Finite Group Databasebeta

← #28373557

Subgroups — #28373557

Order 13952.

340 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 137 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#20295653C212yesmaximal
2#20295433C212yesmaximal, complemented
3#20295432C212yesmaximal, complemented
4#20295428C211yesmaximal, characteristic, complemented
5#20295426C211yesmaximal, characteristic, complemented
6#20295423C211yesmaximal, characteristic, complemented
7#20295065C212yesmaximal
8#20295061C211yesmaximal, characteristic
9#106×D109C211yesmaximal, characteristic
10#20294980C211yesmaximal, characteristic
11#498×C109C211yesmaximal, characteristic, complemented
12C2×#13182012C2211yescharacteristic
13C2×C8×D109C2211yescharacteristic
14#16812698C2212yes—
15#16812697—22no—
16#16812691C2212yes—
17#16812690C2212yes—
18#16812689—22no—
19#16812677—22no—
20#16812675—22no—
21#16812657C2212yes—
22#16812656C2211yescharacteristic
23C4⋊C4×D109C2212yes—
24#16812633C2212yes—
25#16812632C2212yes—
26#16812632C2212yes—
27#16812631C2211yescharacteristic
28#16812631C2211yescharacteristic
29#16812630C2212yes—
30#16812630C2212yes—
31#16812610C2211yescharacteristic
32#16812609C2211yescharacteristic
33C8×Dic109C2211yescharacteristic
34C4⋊Q8×C109C2212yes—
35#106×C109C2211yescharacteristic
36#106×C109C2212yes—
37#106×C109C2211yescharacteristic
38C4×C8×C109C2211yescharacteristic
39C2×Dic218—22no—
40C2×C4×D109—22no—
41C2×C4×D109C2311yescharacteristic
42#13182012Q814yescomplemented
43C2×#11724784C2311yescharacteristic
44C2×#11724784C2311yescharacteristic
45C8×D109Q814yescomplemented
46C4⋊Dic109—22no—
47C4⋊Dic109C2312yes—
48C4⋊Dic109C2312yes—
49#13182008—22no—
50#13182008—22no—
51#13182008—22no—
52#13182008—22no—
53#13182008—22no—
54#13182008D412yes—
55#13182006—22no—
56#13182006D412yes—
57C4×Dic109—22no—
58C4×Dic109C2311yescharacteristic
59C2×Q8×C109—22no—
60C2×C8×C109C2311yescharacteristic
61C2×C8×C109C2311yescharacteristic
62C4⋊C4×C109—22no—
63C4⋊C4×C109C2312yes—
64C4⋊C4×C109C2312yes—
65C42×C109C2311yescharacteristic
66C22×D109C2×D411yescharacteristic
67Q8×C109—82no—
68C2×C4×C109—22no—
69C2×C4×C109—22no—
70C2×C4×C109C2×D412yes—
71C2×C4×C109C2411yescharacteristic
72C2×Dic109—22no—
73C2×Dic109—22no—
74C2×Dic109C2×D412yes—
75C2×Dic109C2×D411yescharacteristic
76C4×D109—82no—
77C4×D109C2×Q812yes—
78C4×D109C2×Q812yes—
79Dic218—82no—
80C8×C109C2×Q812yes—
81C8×C109C2×Q812yes—
82#11724784C2×Q812yes—
83#11724784C2×Q812yes—
84C22×C109C22×D411yescharacteristic
85C2×D109—22no—
86C2×D109C4⋊Q812yes—
87C4×C109—22no—
88C4×C109—42no—
89C4×C109—42no—
90C4×C109C22×Q811yescharacteristic
91C4×C109C22×Q811yescharacteristic
92Dic109—22no—
93Dic109—42no—
94Dic109—42no—
95Dic109C4⋊Q812yes—
96C2×C109C2×C4⋊Q811yescharacteristic
97C2×C109#57712yes—
98D109—22no—
99#3163—1091nomaximal, Sylow
100C109#316311yescharacteristic, complemented, Sylow
101#544—1092no—
102#501—1092no—
103#501—1092no—
104#499—1091no—
105#499—1091no—
106#498—1091no—
107#498D10911yescharacteristic, complemented
108#428—1092no—
109#427—1091no—
110C2×#106—1091no—
111#405—1091no—
112C2×M16—1091no—
113C22×C8—1091no—
114C4⋊Q8—1092no—
115C4⋊Q8—2182no—
116C4⋊Q8C2×D10912yes—
117#125—1092no—
118#125—1092no—
119#125—2182no—
120C22⋊Q8—2182no—
121C22⋊Q8—2182no—
122#117—1091no—
123#117—1092no—
124C2×C4⋊C4—1092no—
125#107—1092no—
126#107—1092no—
127#107—1092no—
128#106—1091no—
129#106—1091no—
130#106—1092no—
131#106—1092no—
132#106C2×D10911yescharacteristic
133#106C2×D10911yescharacteristic
134#106C2×D10912yes—
135#97—1091no—
136#97—1091no—
137C4×C8—1091no—
138C4×C8C2×D10911yescharacteristic
139C2×Q8—22no—
140C2×Q8—2182no—
141C22×C4—1091no—
142C22×C4—2182no—
143M16—1094no—
144C2×C8—1091no—
145C2×C8—1091no—
146C2×C8—1094no—
147C2×C8C22×D10911yescharacteristic
148C2×C8C22×D10911yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—1092no—
151C4⋊C4—1092no—
152C4⋊C4—1092no—
153C4⋊C4—2182no—
154C4⋊C4—2182no—
155C4⋊C4—2182no—
156C4⋊C4—2182no—
157C4⋊C4—2182no—
158C4⋊C4—2182no—
159C4⋊C4C22×D10912yes—
160C4⋊C4C22×D10912yes—
161C22⋊C4—1092no—
162C22⋊C4—2182no—
163C42—1091no—
164C42—2182no—
165C42C22×D10911yescharacteristic
166C23—1091no—
167Q8—82no—
168Q8—8722no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—8722no—
172C2×C4—1091no—
173C2×C4—1092no—
174C2×C4—1092no—
175C2×C4—1092no—
176C2×C4—2182no—
177C2×C4—2182no—
178C2×C4D4×D10912yes—
179C2×C4C23×D10911yescharacteristic
180C8—1092no—
181C8—1092no—
182C8Q8×D10912yes—
183C8Q8×D10912yes—
184C22—1092no—
185C22—2182no—
186C22C2×D4×D10911yescharacteristic
187C4—22no—
188C4—42no—
189C4—42no—
190C4—1092no—
191C4—4362no—
192C4—4362no—
193C4—2182no—
194C4C2×Q8×D10911yescharacteristic
195C4C2×Q8×D10911yescharacteristic
196C2—2182no—
197C2C4⋊Q8×D10911yescharacteristic
198C2#2029583212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#498×C109C2
socleC22×C109C22×D4
prime radicalC22×D109C2×D4
hypercenter#498D109
hypocenterC109#3163
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #20295433C2semidirect
3. #20295432C2semidirect
4. #20295428C2semidirect
5. #20295426C2semidirect
6. #20295423C2semidirect
11. #498×C109C2semidirect
42. #13182012Q8semidirect
45. C8×D109Q8semidirect
100. C109#3163semidirect
107. #498D109semidirect

Maximal subgroups

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

subgroupindex
#202956532
#202954332
#202954322
#202954282
#202954262
#202954232
#202950652
#202950612
#106×D1092
#202949802
#498×C1092
#3163109