Finite Group Databasebeta

← #28362991

Subgroups — #28362991

Order 13952.

552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#16812626C212yesmaximal, complemented
2#20295027C212yesmaximal, complemented
3C2×#16812618C211yesmaximal, characteristic, complemented
4#20294933C211yesmaximal, characteristic, complemented
5C2×2−1+4×C109C211yesmaximal, characteristic
6C2×#13182013C2212yes—
7C2×C4⋊Dic109C2211yescharacteristic
8C2×C4×Dic109C2211yescharacteristic
9#16812626—24no—
10#16812623—22no—
11#16812620—22no—
12#16812618—22no—
13#16812606—22no—
142−1+4×C109—24no—
152−1+4×C109—22no—
162−1+4×C109C414yescomplemented
17C2×#41×C109—22no—
18C2×#41×C109—22no—
19C2×#41×C109C2212yes—
20C22×Q8×C109—22no—
21C22×Q8×C109C2211yescharacteristic
22C22×Dic109—21no—
23C22×Dic109—21no—
24#13182013—22no—
25#13182013—22no—
26C2×#11724784—22no—
27C4⋊Dic109—22no—
28C4×Dic109—22no—
29#41×C109—28no—
30#41×C109—28no—
31#41×C109—28no—
32#41×C109—28no—
33#41×C109—22no—
34#41×C109—22no—
35#41×C109C2×C414yes—
36#41×C109D414yes—
37C2×Q8×C109—22no—
38C2×Q8×C109—28no—
39C2×Q8×C109—28no—
40C2×Q8×C109—22no—
41C2×Q8×C109—22no—
42C2×Q8×C109C2×C412yes—
43C2×Q8×C109D412yes—
44C2×Q8×C109D412yes—
45C2×D4×C109—22no—
46C2×D4×C109—22no—
47C2×D4×C109D412yes—
48C22×C4×C109—24no—
49C22×C4×C109—22no—
50C22×C4×C109D412yes—
51C22×C4×C109C2311yescharacteristic
52C23×C109—22no—
53C23×C109C2×D411yescharacteristic
54Q8×C109—28no—
55Q8×C109—24no—
56Q8×C109—22no—
57Q8×C109—24no—
58Q8×C109C22⋊C414yes—
59D4×C109—28no—
60D4×C109—24no—
61D4×C109—24no—
62D4×C109—22no—
63D4×C109C22⋊C414yes—
64C2×C4×C109—216no—
65C2×C4×C109—22no—
66C2×C4×C109—22no—
67C2×C4×C109—22no—
68C2×C4×C109—24no—
69C2×C4×C109—24no—
70C2×C4×C109C22⋊C414yes—
71C2×C4×C109C22×C411yescharacteristic
72C2×C4×C109C2×D412yes—
73C2×C4×C109C2×D412yes—
74C2×C4×C109C2×D411yescharacteristic
75C2×Dic109—22no—
76C2×Dic109—41no—
77C2×Dic109—41no—
78C2×Dic109—42no—
79C2×Dic109—41no—
80#11724784—42no—
81C22×C109—21no—
82C22×C109—21no—
83C22×C109—24no—
84C22×C109—42no—
85C22×C109—42no—
86C22×C109C2×C22⋊C411yescharacteristic
87C22×C109#12011yescharacteristic
88C22×C109#12011yescharacteristic
89C4×C109—24no—
90C4×C109—24no—
91C4×C109C2×C22⋊C412yes—
92C4×C109#12012yes—
93Dic109—42no—
94Dic109—81no—
95C2×C109—21no—
96C2×C109—21no—
97C2×C109—44no—
98C2×C109#37911yescharacteristic
99C2×C109#45411yescharacteristic
100C2×C109#45611yescharacteristic
101#1793—1091nomaximal, Sylow
102C109#179311yescharacteristic, complemented, Sylow
103C2×2−1+4D10911yescharacteristic
104C2×#104—1092no—
105#417—1092no—
106C2×#101—1091no—
107#391—1091no—
1082−1+4—24no—
1092−1+4—22no—
1102−1+4Dic10914yescomplemented
111C2×#41—22no—
112C2×#41—22no—
113C2×#41C2×D10912yes—
114C22×Q8—22no—
115C22×Q8C2×D10911yescharacteristic
116C2×M16—1092no—
117C2×C4⋊C4—1091no—
118C2×C42—1091no—
119#104—2184no—
120#103—2182no—
121#102—2182no—
122#101—2182no—
123#95—2182no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C2×Dic10914yes—
131#41C22⋊D10914yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×Dic10912yes—
138C2×Q8C22⋊D10912yes—
139C2×Q8C22⋊D10912yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22⋊D10912yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—2181no—
146C22×C4—2181no—
147C22×C4C22⋊D10912yes—
148C22×C4C22×D10911yescharacteristic
149M16—2182no—
150M16—2182no—
151C2×C8—2182no—
152C4⋊C4—2182no—
153C42—2182no—
154C23—22no—
155C23C2×C22⋊D10911yescharacteristic
156Q8—28no—
157Q8—24no—
158Q8—22no—
159Q8—24no—
160Q8C22⋊Dic10914yes—
161D4—28no—
162D4—24no—
163D4—24no—
164D4—22no—
165D4C22⋊Dic10914yes—
166C2×C4—216no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—4362no—
173C2×C4—4361no—
174C2×C4—4361no—
175C2×C4—4361no—
176C2×C4—2182no—
177C2×C4C22⋊Dic10914yes—
178C2×C4C22×Dic10911yescharacteristic
179C2×C4C2×C22⋊D10912yes—
180C2×C4C2×C22⋊D10912yes—
181C2×C4C2×C22⋊D10911yescharacteristic
182C8—4362no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22C2×C22⋊Dic10911yescharacteristic
189C22#1681266811yescharacteristic
190C22#1681266811yescharacteristic
191C4—24no—
192C4—24no—
193C4—8721no—
194C4—4362no—
195C4C2×C22⋊Dic10912yes—
196C4#1681266812yes—
197C2—21no—
198C2—21no—
199C2—44no—
200C2#2029488211yescharacteristic
201C2#2029519911yescharacteristic
202C2#2029520911yescharacteristic
subgroupG/H
centerC22#16812668
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting, prime radicalC2×2−1+4×C109C2
socleC22×C109#120
cosocleC22×C4×C109C23
hypercenterC2×2−1+4D109
hypocenterC109#1793
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16812626C2semidirect
2. #20295027C2semidirect
3. C2×#16812618C2semidirect
4. #20294933C2semidirect
16. 2−1+4×C109C4semidirect
102. C109#1793semidirect
110. 2−1+4Dic109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126262
#202950272
C2×#168126182
#202949332
C2×2−1+4×C1092
#1793109