Finite Group Databasebeta

← #28366305

Subgroups — #28366305

Order 13952.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#16812664C211yesmaximal, characteristic, complemented
2C2×#16812657C211yesmaximal, characteristic
3#20294961C211yesmaximal, characteristic
4#20294960C212yesmaximal, complemented
5#20294950C212yesmaximal
6#20294946C211yesmaximal, characteristic, complemented
7#20294939C211yesmaximal, characteristic
8#20294877C211yesmaximal, characteristic, complemented
9#20294877C212yesmaximal, complemented
10#20294873C212yesmaximal
11#400×C109C211yesmaximal, characteristic, complemented
12C22×Dic218C2211yescharacteristic
13C22×C4×D109C2211yescharacteristic
14#16812664—24no—
15#16812657—24no—
16C2×C4⋊Dic109C2211yescharacteristic
17C2×#13182008C2212yes—
18C2×#13182008C2212yes—
19C2×#13182008C2212yes—
20C2×#13182008C2212yes—
21C2×#13182006C2212yes—
22C2×#13182006C2211yescharacteristic
23C2×#13182006C2211yescharacteristic
24C2×#13182006C2212yes—
25C2×C4×Dic109C2211yescharacteristic
26C2×C4×Dic109C2212yes—
27C2×C4×Dic109C2212yes—
28C2×C4×Dic109C2211yescharacteristic
29#16812606C2212yes—
30#16812606C2212yes—
31#16812606C2211yescharacteristic
32C2×C4⋊C4×C109C2211yescharacteristic
33C2×C4⋊C4×C109C2211yescharacteristic
34#95×C109C2211yescharacteristic
35#95×C109C2212yes—
36#95×C109C2212yes—
37C23×D109C2311yescharacteristic
38C2×Dic218—22no—
39C2×Dic218—24no—
40C22×Dic109C2311yescharacteristic
41C22×Dic109C2311yescharacteristic
42C22×Dic109C2312yes—
43C22×Dic109C2312yes—
44C22×Dic109C2311yescharacteristic
45C2×C4×D109—22no—
46C2×C4×D109—22no—
47C2×C4×D109Q814yescomplemented
48C4⋊Dic109—22no—
49#13182008—24no—
50#13182008—24no—
51#13182008—24no—
52#13182008—24no—
53#13182006—22no—
54#13182006—22no—
55#13182006—24no—
56#13182006—24no—
57C4×Dic109—22no—
58C4×Dic109—24no—
59C4×Dic109—24no—
60C4×Dic109D414yes—
61C22×C4×C109C2312yes—
62C22×C4×C109C2311yescharacteristic
63C22×C4×C109C2312yes—
64C22×C4×C109C2311yescharacteristic
65C22×C4×C109C2311yescharacteristic
66C4⋊C4×C109—22no—
67C4⋊C4×C109—22no—
68C23×C109C2411yescharacteristic
69C22×D109—21no—
70C22×D109—22no—
71C22×D109—22no—
72C22×D109—21no—
73C22×D109C2×Q812yes—
74C2×C4×C109—21no—
75C2×C4×C109—21no—
76C2×C4×C109—22no—
77C2×C4×C109—21no—
78C2×C4×C109—22no—
79C2×C4×C109—22no—
80C2×C4×C109—22no—
81C2×C4×C109—22no—
82C2×C4×C109—22no—
83C2×C4×C109—22no—
84C2×C4×C109—22no—
85C2×C4×C109C2×D412yes—
86C2×C4×C109C2×Q812yes—
87C2×Dic109—22no—
88C2×Dic109—22no—
89C2×Dic109—22no—
90C2×Dic109—22no—
91C2×Dic109—22no—
92C2×Dic109—21no—
93C2×Dic109—22no—
94C2×Dic109C2×D412yes—
95C2×Dic109C2×D412yes—
96C2×Dic109C2×Q812yes—
97C2×Dic109#4114yes—
98C2×Dic109#4114yes—
99C2×Dic109#4112yes—
100C4×D109—28no—
101Dic218—28no—
102C22×C109C22×D411yescharacteristic
103C22×C109C22×Q811yescharacteristic
104C22×C109C2×#4111yescharacteristic
105C22×C109C2×#4112yes—
106C22×C109C2×#4112yes—
107C2×D109—22no—
108C2×D109—24no—
109C2×D109—41no—
110C2×D109—42no—
111C2×D109—41no—
112C4×C109—22no—
113C4×C109—22no—
114C4×C109—41no—
115C4×C109—42no—
116C4×C109—42no—
117Dic109—24no—
118Dic109—24no—
119Dic109—22no—
120Dic109—41no—
121Dic109C22⋊Q814yes—
122C2×C109C2×C22⋊Q811yescharacteristic
123C2×C109#52511yescharacteristic
124C2×C109#52912yes—
125C2×C109#53211yescharacteristic
126C2×C109#53312yes—
127D109—42no—
128#2396—1091nomaximal, Sylow
129C109#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1091no—
131C2×#117—1091no—
132#400D10911yescharacteristic, complemented
133#399—1091no—
134#398—1092no—
135#396—1092no—
136#395—1091no—
137#393—1091no—
138C4×C4⋊C4—1092no—
139C4×C4⋊C4—1091no—
140C4×C22⋊C4—1092no—
141C22×Q8—1091no—
142C23×C4—1091no—
143C4×Q8—2184no—
144#117—2184no—
145C2×C4⋊C4—1091no—
146C2×C4⋊C4—1092no—
147C2×C4⋊C4—1092no—
148C2×C4⋊C4—1092no—
149C2×C4⋊C4—1092no—
150C2×C4⋊C4C2×D10911yescharacteristic
151C2×C4⋊C4C2×D10911yescharacteristic
152C2×C22⋊C4—1091no—
153C2×C22⋊C4—1091no—
154C2×C22⋊C4—1092no—
155C2×C22⋊C4—1092no—
156C2×C42—1091no—
157C2×C42—1091no—
158C2×C42—1092no—
159C2×C42—1092no—
160#95—1092no—
161#95—1092no—
162#95—1091no—
163#95C2×D10912yes—
164#95C2×D10911yescharacteristic
165#95C2×D10912yes—
166C24—1091no—
167C2×Q8—2184no—
168C2×Q8—2182no—
169C22×C4—1091no—
170C22×C4—1092no—
171C22×C4—1092no—
172C22×C4—1091no—
173C22×C4—1094no—
174C22×C4—1091no—
175C22×C4—2182no—
176C22×C4—2182no—
177C22×C4C22×D10912yes—
178C22×C4C22×D10911yescharacteristic
179C22×C4C22×D10912yes—
180C22×C4C22×D10911yescharacteristic
181C22×C4C22×D10911yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—2184no—
185C4⋊C4—2184no—
186C4⋊C4—2184no—
187C4⋊C4—2184no—
188C4⋊C4—2182no—
189C22⋊C4—2184no—
190C22⋊C4—2184no—
191C22⋊C4—2182no—
192C22⋊C4—2182no—
193C42—1094no—
194C42—2184no—
195C42—2182no—
196C42—2184no—
197C23—1092no—
198C23—2181no—
199C23—2182no—
200C23—2182no—
201C23—2181no—
202C23C23×D10911yescharacteristic
203Q8—2188no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1092no—
216C2×C4—1092no—
217C2×C4—1092no—
218C2×C4—1092no—
219C2×C4—1094no—
220C2×C4—1094no—
221C2×C4—2188no—
222C2×C4—2181no—
223C2×C4—2182no—
224C2×C4—2182no—
225C2×C4—2182no—
226C2×C4—2182no—
227C2×C4—2182no—
228C2×C4—2182no—
229C2×C4D4×D10912yes—
230C2×C4Q8×D10912yes—
231C22—4361no—
232C22—4362no—
233C22—4361no—
234C22—2182no—
235C22—2184no—
236C22C2×D4×D10911yescharacteristic
237C22C2×Q8×D10911yescharacteristic
238C22#41×D10911yescharacteristic
239C22#41×D10912yes—
240C22#41×D10912yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—1094no—
247C4—4361no—
248C4—2184no—
249C4—2184no—
250C4—2182no—
251C2—4362no—
252C2C22⋊Q8×D10911yescharacteristic
253C2#2029553411yescharacteristic
254C2#2029555912yes—
255C2#2029557511yescharacteristic
256C2#2029558212yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
Fitting#400×C109C2
prime radicalC23×D109C23
hypercenter#400D109
hypocenterC109#2396
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16812664C2semidirect
4. #20294960C2semidirect
6. #20294946C2semidirect
8. #20294877C2semidirect
9. #20294877C2semidirect
11. #400×C109C2semidirect
47. C2×C4×D109Q8semidirect
129. C109#2396semidirect
132. #400D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126642
C2×#168126572
#202949612
#202949602
#202949502
#202949462
#202949392
#202948772
#202948732
#400×C1092
#2396109