Finite Group Databasebeta

← #28373323

Subgroups — #28373323

Order 13952.

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

normal subgroups only

This group has 382 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
1#20295631C211yesmaximal, characteristic
2#20295350C212yesmaximal, complemented
3#20295333C212yesmaximal, complemented
4#20295226C212yesmaximal, complemented
5#20295165C212yesmaximal, complemented
6#20295124C212yesmaximal, complemented
7C4×#13182016C212yesmaximal, complemented
8#20295087C211yesmaximal, characteristic, complemented
9#519×C109C211yesmaximal, characteristic, complemented
10C2×#13182016C2212yes—
11C2×#13182016C2212yes—
12#16812696C2211yescharacteristic
13#16812691C2211yescharacteristic
14#16812684—22no—
15#16812673C2212yes—
16#16812672—22no—
17C4×D436C2212yes—
18#16812632C2212yes—
19#16812630C2212yes—
20C4⋊#11724784C2212yes—
21#16812623C2212yes—
22#16812623C2212yes—
23#16812619C2212yes—
24#16812619C2212yes—
25C22⋊#11724784C2212yes—
26#16812610C2211yescharacteristic
27C4×#11724784C2211yescharacteristic
28C22×Q8×C109C2211yescharacteristic
29C4×Q8×C109—22no—
30C4×Q8×C109C2212yes—
31C4×Q8×C109C2212yes—
32#117×C109—22no—
33#117×C109C2211yescharacteristic
34#117×C109C2211yescharacteristic
35C2×D436—22no—
36C2×D436C2312yes—
37C2×C22⋊D109—22no—
38C2×C4×D109—22no—
39#13182016—24no—
40#13182016—24no—
41C2×#11724784C2312yes—
42C2×#11724784C2312yes—
43C4⋊Dic109C2312yes—
44#13182008—22no—
45#13182006—22no—
46#13182006—22no—
47C2×Q8×C109—24no—
48C2×Q8×C109—21no—
49C2×Q8×C109C2312yes—
50C22×C4×C109—21no—
51C22×C4×C109C2311yescharacteristic
52C4⋊C4×C109—22no—
53C4⋊C4×C109—22no—
54C4⋊C4×C109D412yes—
55C4⋊C4×C109C2312yes—
56C22⋊C4×C109—21no—
57C22⋊C4×C109D412yes—
58C42×C109—22no—
59C42×C109—22no—
60C42×C109—21no—
61C42×C109C2311yescharacteristic
62C42×C109C2311yescharacteristic
63C23×C109C2×D411yescharacteristic
64C22×D109—22no—
65Q8×C109—24no—
66Q8×C109—22no—
67Q8×C109#4114yes—
68C2×C4×C109—24no—
69C2×C4×C109—22no—
70C2×C4×C109—22no—
71C2×C4×C109—22no—
72C2×C4×C109C2×D412yes—
73C2×C4×C109C2×D412yes—
74C2×C4×C109C2×D411yescharacteristic
75C2×C4×C109C2411yescharacteristic
76C22⋊D109—82no—
77C2×Dic109—22no—
78D436—22no—
79D436—22no—
80D436—82no—
81C4×D109—82no—
82#11724784—22no—
83#11724784—22no—
84C22×C109—21no—
85C22×C109—22no—
86C22×C109C22×D411yescharacteristic
87C2×D109—42no—
88C2×D109—42no—
89C2×D109—42no—
90C4×C109—24no—
91C4×C109—21no—
92C4×C109—22no—
93C4×C109—22no—
94C4×C109—42no—
95C4×C109C2×#4112yes—
96Dic109—42no—
97C2×C109—41no—
98C2×C109#52511yescharacteristic
99C2×C109#57712yes—
100D109—82no—
101#3126—1091nomaximal, Sylow
102C109#312611yescharacteristic, complemented, Sylow
103#540—1091no—
104#519D10911yescharacteristic, complemented
105#482—1092no—
106#479—1092no—
107#460—1092no—
108#448—1092no—
109#440—1092no—
110C4×SD16—1092no—
111#433—1091no—
112C22×Q8C2×D10911yescharacteristic
113C2×SD16—1092no—
114C2×SD16—1092no—
115#127—1091no—
116#125—1091no—
117#123—2182no—
118#121—1092no—
119#121—2182no—
120C4×Q8—22no—
121C4×Q8C2×D10912yes—
122C4×Q8C2×D10912yes—
123C4×D4—1092no—
124#117—22no—
125#117C2×D10911yescharacteristic
126#117C2×D10911yescharacteristic
127#107—1092no—
128#106—1092no—
129C4⋊C8—1092no—
130#103—1092no—
131#103—1092no—
132#102—1092no—
133#102—1092no—
134C22⋊C8—1092no—
135#97—1091no—
136C4×C8—1091no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D10912yes—
140C2×D4—1092no—
141C2×D4—2182no—
142C2×D4—2182no—
143C22×C4—21no—
144C22×C4—2182no—
145C22×C4C22×D10911yescharacteristic
146SD16—2184no—
147SD16—2184no—
148C2×C8—1092no—
149C2×C8—1092no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1092no—
153C4⋊C4—2182no—
154C4⋊C4C22⋊D10912yes—
155C4⋊C4C22×D10912yes—
156C22⋊C4—21no—
157C22⋊C4—2182no—
158C22⋊C4—2182no—
159C22⋊C4C22⋊D10912yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D10911yescharacteristic
164C42C22×D10911yescharacteristic
165C23—2182no—
166C23C2×C22⋊D10911yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1318202914yes—
170D4—8722no—
171D4—8722no—
172D4—2182no—
173D4—2182no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—8722no—
179C2×C4—2182no—
180C2×C4C2×C22⋊D10912yes—
181C2×C4C2×C22⋊D10912yes—
182C2×C4C2×C22⋊D10911yescharacteristic
183C2×C4C23×D10911yescharacteristic
184C8—2182no—
185C8—2182no—
186C22—21no—
187C22—22no—
188C22—4362no—
189C22—4362no—
190C22—4362no—
191C22C22×C22⋊D10911yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—4362no—
198C4C2×#1318202912yes—
199C2—41no—
200C2—8722no—
201C2#2029553711yescharacteristic
202C2#2029583412yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#519×C109C2
socleC22×C109C22×D4
prime radical#20295631C2
hypercenter#519D109
hypocenterC109#3126
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20295350C2semidirect
3. #20295333C2semidirect
4. #20295226C2semidirect
5. #20295165C2semidirect
6. #20295124C2semidirect
7. C4×#13182016C2semidirect
8. #20295087C2semidirect
9. #519×C109C2semidirect
102. C109#3126semidirect
104. #519D109semidirect

Maximal subgroups

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

subgroupindex
#202956312
#202953502
#202953332
#202952262
#202951652
#202951242
C4×#131820162
#202950872
#519×C1092
#3126109