Finite Group Databasebeta

← #28373562

Subgroups — #28373562

Order 13952.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#20295657C211yesmaximal, characteristic
2#20295654C211yesmaximal, characteristic
3#20295434C212yesmaximal, complemented
4#20295433C212yesmaximal, complemented
5#20295427C212yesmaximal, complemented
6#20295423C212yesmaximal, complemented
7#20295068C211yesmaximal, characteristic
8#20295062C211yesmaximal, characteristic
9C2×#16812630C211yesmaximal, characteristic
10#20294984C211yesmaximal, characteristic
11#428×C109C211yesmaximal, characteristic
12C22×#11724784C2211yescharacteristic
13#16812699—21no—
14#16812698—21no—
15#16812698C2212yes—
16#16812698C2212yes—
17#16812691—21no—
18#16812690—21no—
19#16812690C2212yes—
20#16812690C2212yes—
21#16812678—22no—
22#16812676—22no—
23#16812658C2211yescharacteristic
24#16812658C2211yescharacteristic
25C2×C4⋊Dic109C2211yescharacteristic
26#16812633C2212yes—
27#16812632C2211yescharacteristic
28#16812632C2211yescharacteristic
29#16812631C2212yes—
30#16812630C2211yescharacteristic
31#16812630C2212yes—
32#16812630C2212yes—
33#16812630C2211yescharacteristic
34#16812609C2212yes—
35C8×Dic109C2212yes—
36C2×M16×C109C2211yescharacteristic
37#117×C109C2211yescharacteristic
38C2×C4⋊C4×C109C2211yescharacteristic
39#107×C109C2212yes—
40#106×C109C2212yes—
41C2×Dic218—22no—
42C2×Dic218—22no—
43C22×Dic109—21no—
44C2×#11724784Q814yescomplemented
45C2×#11724784C2311yescharacteristic
46C2×#11724784C2311yescharacteristic
47C4⋊Dic109—21no—
48C4⋊Dic109—22no—
49C4⋊Dic109—22no—
50C4⋊Dic109D411yescharacteristic
51C4⋊Dic109D411yescharacteristic
52C4⋊Dic109C2312yes—
53C4⋊Dic109C2311yescharacteristic
54C4⋊Dic109C2311yescharacteristic
55#13182008—22no—
56#13182008—22no—
57#13182008—22no—
58#13182008—22no—
59C22⋊Dic109—21no—
60C22⋊Dic109D412yes—
61C4×Dic109—21no—
62C4×Dic109C2312yes—
63C22×C4×C109—21no—
64C22×C4×C109C2311yescharacteristic
65M16×C109Q814yescomplemented
66C2×C8×C109C2312yes—
67C4⋊C4×C109—21no—
68C4⋊C4×C109C2311yescharacteristic
69C4⋊C4×C109C2312yes—
70C4⋊C4×C109C2311yescharacteristic
71C22⋊C4×C109—21no—
72C42×C109—21no—
73C23×C109C2×D411yescharacteristic
74C2×C4×C109—21no—
75C2×C4×C109—21no—
76C2×C4×C109—22no—
77C2×C4×C109—81no—
78C2×C4×C109C2×D411yescharacteristic
79C2×C4×C109C2×Q812yes—
80C2×C4×C109C2×Q812yes—
81C2×C4×C109C2411yescharacteristic
82C2×Dic109—21no—
83C2×Dic109—21no—
84C2×Dic109—22no—
85C2×Dic109—81no—
86C2×Dic109C2×D412yes—
87C2×Dic109C2×D412yes—
88Dic218—82no—
89Dic218—82no—
90C8×C109C2×Q814yes—
91#11724784C2×Q812yes—
92#11724784C2×Q812yes—
93C22×C109—21no—
94C22×C109—21no—
95C22×C109C4⋊Q812yes—
96C22×C109C22×D411yescharacteristic
97C4×C109—41no—
98C4×C109—42no—
99C4×C109—41no—
100C4×C109C4⋊Q812yes—
101C4×C109C22×Q811yescharacteristic
102C4×C109C22×Q811yescharacteristic
103Dic109—22no—
104Dic109—22no—
105Dic109—41no—
106Dic109—42no—
107Dic109—41no—
108C2×C109—22no—
109C2×C109C2×C4⋊Q811yescharacteristic
110C2×C109#57711yescharacteristic
111C2×C109#57711yescharacteristic
112#3163—1091nomaximal, Sylow
113C109#316311yescharacteristic, complemented, Sylow
114#544—1091no—
115#544—1091no—
116#501—1092no—
117#501—1092no—
118#499—1092no—
119#498—1092no—
120#428—1091no—
121#428D10911yescharacteristic
122#427—1091no—
123C2×#106—1091no—
124#405—1091no—
125C2×M16C2×D10911yescharacteristic
126C22×C8—1091no—
127C4⋊Q8—1092no—
128C4⋊Q8—1092no—
129C4⋊Q8—2181no—
130C4⋊Q8—2181no—
131#125—1092no—
132#125—1092no—
133#125—2181no—
134#125—2181no—
135C22⋊Q8—2182no—
136C22⋊Q8—2182no—
137#117—1091no—
138#117—1091no—
139#117C2×D10911yescharacteristic
140C2×C4⋊C4—1091no—
141C2×C4⋊C4C2×D10911yescharacteristic
142#107—1092no—
143#107—1091no—
144#107—1091no—
145#107C2×D10912yes—
146#106—1092no—
147#106—1092no—
148#106—1092no—
149#106—1091no—
150#106—1091no—
151#106C2×D10912yes—
152#97—1092no—
153C4×C8—1092no—
154C2×Q8—2182no—
155C2×Q8—2182no—
156C22×C4—21no—
157C22×C4—2181no—
158C22×C4C22×D10911yescharacteristic
159M16Dic21814yescomplemented
160C2×C8—1094no—
161C2×C8—1091no—
162C2×C8—1091no—
163C2×C8C22×D10912yes—
164C4⋊C4—21no—
165C4⋊C4—1091no—
166C4⋊C4—1091no—
167C4⋊C4—1092no—
168C4⋊C4—1091no—
169C4⋊C4—1091no—
170C4⋊C4—2182no—
171C4⋊C4—2182no—
172C4⋊C4—2182no—
173C4⋊C4—2182no—
174C4⋊C4—2182no—
175C4⋊C4—2182no—
176C4⋊C4—2181no—
177C4⋊C4C22×D10912yes—
178C4⋊C4C22×D10911yescharacteristic
179C4⋊C4C22×D10911yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1092no—
182C22⋊C4—2181no—
183C42—21no—
184C42—1092no—
185C42—2181no—
186C23D4×D10911yescharacteristic
187Q8—8722no—
188Q8—8722no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—8721no—
194C2×C4—1092no—
195C2×C4—1092no—
196C2×C4—2181no—
197C2×C4—2181no—
198C2×C4—2182no—
199C2×C4D4×D10911yescharacteristic
200C2×C4Q8×D10912yes—
201C2×C4C2×Dic21812yes—
202C2×C4C23×D10911yescharacteristic
203C8—1092no—
204C8—1092no—
205C8C2×Dic21814yes—
206C22—21no—
207C22—21no—
208C22#1681269812yes—
209C22C2×D4×D10911yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—4362no—
214C4—4361no—
215C4—4361no—
216C4—2182no—
217C4—2182no—
218C4#1681269812yes—
219C4C2×Q8×D10911yescharacteristic
220C4C22×Dic21811yescharacteristic
221C2—22no—
222C2C2×#1681269811yescharacteristic
223C2#2029583511yescharacteristic
224C2#2029583611yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
prime radicalC23×C109C2×D4
hypercenter#428D109
hypocenterC109#3163
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20295434C2semidirect
4. #20295433C2semidirect
5. #20295427C2semidirect
6. #20295423C2semidirect
44. C2×#11724784Q8semidirect
65. M16×C109Q8semidirect
113. C109#3163semidirect
159. M16Dic218semidirect

Maximal subgroups

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

subgroupindex
#202956572
#202956542
#202954342
#202954332
#202954272
#202954232
#202950682
#202950622
C2×#168126302
#202949842
#428×C1092
#3163109