteristic, complemented
12#16812695C2211yescharacteristic
13C22⋊C4×D109—22no—
14C42×D109C411yescharacteristic, complemented
15#16812614C2212yescomplemented
16#16812613—22no—
17#127×C109C2211yescharacteristic
18#99×C109C2212yescomplemented
19C23×D109—21no—
20C23×D109D412yes—
21C2×D436—22no—
22C2×D436C2×C411yescharacteristic
23C2×D436C2×C411yescharacteristic, complemented
24C2×C22⋊D109—22no—
25C2×C22⋊D109—22no—
26C2×C22⋊D109—21no—
27C2×C22⋊D109D412yes—
28D4×D109—41no—
29D4×D109—41no—
30D4×D109—41no—
31D4×D109—41no—
32D4×D109—48no—
33C2×C4×D109—21no—
34C2×C4×D109—42no—
35C2×C4×D109C2×C411yescharacteristic
36C22⋊Dic109—22no—
37#13182006—42no—
38C4×Dic109—21no—
39C4×Dic109C2×C411yescharacteristic, complemented
40C2×D4×C109—22no—
41C2×D4×C109C2×C411yescharacteristic, complemented
42C2×D4×C109C2311yescharacteristic
43C22⋊C4×C109—22no—
44C42×C109C2×C411yescharacteristic, complemented
45C23×C109—21no—
46C23×C109C2×D412yes—
47C22×D109—22no—
48C22×D109—22no—
49C22×D109—21no—
50C22×D109—44no—
51C22×D109—44no—
52C22×D109—41no—
53C22×D109—41no—
54C22×D109C22⋊C411yescharacteristic
55C22×D109C22⋊C412yes—
56D4×C109—41no—
57D4×C109—41no—
58D4×C109—44no—
59C2×C4×C109—21no—
60C2×C4×C109—42no—
61C2×C4×C109C22×C411yescharacteristic
62C22⋊D109—42no—
63C22⋊D109—44no—
64C22⋊D109—44no—
65C22⋊D109—42no—
66C2×Dic109—21no—
67C2×Dic109—42no—
68C2×Dic109C22⋊C411yescharacteristic
69D436—41no—
70D436—41no—
71D436—44no—
72C4×D109—24no—
73C4×D109—21no—
74C4×D109—21no—
75C4×D109—84no—
76C22×C109—22no—
77C22×C109—42no—
78C22×C109—42no—
79C22×C109—41no—
80C22×C109C2×C22⋊C411yescharacteristic
81C2×D109—22no—
82C2×D109—22no—
83C2×D109—44no—
84C2×D109—42no—
85C2×D109—42no—
86C2×D109—41no—
87C2×D109—82no—
88C2×D109—82no—
89C2×D109—82no—
90C2×D109#9911yescharacteristic
91C2×D109#9911yescharacteristic
92C4×C109—22no—
93C4×C109—21no—
94C4×C109—82no—
95Dic109—22no—
96Dic109—21no—
97Dic109—82no—
98C2×C109—21no—
99C2×C109—42no—
100C2×C109—81no—
101C2×C109C2×#9911yescharacteristic
102D109—21no—
103D109—42no—
104D109—81no—
105D109#35312yesdirect product factor
106C2×#353—1091nomaximal, Sylow
107C109C2×#35311yescharacteristic, complemented, Sylow
108C2×#127—1091no—
109C2×#99—1092no—
110#353—1091no—
111#353—1092no—
112#353D10911yescharacteristic, direct product factor
113C22×D4—1091no—
114C22×D4—1091no—
115C22×D4—2182no—
116#127—1091no—
117#127—1091no—
118#127—1091no—
119#127—2182no—
120#127C2×D10911yescharacteristic
121C2×C22⋊C4—2182no—
122C2×C42—1091no—
123#99—1092no—
124#99—2182no—
125#99C2×D10912yescomplemented
126C24—1092no—
127C24—2181no—
128C2×D4—22no—
129C2×D4—1092no—
130C2×D4—1091no—
131C2×D4—1091no—
132C2×D4—4361no—
133C2×D4—4361no—
134C2×D4—4368no—
135C2×D4—4361no—
136C2×D4—4361no—
137C2×D4—2182no—
138C2×D4—2181no—
139C2×D4—2182no—
140C2×D4—2182no—
141C2×D4C4×D10911yescharacteristic, complemented
142C2×D4C22×D10911yescharacteristic
143C22×C4—1091no—
144C22×C4—4362no—
145C22×C4—2181no—
146C22⋊C4—22no—
147C22⋊C4—4362no—
148C22⋊C4—2182no—
149C42—1091no—
150C42—2181no—
151C42C4×D10911yescharacteristic, complemented
152C23—21no—
153C23—1091no—
154C23—1092no—
155C23—4361no—
156C23—4361no—
157C23—4364no—
158C23—4364no—
159C23—2182no—
160C23—2182no—
161C23—2181no—
162C23D4×D10912yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—4361no—
167D4—4364no—
168D4—4361no—
169D4—4362no—
170D4—4364no—
171D4—4364no—
172D4—4362no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—8724no—
176C2×C4—1091no—
177C2×C4—4362no—
178C2×C4—2181no—
179C2×C4—2181no—
180C2×C4—2184no—
181C2×C4—2181no—
182C2×C4C2×C4×D10911yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—8722no—
188C22—8722no—
189C22—8722no—
190C22—1091no—
191C22—1091no—
192C22—4362no—
193C22—4362no—
194C22—4364no—
195C22—4361no—
196C22—2182no—
197C22—2182no—
198C22C22⋊C4×D10911yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—8722no—
203C4—2182no—
204C4—2181no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—8721no—
209C2—1092no—
210C2—4362no—
211C2—2181no—
212C2#99×D10911yescharacteristic
subgroupG/H
centerC2#99×D109
commutatorC2×C4×C109C22×C4
FrattiniC2×D4C22×D109
Fitting#353×C109C2
socleC2×C109C2×#99
prime radical#127×D109C2
cosocleC2×D4×C109C23
hypercenter#353D109
hypocenterC109C2×#353
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D109C2semidirect
3. #20294819C2semidirect
4. #20294818C2semidirect
5. #353×C109C2semidirect
7. C2×D4×D109C4semidirect
9. #16812696C4semidirect
11. #16812695C4semidirect
14. C42×D109C4semidirect
15. #16812614C22semidirect
18. #99×C109C22semidirect
23. C2×D436C2×C4semidirect
39. C4×Dic109C2×C4semidirect
41. C2×D4×C109C2×C4semidirect
44. C42×C109C2×C4semidirect
105. D109#353direct (N × H)
107. C109C2×#353semidirect
112. #353D109direct (N × H)
125. #99C2×D109semidirect
141. C2×D4C4×D109semidirect
151. C42C4×D109semidirect

Maximal subgroups

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

subgroupindex
#127×D1092
#99×D1092
#202948192
#202948182
#353×C1092
C2×#353109