Finite Group Databasebeta

← #34009937

Subgroups — #34009937

Order 18688.

386 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 86 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#22626159C212yesmaximal, complemented
2#22624869C212yesmaximal, complemented
3#22624854C212yesmaximal, complemented
4#22618627C211yesmaximal, characteristic
5C2×#14556666—22no—
6#17989482—22no—
7C22⋊#11924031C2214yes—
8C2×#14556597C411yescharacteristic
9C22×C8×D73C2211yescharacteristic
10#17989099C411yescharacteristic
11#17988660—22no—
12C8×D292—22no—
13#17988614C2211yescharacteristic
14#17988555C2211yescharacteristic
15#17988550C411yescharacteristic
16#405×C73C411yescharacteristic
17#14556666—44no—
18C2×#11924031—24no—
19C2×#11924028C2×C411yescharacteristic
20C22×C4×D73C2×C411yescharacteristic
21#14556597—42no—
22C2×#11924012C2×C411yescharacteristic
23C2×#11924011—21no—
24C22×#10596831C2311yescharacteristic
25C2×C8×D73—21no—
26C2×C8×D73—42no—
27C2×C8×D73D412yes—
28C2×C8×D73D412yes—
29#14556575—21no—
30#14556571C812yes—
31#14556566C812yes—
32#14556561—21no—
33C4×D292—22no—
34C4⋊#10596831—22no—
35#14556503—22no—
36#14556503—22no—
37C2×M16×C73C2×C411yescharacteristic
38C22×C8×C73C2×C411yescharacteristic
39#117×C73C2×C411yescharacteristic
40#97×C73—21no—
41C4×C8×C73—21no—
42#11924031—44no—
43C23×D73M1611yescharacteristic
44#11924028—41no—
45#11924028—41no—
46C2×Dic146C2×C811yescharacteristic
47C2×D292—22no—
48C2×D292C2×C811yescharacteristic
49C2×C22⋊D73—21no—
50C22×Dic73M1611yescharacteristic
51C2×C4×D73—21no—
52C2×C4×D73—41no—
53C2×C4×D73—41no—
54C2×C4×D73C22⋊C412yes—
55C2×C4×D73C2×C812yes—
56#11924012—41no—
57#11924011—42no—
58C2×#10596831—21no—
59C2×#10596831—41no—
60C2×#10596831C2×D412yes—
61C8×D73—24no—
62C8×D73—42no—
63C8×D73—42no—
64C4⋊Dic73—22no—
65#11924005—22no—
66#11924005—22no—
67C22×C4×C73C22×C411yescharacteristic
68M16×C73—41no—
69C2×C8×C73—22no—
70C2×C8×C73—21no—
71C2×C8×C73C22⋊C411yescharacteristic
72C2×C8×C73C22⋊C411yescharacteristic
73C4⋊C4×C73C2×C812yes—
74C22⋊C4×C73—21no—
75C42×C73—21no—
76C23×C73C2×M1611yescharacteristic
77C22×D73—21no—
78C22×D73—41no—
79C22×D73—42no—
80C22×D73C22⋊C812yes—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—22no—
84C2×C4×C73C2×C22⋊C411yescharacteristic
85C2×C4×C73C22×C811yescharacteristic
86C22⋊D73—42no—
87C2×Dic73—21no—
88C2×Dic73—41no—
89C2×Dic73C22⋊C812yes—
90D292—41no—
91D292—44no—
92C4×D73—22no—
93C4×D73—22no—
94C4×D73—42no—
95C4×D73—41no—
96C4×D73—41no—
97Dic146—41no—
98C8×C73—21no—
99C8×C73—41no—
100C8×C73#9912yes—
101#10596831—22no—
102#10596831—41no—
103C22×C73—22no—
104C22×C73—21no—
105C22×C73C2×C22⋊C811yescharacteristic
106C2×D73—22no—
107C2×D73—22no—
108C2×D73—22no—
109C2×D73—44no—
110C2×D73—42no—
111C2×D73—41no—
112C4×C73—21no—
113C4×C73—42no—
114C4×C73C2×#9911yescharacteristic
115C4×C73#41311yescharacteristic
116Dic73—22no—
117Dic73—41no—
118#11469—731nomaximal, Sylow
119C2×C73—41no—
120C2×C73#145911yescharacteristic
121C2×C73#211712yes—
122D73—42no—
123D73—81no—
124#2929—731no—
125#2112—732no—
126#1323—732no—
127#1315—732no—
128C73#1146911yescharacteristic, complemented, Sylow
129C2×#131—731no—
130C23×C8—731no—
131#525—731no—
132C2×M32—1462no—
133#435—1462no—
134C8×D4—1462no—
135#424—731no—
136#408—731no—
137#407—731no—
138#405#6440811yescharacteristic
139C22⋊C16—734no—
140C22⋊C16—1462no—
141C2×#41—731no—
142C23×C4—731no—
143#131—2922no—
144C2×M16—731no—
145C2×M16—1461no—
146C2×M16C2×#6440811yescharacteristic
147C22×C8—2922no—
148C22×C8—731no—
149C22×C8—732no—
150C22×C8—732no—
151C22×C8—1461no—
152C22×C8C2×#6440811yescharacteristic
153#123—1461no—
154C22⋊Q8—732no—
155#121—732no—
156#120—1461no—
157C4×D4—1462no—
158#117C2×#6440811yescharacteristic
159M32—2924no—
160C2×C16—1464no—
161C4⋊C8—1462no—
162C22⋊C8—1462no—
163C22⋊C8—1462no—
164#97—21no—
165C4×C8—21no—
166C24—731no—
167#41—2921no—
168#41—2921no—
169C2×Q8—731no—
170C2×D4—731no—
171C2×D4—1462no—
172C2×D4—1461no—
173C22×C4—2921no—
174C22×C4—2921no—
175C22×C4—731no—
176C22×C4—732no—
177C22×C4—732no—
178C22×C4—1461no—
179C22×C4C22×#6440811yescharacteristic
180M16—41no—
181M16—2922no—
182M16—2921no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—2922no—
186C2×C8—2922no—
187C2×C8—2921no—
188C2×C8—732no—
189C2×C8—1461no—
190C2×C8—1464no—
191C2×C8C22⋊#6440811yescharacteristic
192C2×C8C22⋊#6440811yescharacteristic
193C4⋊C4—1462no—
194C4⋊C4#1192403612yes—
195C22⋊C4—21no—
196C22⋊C4—1462no—
197C22⋊C4—1462no—
198C42—21no—
199C16—2924no—
200C23—2921no—
201C23—2922no—
202C23—732no—
203C23—1461no—
204C23D73⋊M1611yescharacteristic
205Q8—2921no—
206D4—2921no—
207D4—2924no—
208D4—2922no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—2922no—
213C2×C4—2921no—
214C2×C4—2921no—
215C2×C4—2921no—
216C2×C4—732no—
217C2×C4—1461no—
218C2×C4—1462no—
219C2×C4—1462no—
220C2×C4C2×C22⋊#6440811yescharacteristic
221C2×C4C2×#1192403611yescharacteristic
222C8—21no—
223C8—41no—
224C8—2921no—
225C8—1462no—
226C8#1455665012yes—
227C22—22no—
228C22—21no—
229C22—2924no—
230C22—2922no—
231C22—2921no—
232C22—1462no—
233C22—1462no—
234C22—1462no—
235C22#1798955511yescharacteristic
236C4—21no—
237C4—42no—
238C4—2921no—
239C4—1462no—
240C4C2×#1455665011yescharacteristic
241C4#1798957311yescharacteristic
242C2—41no—
243C2—2922no—
244C2—5841no—
245C2#2262516811yescharacteristic
246C2#2262617412yes—
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#405×C73C4
socleC22×C73C2×C22⋊C8
prime radical#17989099C4
cosocleC22×#10596831C23
hypercenter#405#64408
hypocenterC73#11469
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #22626159C2semidirect
2. #22624869C2semidirect
3. #22624854C2semidirect
128. C73#11469semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226248692
#226248542
#226186272
#1146973