Finite Group Databasebeta

← #34009995

Subgroups — #34009995

Order 18688.

368 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 82 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#22626168C212yesmaximal, complemented
2#22624880C212yesmaximal, complemented
3#22624863C212yesmaximal, complemented
4#22618634C211yesmaximal, characteristic
5#17989643—22no—
6#17989484—22no—
7#17989483C2214yes—
8C2×#14556597C2211yescharacteristic
9#17989098C411yescharacteristic
10#17988663—22no—
11#17988656—22no—
12#17988613C2211yescharacteristic
13#17988553C411yescharacteristic
14#17988551C2211yescharacteristic
15#17988542C411yescharacteristic
16C22×M16×C73C411yescharacteristic
17#14556666—42no—
18#14556665—24no—
19#14556664—42no—
20C2×#11924028C2×C411yescharacteristic
21#14556597—42no—
22C2×#11924012C2×C411yescharacteristic
23C2×#11924012C2311yescharacteristic
24C2×#11924011D412yes—
25C2×C8×D73D412yes—
26#14556574C812yes—
27#14556570—21no—
28#14556565—21no—
29#14556560C812yes—
30C4×C22⋊D73—22no—
31#14556550C2×C411yescharacteristic
32#14556520—22no—
33C22⋊#10596831—22no—
34#14556503—22no—
35#14556501—21no—
36C8×Dic73—21no—
37C23×C4×C73C2×C411yescharacteristic
38C2×M16×C73—21no—
39C2×M16×C73—21no—
40C2×M16×C73—42no—
41C22×C8×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43#11924031—44no—
44#11924028—41no—
45#11924028—41no—
46C2×Dic146M1611yescharacteristic
47C2×D292M1611yescharacteristic
48C2×C22⋊D73—22no—
49C2×C22⋊D73C2×C812yes—
50C2×C4×D73C22⋊C412yes—
51#11924012—41no—
52#11924012—41no—
53#11924011—42no—
54C2×#10596831—21no—
55C2×#10596831C2×D412yes—
56C8×D73—22no—
57C8×D73—22no—
58C4⋊Dic73—21no—
59#11924007—22no—
60C22⋊Dic73—22no—
61C22⋊Dic73C2×C812yes—
62#11924005—22no—
63C4×Dic73—21no—
64C24×C73C2×C811yescharacteristic
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—41no—
68C22×C4×C73—41no—
69C22×C4×C73C2×C811yescharacteristic
70C22×C4×C73C22×C411yescharacteristic
71M16×C73—42no—
72M16×C73—42no—
73C2×C8×C73—22no—
74C2×C8×C73—21no—
75C2×C8×C73—41no—
76C2×C8×C73C22⋊C411yescharacteristic
77C2×C8×C73C22⋊C411yescharacteristic
78C23×C73—21no—
79C23×C73—41no—
80C23×C73—81no—
81C23×C73C22×C811yescharacteristic
82C22×D73C22⋊C812yes—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—42no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73C2×C22⋊C411yescharacteristic
91C2×C4×C73C2×M1611yescharacteristic
92C22⋊D73—42no—
93C22⋊D73—82no—
94C2×Dic73—22no—
95C2×Dic73C22⋊C812yes—
96D292—41no—
97C4×D73—22no—
98C4×D73—22no—
99Dic146—41no—
100C8×C73—21no—
101C8×C73—41no—
102C8×C73#9912yes—
103#10596831—22no—
104#10596831—41no—
105C22×C73—21no—
106C22×C73—41no—
107C22×C73—41no—
108C22×C73—41no—
109C22×C73—41no—
110C22×C73—82no—
111C22×C73C2×C22⋊C811yescharacteristic
112C2×D73—22no—
113C2×D73—42no—
114C4×C73—21no—
115C4×C73—41no—
116C4×C73C2×#9911yescharacteristic
117C4×C73#41311yescharacteristic
118Dic73—22no—
119Dic73—42no—
120#11484—731nomaximal, Sylow
121C2×C73—21no—
122C2×C73—41no—
123C2×C73—81no—
124C2×C73#145911yescharacteristic
125D73—42no—
126#2930—731no—
127#2115—732no—
128#1328—732no—
129#1320—732no—
130C73#1148411yescharacteristic, complemented, Sylow
131C2×#131—731no—
132C22×M16#6440811yescharacteristic
133#525—731no—
134C2×M32—1462no—
135#436—1462no—
136#435—1462no—
137#424—731no—
138#408—731no—
139#407—731no—
140#405—731no—
141#349—734no—
142#349—1462no—
143C2×#41—731no—
144C23×C4C2×#6440811yescharacteristic
145#131—2922no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—42no—
149C2×M16—731no—
150C2×M16—732no—
151C2×M16—731no—
152C22×C8—732no—
153C22×C8C2×#6440811yescharacteristic
154C22×C8C2×#6440811yescharacteristic
155#123—732no—
156C22⋊Q8—1461no—
157#121—1461no—
158#120—732no—
159C4×D4—1462no—
160#117—731no—
161M32—2922no—
162M32—1464no—
163C2×C16—2922no—
164C4⋊C8—1462no—
165C22⋊C8—1462no—
166C22⋊C8—1462no—
167#97—1461no—
168C4×C8—1461no—
169C24#1192403611yescharacteristic
170#41—2921no—
171#41—2921no—
172C2×Q8—731no—
173C2×D4—732no—
174C2×D4—731no—
175C2×D4—1462no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—732no—
181C22×C4#1192403611yescharacteristic
182C22×C4C22×#6440811yescharacteristic
183M16—42no—
184M16—42no—
185M16—2922no—
186M16—2921no—
187M16—2921no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—732no—
192C2×C8—1461no—
193C2×C8—1462no—
194C2×C8—1462no—
195C2×C8C22⋊#6440811yescharacteristic
196C2×C8C22⋊#6440811yescharacteristic
197C4⋊C4—1462no—
198C4⋊C4—1461no—
199C22⋊C4—732no—
200C22⋊C4—1462no—
201C22⋊C4—1462no—
202C42—1461no—
203C16—2924no—
204C23—21no—
205C23—41no—
206C23—81no—
207C23—732no—
208C23C2×#1192403611yescharacteristic
209Q8—2921no—
210D4—2921no—
211D4—2922no—
212D4—5842no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—732no—
221C2×C4—1462no—
222C2×C4—1462no—
223C2×C4—1462no—
224C2×C4C2×C22⋊#6440811yescharacteristic
225C2×C4D73⋊M1611yescharacteristic
226C8—21no—
227C8—41no—
228C8—2921no—
229C8—1462no—
230C8#1455664912yes—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—82no—
237C22—2922no—
238C22—1462no—
239C22#1798955511yescharacteristic
240C4—21no—
241C4—41no—
242C4—2922no—
243C4—1462no—
244C4C2×#1455664911yescharacteristic
245C4#1798957311yescharacteristic
246C2—21no—
247C2—41no—
248C2—81no—
249C2—2922no—
250C2#2262516611yescharacteristic
subgroupG/H
centerC4C2×#14556649
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×M16×C73C4
socleC2×C73#1459
prime radical#17989098C4
cosocleC2×#11924012C23
hypercenterC22×M16#64408
hypocenterC73#11484
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #22626168C2semidirect
2. #22624880C2semidirect
3. #22624863C2semidirect
130. C73#11484semidirect

Maximal subgroups

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

subgroupindex
#226261682
#226248802
#226248632
#226186342
#1148473