Finite Group Databasebeta

← #34009934

Subgroups — #34009934

Order 18688.

398 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 98 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×C22⋊#11924031C212yesmaximal, complemented
2#22624870C212yesmaximal, complemented
3#22624854C212yesmaximal, complemented
4#22618621C211yesmaximal, characteristic
5C22×#11924031—22no—
6C22⋊#11924031—22no—
7C22⋊#11924031C2214yes—
8C2×#14556599C411yescharacteristic
9C23×#10596831C2211yescharacteristic
10#17989098C411yescharacteristic
11#17988655—22no—
12#17988650—22no—
13#17988615C2211yescharacteristic
14#17988555C2211yescharacteristic
15#17988545C411yescharacteristic
16#407×C73C411yescharacteristic
17C2×#11924031—24no—
18C2×#11924031—44no—
19C2×#11924028C2×C411yescharacteristic
20#14556599—42no—
21C2×#11924011—21no—
22C22×#10596831—42no—
23C22×#10596831C2×C411yescharacteristic
24C22×#10596831D412yes—
25C22×#10596831D412yes—
26C22×#10596831C2311yescharacteristic
27C2×C8×D73—21no—
28#14556574—21no—
29#14556570C812yes—
30#14556565C812yes—
31#14556560—21no—
32C4×C22⋊D73—22no—
33#14556550C2×C411yescharacteristic
34#14556520—22no—
35#14556503—22no—
36#14556501—21no—
37C8×Dic73—21no—
38C23×C4×C73C2×C411yescharacteristic
39C2×M16×C73C2×C411yescharacteristic
40C2×M16×C73C2×C411yescharacteristic
41C22⋊C8×C73—22no—
42#11924031—44no—
43#11924028—41no—
44#11924028—41no—
45C2×Dic146C2×C811yescharacteristic
46C2×D292C2×C811yescharacteristic
47C2×C22⋊D73—22no—
48C2×C22⋊D73—21no—
49C2×C4×D73—21no—
50#11924011—42no—
51C2×#10596831—22no—
52C2×#10596831—24no—
53C2×#10596831—42no—
54C2×#10596831—41no—
55C2×#10596831C22⋊C411yescharacteristic
56C2×#10596831C22⋊C411yescharacteristic
57C2×#10596831C2×D412yes—
58C8×D73—42no—
59C4⋊Dic73C2×C812yes—
60#11924007—22no—
61C22⋊Dic73—22no—
62C22⋊Dic73—21no—
63#11924005—22no—
64C4×Dic73—21no—
65C24×C73M1611yescharacteristic
66C22×C4×C73—41no—
67C22×C4×C73—41no—
68C22×C4×C73C22⋊C412yes—
69C22×C4×C73C2×C812yes—
70C22×C4×C73M1611yescharacteristic
71C22×C4×C73C22×C411yescharacteristic
72M16×C73—41no—
73M16×C73—41no—
74C2×C8×C73—21no—
75C2×C8×C73—21no—
76C23×C73—24no—
77C23×C73—41no—
78C23×C73C22⋊C812yes—
79C23×C73C2×M1611yescharacteristic
80C22×D73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—22no—
83C2×C4×C73—21no—
84C2×C4×C73—22no—
85C2×C4×C73—42no—
86C2×C4×C73—41no—
87C2×C4×C73C22⋊C812yes—
88C2×C4×C73C2×C22⋊C411yescharacteristic
89C2×C4×C73C22×C811yescharacteristic
90C22⋊D73—42no—
91C22⋊D73—44no—
92C2×Dic73—22no—
93C2×Dic73—21no—
94D292—41no—
95C4×D73—41no—
96C4×D73—41no—
97Dic146—41no—
98C8×C73—41no—
99C8×C73—41no—
100#10596831—21no—
101#10596831—22no—
102#10596831#9912yes—
103C22×C73—22no—
104C22×C73—22no—
105C22×C73—22no—
106C22×C73—21no—
107C22×C73—44no—
108C22×C73#35014yes—
109C22×C73C2×C22⋊C811yescharacteristic
110C2×D73—42no—
111C2×D73—41no—
112C4×C73—21no—
113C4×C73—22no—
114C4×C73C2×#9911yescharacteristic
115C4×C73#41311yescharacteristic
116Dic73—42no—
117Dic73—41no—
118#11468—731nomaximal, Sylow
119C2×C73—24no—
120C2×C73—41no—
121C2×C73#145911yescharacteristic
122C2×C73C2×#35012yes—
123D73—81no—
124#2929—731no—
125C2×C22⋊C16—732no—
126#1323—732no—
127#1315—732no—
128C73#1146811yescharacteristic, complemented, Sylow
129C2×#131—731no—
130C23×C8—731no—
131#525—731no—
132C22×C16—1462no—
133#435—1462no—
134C8×D4—1462no—
135#424—731no—
136#408—731no—
137#407#6440811yescharacteristic
138#405—731no—
139C22⋊C16—734no—
140C22⋊C16—1462no—
141C2×#41—731no—
142C23×C4C2×#6440811yescharacteristic
143#131—2922no—
144C2×M16—1461no—
145C2×M16C2×#6440811yescharacteristic
146C2×M16C2×#6440811yescharacteristic
147C22×C8—2922no—
148C22×C8—732no—
149C22×C8—731no—
150C22×C8—732no—
151C22×C8—731no—
152C22×C8—1461no—
153#123—1461no—
154C22⋊Q8—732no—
155#121—732no—
156#120—1461no—
157C4×D4—1462no—
158#117—731no—
159C2×C16—2924no—
160C2×C16—1464no—
161C4⋊C8—1462no—
162C22⋊C8—22no—
163C22⋊C8—1462no—
164#97—1461no—
165C4×C8—1461no—
166C24#1192403811yescharacteristic
167#41—2921no—
168#41—2921no—
169C2×Q8—731no—
170C2×D4—731no—
171C2×D4—1461no—
172C2×D4—1462no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—1461no—
176C22×C4C22⋊#6440812yes—
177C22×C4#1192403612yes—
178C22×C4#1192403811yescharacteristic
179C22×C4C22×#6440811yescharacteristic
180M16—41no—
181M16—41no—
182M16—2922no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—2922no—
186C2×C8—2921no—
187C2×C8—2922no—
188C2×C8—731no—
189C2×C8—731no—
190C2×C8—732no—
191C2×C8—1464no—
192C2×C8—1462no—
193C4⋊C4—732no—
194C4⋊C4—1462no—
195C22⋊C4—1461no—
196C22⋊C4—1462no—
197C22⋊C4—1462no—
198C42—1461no—
199C16—2924no—
200C23—24no—
201C23—41no—
202C23—1461no—
203C23#1455664812yes—
204C23C2×#1192403811yescharacteristic
205Q8—2921no—
206D4—2921no—
207D4—2922no—
208D4—2924no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—2921no—
216C2×C4—2921no—
217C2×C4—1462no—
218C2×C4—1461no—
219C2×C4#1455664812yes—
220C2×C4C2×C22⋊#6440811yescharacteristic
221C2×C4C2×#1192403611yescharacteristic
222C8—41no—
223C8—41no—
224C8—732no—
225C8—1462no—
226C8—1461no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—21no—
231C22—44no—
232C22—2922no—
233C22—2921no—
234C22#1798948614yes—
235C22C2×#1455664811yescharacteristic
236C4—21no—
237C4—22no—
238C4—2922no—
239C4—2921no—
240C4D73⋊#9911yescharacteristic
241C4#1798957411yescharacteristic
242C2—24no—
243C2—41no—
244C2—5841no—
245C2#2262516911yescharacteristic
246C2C2×#1798948612yes—
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#407×C73C4
socleC22×C73C2×C22⋊C8
prime radical#17989098C4
cosocleC22×#10596831C23
hypercenter#407#64408
hypocenterC73#11468
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#11924031C2semidirect
2. #22624870C2semidirect
3. #22624854C2semidirect
128. C73#11468semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#119240312
#226248702
#226248542
#226186212
#1146873