Finite Group Databasebeta

← #34009890

Subgroups — #34009890

Order 18688.

378 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 88 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#22626165C212yesmaximal
2#22624880C212yesmaximal, complemented
3#22624879C212yesmaximal, complemented
4#424×D73C211yesmaximal, characteristic
5#17989643—22no—
6#17989484—22no—
7#17989483C2214yes—
8C2×M16×D73C2211yescharacteristic
9C22×C8×D73C411yescharacteristic
10#117×D73C411yescharacteristic
11#17988615C411yescharacteristic
12#17988614C2211yescharacteristic
13#17988613C2211yescharacteristic
14#17988612—22no—
15C4⋊C8×D73—22no—
16#424×C73C411yescharacteristic
17#14556666—42no—
18#14556665—24no—
19#14556664—42no—
20C22×C4×D73C2×C411yescharacteristic
21C2×#11924012C2311yescharacteristic
22C2×#11924011D412yes—
23M16×D73—41no—
24M16×D73—41no—
25C22×#10596831C2×C411yescharacteristic
26C2×C8×D73—21no—
27C2×C8×D73—21no—
28C2×C8×D73—41no—
29C2×C8×D73—41no—
30C2×C8×D73D412yes—
31#14556550C2×C411yescharacteristic
32#14556549—21no—
33#14556548—21no—
34#14556547C812yes—
35C4⋊C4×D73—21no—
36C22⋊C4×D73C812yes—
37C42×D73—21no—
38C4⋊#10596831—22no—
39#14556520—22no—
40#14556520—22no—
41C2×M16×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43#117×C73C2×C411yescharacteristic
44C4⋊C8×C73—22no—
45#11924031—44no—
46C23×D73C2×C811yescharacteristic
47C22×Dic73C2×C811yescharacteristic
48C2×C4×D73—21no—
49C2×C4×D73—21no—
50C2×C4×D73—21no—
51C2×C4×D73—21no—
52C2×C4×D73—22no—
53C2×C4×D73C22⋊C411yescharacteristic
54C2×C4×D73C22⋊C411yescharacteristic
55C2×C4×D73M1611yescharacteristic
56C2×C4×D73M1611yescharacteristic
57#11924012—41no—
58#11924011—42no—
59C2×#10596831—21no—
60C2×#10596831—41no—
61C2×#10596831C2×D412yes—
62C8×D73—22no—
63C8×D73—22no—
64C8×D73—42no—
65C8×D73—41no—
66C8×D73—41no—
67C4⋊Dic73—21no—
68#11924007—22no—
69C22⋊Dic73C2×C812yes—
70#11924005—22no—
71C4×Dic73—21no—
72C4×Dic73—22no—
73C22×C4×C73C22×C411yescharacteristic
74M16×C73—41no—
75C2×C8×C73—21no—
76C2×C8×C73—41no—
77C2×C8×C73C22⋊C412yes—
78C4⋊C4×C73—21no—
79C22⋊C4×C73C2×C812yes—
80C42×C73—21no—
81C23×C73C22×C811yescharacteristic
82C22×D73—21no—
83C22×D73—21no—
84C22×D73—42no—
85C22×D73C22⋊C811yescharacteristic
86C22×D73C22⋊C811yescharacteristic
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
90C2×C4×C73C2×C22⋊C411yescharacteristic
91C2×C4×C73C2×M1611yescharacteristic
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—22no—
95C2×Dic73C22⋊C811yescharacteristic
96C2×Dic73C22⋊C811yescharacteristic
97C4×D73—21no—
98C4×D73—21no—
99C4×D73—21no—
100C4×D73—21no—
101C4×D73—21no—
102C4×D73—21no—
103C4×D73—44no—
104C4×D73#9911yescharacteristic
105C4×D73#9911yescharacteristic
106C4×D73#10111yescharacteristic
107C4×D73#10111yescharacteristic
108C8×C73—22no—
109C8×C73—41no—
110#10596831—22no—
111#10596831—41no—
112C22×C73—21no—
113C22×C73—41no—
114C22×C73C2×C22⋊C811yescharacteristic
115C2×D73—22no—
116C2×D73—21no—
117C2×D73—42no—
118C2×D73—42no—
119C2×D73—41no—
120C2×D73#32411yescharacteristic
121C2×D73#32411yescharacteristic
122C4×C73—21no—
123C4×C73—42no—
124C4×C73C2×#9911yescharacteristic
125C4×C73C2×#10111yescharacteristic
126Dic73—21no—
127Dic73—42no—
128Dic73#32411yescharacteristic
129Dic73#32411yescharacteristic
130C2×#1328—731nomaximal, Sylow
131C2×C73—21no—
132C2×C73—41no—
133C2×C73C2×#32411yescharacteristic
134D73—21no—
135D73—41no—
136D73#132812yescomplemented
137C2×#424—731no—
138C2×#349—732no—
139#1328—732no—
140#1328—732no—
141C73C2×#132811yescharacteristic, complemented, Sylow
142C22×M16—731no—
143C23×C8—731no—
144C2×#117—731no—
145C2×M32—1462no—
146#424—731no—
147#424—731no—
148#424—731no—
149#424—1462no—
150#424#6440811yescharacteristic
151C2×C4⋊C8—1462no—
152#349—734no—
153#349—1462no—
154C23×C4—731no—
155C2×M16—2921no—
156C2×M16—2921no—
157C2×M16—731no—
158C2×M16—732no—
159C2×M16C2×#6440811yescharacteristic
160C22×C8—2921no—
161C22×C8—2921no—
162C22×C8—731no—
163C22×C8—732no—
164C22×C8—1461no—
165C22×C8—1461no—
166C22×C8C2×#6440811yescharacteristic
167#117—731no—
168#117—732no—
169#117—1461no—
170#117—1461no—
171#117C2×#6440811yescharacteristic
172C2×C4⋊C4—1461no—
173C2×C22⋊C4—732no—
174C2×C42—1461no—
175M32—2922no—
176M32—1464no—
177C2×C16—2922no—
178C4⋊C8—22no—
179C4⋊C8—1462no—
180C4⋊C8—1462no—
181C4⋊C8—1462no—
182C24—731no—
183C22×C4—731no—
184C22×C4—731no—
185C22×C4—731no—
186C22×C4—731no—
187C22×C4—731no—
188C22×C4—1462no—
189C22×C4—1461no—
190C22×C4—1461no—
191C22×C4—1461no—
192C22×C4—1461no—
193C22×C4C22×#6440811yescharacteristic
194M16—41no—
195M16—2922no—
196M16—2921no—
197C2×C8—21no—
198C2×C8—41no—
199C2×C8—2921no—
200C2×C8—2921no—
201C2×C8—2922no—
202C2×C8—2921no—
203C2×C8—732no—
204C2×C8—1462no—
205C2×C8—1462no—
206C2×C8—1461no—
207C2×C8C22⋊#6440812yes—
208C4⋊C4—21no—
209C4⋊C4—1462no—
210C4⋊C4—1461no—
211C22⋊C4—732no—
212C22⋊C4—1462no—
213C22⋊C4#1192403612yes—
214C42—21no—
215C42—1462no—
216C42—1461no—
217C16—2924no—
218C23—2922no—
219C23—731no—
220C23—731no—
221C23—1461no—
222C23—1461no—
223C23C2×#1192403611yescharacteristic
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—2924no—
228C2×C4—731no—
229C2×C4—731no—
230C2×C4—731no—
231C2×C4—731no—
232C2×C4—731no—
233C2×C4—731no—
234C2×C4—1461no—
235C2×C4—1461no—
236C2×C4—1461no—
237C2×C4—1461no—
238C2×C4—1462no—
239C2×C4—1461no—
240C2×C4—1461no—
241C2×C4—1461no—
242C2×C4—1461no—
243C2×C4C2×C22⋊#6440811yescharacteristic
244C2×C4D73⋊M1611yescharacteristic
245C8—22no—
246C8—41no—
247C8—2921no—
248C8—1462no—
249C22—21no—
250C22—41no—
251C22—2921no—
252C22—2922no—
253C22—2922no—
254C22—731no—
255C22—731no—
256C22—1462no—
257C22—1461no—
258C22#1798955511yescharacteristic
259C4—21no—
260C4—42no—
261C4—2922no—
262C4—731no—
263C4—731no—
264C4—1461no—
265C4D73⋊#9911yescharacteristic
266C4D73⋊#10111yescharacteristic
267C2—21no—
268C2—41no—
269C2—2921no—
270C2—732no—
271C2—1461no—
272C2#2262516011yescharacteristic
subgroupG/H
centerC4D73⋊#99
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#424×C73C4
socleC2×C73C2×#324
prime radicalC23×D73C2×C8
cosocleC2×#11924012C23
hypercenter#424#64408
hypocenterC73C2×#1328
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #22624880C2semidirect
3. #22624879C2semidirect
136. D73#1328semidirect
141. C73C2×#1328semidirect

Maximal subgroups

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

subgroupindex
#226261652
#226248802
#226248792
#424×D732
C2×#132873