Finite Group Databasebeta

← #34009991

Subgroups — #34009991

Order 18688.

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

normal subgroups only

This group has 382 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#22618625C211yesmaximal, characteristic
5#17989643—22no—
6#17989484—22no—
7#17989483C2214yes—
8C2×#14556597C2211yescharacteristic
9#17989098C411yescharacteristic
10#17988658—22no—
11C8×C22⋊D73—22no—
12#17988613C2211yescharacteristic
13#17988553C411yescharacteristic
14#17988551C2211yescharacteristic
15#17988542C411yescharacteristic
16C23×C8×C73C411yescharacteristic
17#14556666—42no—
18#14556665—24no—
19#14556664—42no—
20C2×#11924028C2×C411yescharacteristic
21#14556597—41no—
22#14556597—41no—
23C2×#11924012C2×C411yescharacteristic
24C2×#11924012C2311yescharacteristic
25C2×#11924011D412yes—
26C2×C8×D73D412yes—
27#14556574C812yes—
28#14556570—21no—
29#14556565—21no—
30#14556560C812yes—
31C4×C22⋊D73—22no—
32#14556550C2×C411yescharacteristic
33#14556520—22no—
34C22⋊#10596831—22no—
35#14556503—22no—
36#14556501—21no—
37C8×Dic73—21no—
38C23×C4×C73C2×C411yescharacteristic
39C22×C8×C73—21no—
40C22×C8×C73—21no—
41C22×C8×C73—41no—
42C22×C8×C73—41no—
43C22×C8×C73C2×C411yescharacteristic
44C22×C8×C73C2×C411yescharacteristic
45#11924031—44no—
46#11924028—41no—
47#11924028—41no—
48C2×Dic146M1611yescharacteristic
49C2×D292M1611yescharacteristic
50C2×C22⋊D73—22no—
51C2×C22⋊D73C2×C812yes—
52C2×C4×D73C22⋊C412yes—
53#11924012—41no—
54#11924012—41no—
55#11924011—42no—
56C2×#10596831—21no—
57C2×#10596831C2×D412yes—
58C8×D73—22no—
59C8×D73—22no—
60C4⋊Dic73—21no—
61#11924007—22no—
62C22⋊Dic73—22no—
63C22⋊Dic73C2×C812yes—
64#11924005—22no—
65C4×Dic73—21no—
66C24×C73C2×C811yescharacteristic
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—41no—
70C22×C4×C73—41no—
71C22×C4×C73C2×C811yescharacteristic
72C22×C4×C73C22×C411yescharacteristic
73C2×C8×C73—21no—
74C2×C8×C73—21no—
75C2×C8×C73—21no—
76C2×C8×C73—42no—
77C2×C8×C73—41no—
78C2×C8×C73—41no—
79C2×C8×C73—41no—
80C2×C8×C73C22⋊C411yescharacteristic
81C2×C8×C73C22⋊C411yescharacteristic
82C23×C73—21no—
83C23×C73—41no—
84C23×C73—42no—
85C23×C73C22×C811yescharacteristic
86C22×D73C22⋊C812yes—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—42no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73C2×C22⋊C411yescharacteristic
95C2×C4×C73C2×M1611yescharacteristic
96C22⋊D73—42no—
97C22⋊D73—44no—
98C2×Dic73—22no—
99C2×Dic73C22⋊C812yes—
100D292—41no—
101C4×D73—22no—
102C4×D73—22no—
103Dic146—41no—
104C8×C73—21no—
105C8×C73—41no—
106C8×C73#9911yescharacteristic
107C8×C73#9911yescharacteristic
108#10596831—22no—
109#10596831—41no—
110C22×C73—22no—
111C22×C73—21no—
112C22×C73—41no—
113C22×C73—41no—
114C22×C73—42no—
115C22×C73—42no—
116C22×C73—41no—
117C22×C73C2×C22⋊C811yescharacteristic
118C2×D73—22no—
119C2×D73—42no—
120C4×C73—21no—
121C4×C73—41no—
122C4×C73C2×#9911yescharacteristic
123C4×C73#41311yescharacteristic
124Dic73—22no—
125Dic73—42no—
126#11483—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—42no—
129C2×C73—41no—
130C2×C73#145911yescharacteristic
131D73—42no—
132#2929—731no—
133#2115—732no—
134#1328—732no—
135#1320—732no—
136C73#1148311yescharacteristic, complemented, Sylow
137C2×#131—731no—
138C23×C8#6440811yescharacteristic
139#525—731no—
140C2×M32—1462no—
141#435—1462no—
142C8×D4—1462no—
143#424—731no—
144#408—731no—
145#407—731no—
146#405—731no—
147#349—734no—
148#349—1462no—
149C2×#41—731no—
150C23×C4C2×#6440811yescharacteristic
151#131—2921no—
152#131—2921no—
153C2×M16—731no—
154C2×M16—732no—
155C2×M16—731no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—732no—
161C22×C8C2×#6440811yescharacteristic
162C22×C8C2×#6440811yescharacteristic
163#123—732no—
164C22⋊Q8—1461no—
165#121—1461no—
166#120—732no—
167C4×D4—1462no—
168#117—731no—
169M32—2922no—
170M32—1464no—
171C2×C16—2922no—
172C4⋊C8—1462no—
173C22⋊C8—1462no—
174C22⋊C8—1462no—
175#97—1461no—
176C4×C8—1461no—
177C24#1192403611yescharacteristic
178#41—2921no—
179#41—2921no—
180C2×Q8—731no—
181C2×D4—732no—
182C2×D4—731no—
183C2×D4—1462no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—732no—
189C22×C4#1192403611yescharacteristic
190C22×C4C22×#6440811yescharacteristic
191M16—2922no—
192M16—2921no—
193M16—2921no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—732no—
202C2×C8—1461no—
203C2×C8—1462no—
204C2×C8—1462no—
205C2×C8C22⋊#6440811yescharacteristic
206C2×C8C22⋊#6440811yescharacteristic
207C4⋊C4—1462no—
208C4⋊C4—1461no—
209C22⋊C4—732no—
210C22⋊C4—1462no—
211C22⋊C4—1462no—
212C42—1461no—
213C16—2924no—
214C23—21no—
215C23—41no—
216C23—42no—
217C23—732no—
218C23C2×#1192403611yescharacteristic
219Q8—2921no—
220D4—2921no—
221D4—2924no—
222D4—2922no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—732no—
231C2×C4—1462no—
232C2×C4—1462no—
233C2×C4—1462no—
234C2×C4C2×C22⋊#6440811yescharacteristic
235C2×C4D73⋊M1611yescharacteristic
236C8—21no—
237C8—41no—
238C8—2921no—
239C8—1462no—
240C8#1455664911yescharacteristic
241C8#1455664911yescharacteristic
242C22—22no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—2922no—
250C22—1462no—
251C22#1798955511yescharacteristic
252C4—21no—
253C4—41no—
254C4—2922no—
255C4—1462no—
256C4C2×#1455664911yescharacteristic
257C4#1798957311yescharacteristic
258C2—21no—
259C2—42no—
260C2—41no—
261C2—2922no—
262C2#2262516611yescharacteristic
subgroupG/H
centerC8#14556649
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC23×C8×C73C4
socleC2×C73#1459
prime radical#17989098C4
cosocleC2×#11924012C23
hypercenterC23×C8#64408
hypocenterC73#11483
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #22626168C2semidirect
2. #22624880C2semidirect
3. #22624863C2semidirect
136. C73#11483semidirect

Maximal subgroups

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

subgroupindex
#226261682
#226248802
#226248632
#226186252
#1148373