Finite Group Databasebeta

← #28361602

Subgroups — #28361602

Order 13952.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D109C211yesmaximal, characteristic
2C22⋊C8×D109C211yesmaximal, characteristic
3#20294885C211yesmaximal, characteristic
4#20294764C211yesmaximal, characteristic, complemented
5#20294763C211yesmaximal, characteristic, complemented
6#20294762C211yesmaximal, characteristic, complemented
7#328×C109C211yesmaximal, characteristic, complemented
8C2×Q8×D109C411yescharacteristic, complemented
9C22×C4×D109C2211yescharacteristic
10C2×C8×D109—21no—
11#16812679C411yescharacteristic, complemented
12#16812678C2211yescharacteristic
13#16812677—21no—
14#16812676—21no—
15#16812675C411yescharacteristic, complemented
16C2×#13182008—21no—
17C4⋊C4×D109—21no—
18C4⋊C4×D109C411yescharacteristic, complemented
19C2×#13182006—21no—
20C22⋊C4×D109—21no—
21C22⋊#11724784C2211yescharacteristic
22#16812611—21no—
23#16812606C2211yescharacteristic
24C22⋊Q8×C109C2211yescharacteristic
25C22⋊C8×C109C2211yescharacteristic
26#95×C109C2211yescharacteristic
27C23×D109D411yescharacteristic
28C2×Dic218—21no—
29C2×Dic218C2×C411yescharacteristic
30Q8×D109—42no—
31C22×Dic109—21no—
32C22×Dic109D411yescharacteristic
33C2×C4×D109—21no—
34C2×C4×D109—21no—
35C2×C4×D109—42no—
36C2×C4×D109C2×C411yescharacteristic
37C2×C4×D109C2×C411yescharacteristic
38C2×C4×D109D411yescharacteristic
39C2×C4×D109D411yescharacteristic
40C2×#11724784—21no—
41C8×D109—42no—
42C4⋊Dic109—21no—
43C4⋊Dic109C2×C411yescharacteristic
44#13182008—21no—
45#13182008—21no—
46#13182008—21no—
47#13182008—42no—
48C22⋊Dic109—21no—
49#13182006—21no—
50#13182006—21no—
51#13182006—42no—
52C2×Q8×C109C2×C411yescharacteristic
53C22×C4×C109—21no—
54C22×C4×C109C2311yescharacteristic
55C2×C8×C109—21no—
56C4⋊C4×C109—21no—
57C4⋊C4×C109C2×C411yescharacteristic
58C22⋊C4×C109—21no—
59C23×C109C2×D411yescharacteristic
60C22×D109—41no—
61C22×D109—41no—
62C22×D109C22⋊C411yescharacteristic
63C22×D109C22⋊C411yescharacteristic
64C22×D109SD1612yes—
65C22×D109Q1612yes—
66Q8×C109—41no—
67C2×C4×C109—21no—
68C2×C4×C109—21no—
69C2×C4×C109—41no—
70C2×C4×C109—41no—
71C2×C4×C109—41no—
72C2×C4×C109—41no—
73C2×C4×C109C22×C411yescharacteristic
74C2×C4×C109C2×D411yescharacteristic
75C2×Dic109—22no—
76C2×Dic109—21no—
77C2×Dic109—21no—
78C2×Dic109—41no—
79C2×Dic109—41no—
80C2×Dic109—41no—
81C2×Dic109C22⋊C411yescharacteristic
82C2×Dic109C22⋊C411yescharacteristic
83C4×D109—22no—
84C4×D109—22no—
85C4×D109—41no—
86C4×D109—41no—
87C4×D109—42no—
88C4×D109—42no—
89Dic218—42no—
90Dic218—41no—
91C8×C109—41no—
92#11724784—41no—
93C22×C109—21no—
94C22×C109—21no—
95C22×C109C2×C22⋊C411yescharacteristic
96C22×C109C2×SD1611yescharacteristic
97C22×C109C2×Q1611yescharacteristic
98C2×D109—21no—
99C2×D109—21no—
100C2×D109—21no—
101C2×D109—21no—
102C2×D109—42no—
103C2×D109—42no—
104C2×D109#10312yes—
105C2×D109#10312yes—
106C4×C109—21no—
107C4×C109—21no—
108C4×C109—41no—
109C4×C109—41no—
110C4×C109—81no—
111Dic109—21no—
112Dic109—41no—
113Dic109—41no—
114Dic109—81no—
115Dic109#9912yes—
116C2×C109—21no—
117C2×C109—21no—
118C2×C109C2×#9911yescharacteristic
119C2×C109C2×#10311yescharacteristic
120C2×C109#42111yescharacteristic
121D109—22no—
122D109—22no—
123#1520—1091nomaximal, Sylow
124C109#152011yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—1091no—
126C2×C22⋊C8—1091no—
127#380—1091no—
128#328—1091no—
129#328—1091no—
130#328—1091no—
131#328D10911yescharacteristic, complemented
132C22×Q8—1091no—
133C23×C4—1091no—
134C22×C8—2181no—
135C22⋊Q8—1091no—
136C22⋊Q8—1091no—
137C22⋊Q8—1091no—
138C22⋊Q8—2181no—
139C22⋊Q8—2181no—
140C22⋊Q8C2×D10911yescharacteristic
141C2×C4⋊C4—1091no—
142C2×C4⋊C4—2181no—
143C2×C4⋊C4—2181no—
144C2×C22⋊C4—2181no—
145C2×C22⋊C4—2181no—
146C22⋊C8—1091no—
147C22⋊C8—2181no—
148C22⋊C8C2×D10911yescharacteristic
149#95—1091no—
150#95C2×D10911yescharacteristic
151C24—1091no—
152C2×Q8—1091no—
153C2×Q8—4362no—
154C2×Q8—2181no—
155C2×Q8C4×D10911yescharacteristic
156C22×C4—21no—
157C22×C4—1091no—
158C22×C4—1091no—
159C22×C4—1091no—
160C22×C4—1091no—
161C22×C4—1091no—
162C22×C4—4362no—
163C22×C4—2181no—
164C22×C4—2181no—
165C22×C4—2181no—
166C22×C4C22×D10911yescharacteristic
167C2×C8—21no—
168C2×C8—4362no—
169C2×C8—2181no—
170C4⋊C4—21no—
171C4⋊C4—1091no—
172C4⋊C4—4362no—
173C4⋊C4—2181no—
174C4⋊C4—2181no—
175C4⋊C4—2181no—
176C4⋊C4—2181no—
177C4⋊C4C4×D10911yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4362no—
180C22⋊C4—2181no—
181C22⋊C4—2181no—
182C22⋊C4—2181no—
183C23—1092no—
184C23—1092no—
185C23—1091no—
186C23—1091no—
187C23—4361no—
188C23—4361no—
189C23D4×D10911yescharacteristic
190Q8—41no—
191Q8—4361no—
192Q8—4362no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1091no—
200C2×C4—1091no—
201C2×C4—4361no—
202C2×C4—4361no—
203C2×C4—4362no—
204C2×C4—4362no—
205C2×C4—4361no—
206C2×C4—4361no—
207C2×C4—4361no—
208C2×C4—2182no—
209C2×C4—2182no—
210C2×C4—2182no—
211C2×C4—2181no—
212C2×C4—2181no—
213C2×C4C2×C4×D10911yescharacteristic
214C2×C4D4×D10911yescharacteristic
215C8—41no—
216C8—4361no—
217C22—21no—
218C22—21no—
219C22—1092no—
220C22—1092no—
221C22—4362no—
222C22—4362no—
223C22—2181no—
224C22—2181no—
225C22—2181no—
226C22—2181no—
227C22C22⋊C4×D10911yescharacteristic
228C22SD16×D10911yescharacteristic
229C22Q16×D10911yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—8721no—
236C4—1092no—
237C4—4361no—
238C4—4361no—
239C4—2181no—
240C2—21no—
241C2—21no—
242C2—2182no—
243C2—2182no—
244C2#99×D10911yescharacteristic
245C2#103×D10911yescharacteristic
246C2#2029504111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D109
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#328×C109C2
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC22×C4×C109C23
hypercenter#328D109
hypocenterC109#1520
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #20294764C2semidirect
5. #20294763C2semidirect
6. #20294762C2semidirect
7. #328×C109C2semidirect
8. C2×Q8×D109C4semidirect
11. #16812679C4semidirect
15. #16812675C4semidirect
18. C4⋊C4×D109C4semidirect
124. C109#1520semidirect
131. #328D109semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1092
C22⋊C8×D1092
#202948852
#202947642
#202947632
#202947622
#328×C1092
#1520109