Finite Group Databasebeta

← #28363561

Subgroups — #28363561

Order 13952.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#20295590C211yesmaximal, characteristic
2#20295500C211yesmaximal, characteristic, complemented
3#20295043C211yesmaximal, characteristic, complemented
4C2×#16812615C211yesmaximal, characteristic, complemented
5#20294783C211yesmaximal, characteristic, complemented
6#20294782C211yesmaximal, characteristic, complemented
7#421×C109C211yesmaximal, characteristic, complemented
8C22×D436C2211yescharacteristic
9C2×#13182013C2211yescharacteristic
10#16812695C412yescomplemented
11#16812688C412yescomplemented
12#16812673—22no—
13#16812667—22no—
14C4×D436—22no—
15#16812661—22no—
16#16812658C2211yescharacteristic
17#16812658C2211yescharacteristic
18C22⋊C4×D109—22no—
19#16812626—22no—
20#16812615—22no—
21C2×#41×C109C2211yescharacteristic
22C2×M16×C109C2211yescharacteristic
23#117×C109C2211yescharacteristic
24#104×C109—22no—
25C23×D109D412yes—
26C2×D436—21no—
27C2×D436—21no—
28C2×D436—21no—
29C2×D436—21no—
30C2×D436C2×C412yes—
31C2×D436D412yes—
32C2×C22⋊D109—24no—
33C2×C4×D109—24no—
34#13182013—21no—
35#13182013—21no—
36C2×#11724784—21no—
37C4⋊Dic109—21no—
38C4⋊Dic109—21no—
39C22⋊Dic109—21no—
40C22⋊Dic109—21no—
41#13182006—24no—
42#13182006—24no—
43C4×Dic109C2×C412yescomplemented
44C4×Dic109D412yescomplemented
45#41×C109—21no—
46#41×C109—21no—
47#41×C109—21no—
48#41×C109—21no—
49C2×Q8×C109C2×C411yescharacteristic, complemented
50C2×D4×C109—21no—
51C2×D4×C109C2×C411yescharacteristic, complemented
52C22×C4×C109—21no—
53C22×C4×C109C2311yescharacteristic
54M16×C109—21no—
55M16×C109—21no—
56C2×C8×C109—21no—
57C4⋊C4×C109—21no—
58C22⋊C4×C109—21no—
59C42×C109D412yescomplemented
60C23×C109—21no—
61C23×C109C2×D411yescharacteristic
62C22×D109—22no—
63C22×D109—22no—
64C22×D109—82no—
65C22×D109C22⋊C414yes—
66Q8×C109—21no—
67Q8×C109—21no—
68D4×C109—21no—
69D4×C109—21no—
70D4×C109—41no—
71D4×C109—41no—
72C2×C4×C109—21no—
73C2×C4×C109—22no—
74C2×C4×C109—41no—
75C2×C4×C109—41no—
76C2×C4×C109C22×C411yescharacteristic
77C2×C4×C109C2×D411yescharacteristic
78C2×C4×C109C2×D411yescharacteristic
79C2×C4×C109C2×D411yescharacteristic
80C2×C4×C109#4111yescharacteristic
81C2×C4×C109#4111yescharacteristic
82C22⋊D109—84no—
83C2×Dic109—22no—
84C2×Dic109—22no—
85D436—22no—
86D436—22no—
87D436—22no—
88D436—22no—
89C4×D109—84no—
90C8×C109—41no—
91#11724784—41no—
92C22×C109—41no—
93C22×C109—42no—
94C22×C109—41no—
95C22×C109C2×C22⋊C411yescharacteristic
96C22×C109#12011yescharacteristic
97C22×C109#12311yescharacteristic
98C2×D109—24no—
99C2×D109—44no—
100C2×D109—44no—
101C2×D109—82no—
102C4×C109—42no—
103C4×C109—41no—
104C4×C109C4×D411yescharacteristic
105C4×C109C4×D411yescharacteristic
106C4×C109#12111yescharacteristic
107C4×C109#12111yescharacteristic
108Dic109—42no—
109Dic109—42no—
110C2×C109—21no—
111C2×C109—21no—
112C2×C109—21no—
113C2×C109—81no—
114C2×C109#38611yescharacteristic
115D109—44no—
116#1911—1091nomaximal, Sylow
117C109#191111yescharacteristic, complemented, Sylow
118#535—1091no—
119#518—1091no—
120#421—1091no—
121#421D10911yescharacteristic, complemented
122C2×#100—1091no—
123H(3,ℤ/4)—1091no—
124H(3,ℤ/4)—1091no—
125C2×#41C2×D10911yescharacteristic
126C22×D4—1091no—
127C2×M16—1091no—
128C2×M16C2×D10911yescharacteristic
129#127—1092no—
130#124—1092no—
131#121—2182no—
132#120—2182no—
133C4×D4—2182no—
134C4×D4—2182no—
135#117—1091no—
136#117—1091no—
137#117C2×D10911yescharacteristic
138C2×C22⋊C4—2182no—
139#104—22no—
140#104—2182no—
141#100—2182no—
142C24—1092no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D10911yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—1092no—
150C2×D4—1092no—
151C2×D4—2184no—
152C2×D4—2181no—
153C2×D4—2181no—
154C2×D4—2181no—
155C2×D4—2181no—
156C2×D4C4×D10911yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—2184no—
159C22×C4C22×D10911yescharacteristic
160M16—21no—
161M16—21no—
162M16—2181no—
163M16—2181no—
164C2×C8—21no—
165C2×C8—2181no—
166C4⋊C4—21no—
167C4⋊C4—2181no—
168C4⋊C4—2181no—
169C22⋊C4—21no—
170C22⋊C4—2184no—
171C22⋊C4—2184no—
172C22⋊C4—2181no—
173C22⋊C4—2181no—
174C42—1092no—
175C42—1092no—
176C42C22⋊D10912yescomplemented
177C23—21no—
178C23—8722no—
179C23—1094no—
180C23—2182no—
181C23—2182no—
182C23D4×D10911yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—8724no—
190D4—2182no—
191D4—2182no—
192D4—2182no—
193D4—2182no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—8724no—
199C2×C4—2182no—
200C2×C4—2182no—
201C2×C4C2×C4×D10911yescharacteristic
202C2×C4D4×D10911yescharacteristic
203C2×C4D4×D10911yescharacteristic
204C2×C4C2×C22⋊D10911yescharacteristic
205C2×C4#1318202811yescharacteristic
206C2×C4#1318202911yescharacteristic
207C8—41no—
208C8—4361no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—8722no—
213C22—4364no—
214C22—4364no—
215C22—2184no—
216C22C22⋊C4×D10911yescharacteristic
217C22#1681266711yescharacteristic
218C22#1681268111yescharacteristic
219C4—42no—
220C4—41no—
221C4—4362no—
222C4—4362no—
223C4#1681265911yescharacteristic
224C4C4×C22⋊D10911yescharacteristic
225C4#1681267011yescharacteristic
226C4#1681267211yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—4364no—
232C2#2029490211yescharacteristic
subgroupG/H
centerC2#20294902
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#421×C109C2
socleC2×C109#386
prime radical#20295590C2
cosocleC22×C4×C109C23
hypercenter#421D109
hypocenterC109#1911
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20295500C2semidirect
3. #20295043C2semidirect
4. C2×#16812615C2semidirect
5. #20294783C2semidirect
6. #20294782C2semidirect
7. #421×C109C2semidirect
10. #16812695C4semidirect
11. #16812688C4semidirect
43. C4×Dic109C2×C4semidirect
44. C4×Dic109D4semidirect
49. C2×Q8×C109C2×C4semidirect
51. C2×D4×C109C2×C4semidirect
59. C42×C109D4semidirect
117. C109#1911semidirect
121. #421D109semidirect
147. C2×Q8C4×D109semidirect
156. C2×D4C4×D109semidirect
176. C42C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955902
#202955002
#202950432
C2×#168126152
#202947832
#202947822
#421×C1092
#1911109