Finite Group Databasebeta

← #33003733

Subgroups — #33003733

Order 18688.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22611697C211yesmaximal, characteristic, complemented
2#22611558C211yesmaximal, characteristic
3C2×#17988370C211yesmaximal, characteristic
4#22611527C211yesmaximal, characteristic, complemented
5#22609732C211yesmaximal, characteristic
6#22608018C211yesmaximal, characteristic, complemented
7C73×#1379C211yesmaximal, characteristic, complemented
8C2×#14556597C2211yescharacteristic
9C2×#14556531C2211yescharacteristic
10C2×#14556531C2211yescharacteristic
11#17988569C411yescharacteristic
12#17988560C411yescharacteristic
13#17988542C2211yescharacteristic
14#17988408—22no—
15#17988370—22no—
16#17988370—21no—
17#17988370—21no—
18#17988362—21no—
19C16×Dic73—21no—
20#17988359C411yescharacteristic
21#17988357C411yescharacteristic
22C2×M32×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24#405×C73C2211yescharacteristic
25C2×#11924028C2×C411yescharacteristic
26#14556597—21no—
27#14556597—21no—
28#14556597—21no—
29#14556597—21no—
30C2×#11924012C2×C411yescharacteristic
31C2×#11924012C2×C411yescharacteristic
32C2×#11924011—21no—
33C2×C8×D73—21no—
34#14556550C2×C411yescharacteristic
35#14556531—21no—
36#14556531—21no—
37#14556531—21no—
38#14556531D412yes—
39C2×#11924003—21no—
40C2×#11924003—21no—
41#14556509C812yes—
42#14556507C812yes—
43#14556505—22no—
44#14556501—21no—
45C8×Dic73—21no—
46C2×M16×C73C2×C411yescharacteristic
47C22×C8×C73C2311yescharacteristic
48#117×C73C2×C411yescharacteristic
49M32×C73—21no—
50M32×C73—21no—
51C2×C16×C73—21no—
52C2×C16×C73—21no—
53C2×C16×C73—21no—
54C2×C16×C73D411yescharacteristic
55C2×C16×C73D411yescharacteristic
56#97×C73—21no—
57C4×C8×C73—21no—
58#11924028—21no—
59#11924028—21no—
60#11924028—21no—
61#11924028—21no—
62C2×Dic146C2×C811yescharacteristic
63C2×D292C2×C811yescharacteristic
64C2×C22⋊D73—21no—
65C2×C4×D73—21no—
66#11924012—21no—
67#11924012—21no—
68#11924012—21no—
69#11924012C2×C812yes—
70#11924011—41no—
71#11924011—41no—
72C2×#10596831—21no—
73C2×#10596831—21no—
74C8×D73—41no—
75C8×D73—41no—
76C4⋊Dic73—21no—
77C22⋊Dic73M1612yes—
78C4×Dic73—21no—
79#11924003—22no—
80#11924003—41no—
81C22×C4×C73C22×C411yescharacteristic
82M16×C73—21no—
83M16×C73C2×C812yes—
84C2×C8×C73—21no—
85C2×C8×C73C2×D411yescharacteristic
86C2×C8×C73C2×D411yescharacteristic
87C2×C8×C73#4111yescharacteristic
88C2×C8×C73#4111yescharacteristic
89C2×C8×C73#4111yescharacteristic
90C2×C8×C73#4111yescharacteristic
91C4⋊C4×C73—21no—
92C22⋊C4×C73—21no—
93C42×C73—21no—
94C16×C73—21no—
95C16×C73—21no—
96C16×C73—41no—
97C23×C73C2×M1611yescharacteristic
98C22×D73—21no—
99C2×C4×C73—41no—
100C2×C4×C73#11711yescharacteristic
101C2×C4×C73C4×D411yescharacteristic
102C2×C4×C73C4×D411yescharacteristic
103C2×C4×C73C22×C811yescharacteristic
104C2×C4×C73#13111yescharacteristic
105C2×C4×C73#13111yescharacteristic
106C22⋊D73—41no—
107C22⋊D73—41no—
108C2×Dic73—21no—
109C2×Dic73—22no—
110D292—21no—
111D292—21no—
112C4×D73—41no—
113C4×D73—41no—
114Dic146—21no—
115Dic146—21no—
116C8×C73—22no—
117C8×C73#12111yescharacteristic
118C8×C73#12311yescharacteristic
119C8×C73#12411yescharacteristic
120C8×C73#12611yescharacteristic
121#10596831—22no—
122#10596831—41no—
123C22×C73—41no—
124C22×C73#43111yescharacteristic
125C22×C73#43511yescharacteristic
126C22×C73#43611yescharacteristic
127C2×D73—41no—
128C2×D73—81no—
129C4×C73—81no—
130C4×C73#38811yescharacteristic
131C4×C73#43311yescharacteristic
132C4×C73C8×D411yescharacteristic
133C4×C73C8×D411yescharacteristic
134Dic73—42no—
135Dic73—41no—
136#13123—731nomaximal, Sylow
137C2×C73—21no—
138C2×C73—21no—
139C2×C73—21no—
140C2×C73#192311yescharacteristic
141D73—81no—
142#2153—731no—
143#2117—731no—
144C2×#350—731no—
145#2108—731no—
146#1757—731no—
147#1379D7311yescharacteristic, complemented
148#1378—731no—
149C73#1312311yescharacteristic, complemented, Sylow
150C2×#131—731no—
151C2×M32—731no—
152C2×M32—731no—
153C2×M32C2×D7311yescharacteristic
154C22×C16C2×D7311yescharacteristic
155#413—731no—
156#410—731no—
157#405—731no—
158#405C2×D7311yescharacteristic
159#364—1462no—
160#350—1462no—
161#350—1461no—
162#350—1461no—
163#346—1461no—
164C4×C16—1461no—
165#344—731no—
166#344—731no—
167C2×#41—731no—
168#131—1461no—
169#131—1461no—
170#131—1461no—
171#131—1461no—
172C2×M16—731no—
173C2×M16—731no—
174C2×M16—1461no—
175C2×M16C4×D7311yescharacteristic
176C22×C8—1461no—
177C22×C8C22×D7311yescharacteristic
178#117—731no—
179#117C4×D7311yescharacteristic
180M32—21no—
181M32—21no—
182M32—732no—
183M32—1461no—
184M32—1461no—
185M32—1461no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—1461no—
190C2×C16—1461no—
191C2×C16D29211yescharacteristic
192C2×C16D29211yescharacteristic
193#101—732no—
194#100—732no—
195#99—1462no—
196#97—21no—
197#97—1461no—
198C4×C8—21no—
199C4×C8—1461no—
200#41—1461no—
201#41—1461no—
202#41—1461no—
203#41—1461no—
204C2×Q8—731no—
205C2×D4—731no—
206C2×D4—1461no—
207C22×C4—1461no—
208C22×C4C2×C4×D7311yescharacteristic
209M16—21no—
210M16—2921no—
211M16—2921no—
212M16—732no—
213M16—1461no—
214M16—1461no—
215M16—1461no—
216M16C8×D7312yes—
217C2×C8—21no—
218C2×C8—2921no—
219C2×C8—2921no—
220C2×C8—1461no—
221C2×C8—1461no—
222C2×C8D4×D7311yescharacteristic
223C2×C8C2×D29211yescharacteristic
224C2×C8#1192402711yescharacteristic
225C2×C8#1192402711yescharacteristic
226C2×C8#1192402811yescharacteristic
227C2×C8#1192402911yescharacteristic
228C4⋊C4—21no—
229C4⋊C4—1461no—
230C22⋊C4—21no—
231C22⋊C4—732no—
232C42—21no—
233C42—1461no—
234C16—21no—
235C16—21no—
236C16—41no—
237C16—2921no—
238C16—1462no—
239C23—1461no—
240C23M16×D7311yescharacteristic
241Q8—1461no—
242Q8—1461no—
243D4—2921no—
244D4—2921no—
245D4—1461no—
246D4—1461no—
247C2×C4—41no—
248C2×C4—2921no—
249C2×C4—2921no—
250C2×C4—1461no—
251C2×C4—1462no—
252C2×C4#1455654911yescharacteristic
253C2×C4#1455655111yescharacteristic
254C2×C4C4×D29211yescharacteristic
255C2×C4C2×C8×D7311yescharacteristic
256C2×C4#1455659711yescharacteristic
257C2×C4#1455659911yescharacteristic
258C8—22no—
259C8—2921no—
260C8—1462no—
261C8#1455656611yescharacteristic
262C8#1455657511yescharacteristic
263C8#1455657711yescharacteristic
264C8#1455658611yescharacteristic
265C22—41no—
266C22—2921no—
267C22—5841no—
268C22#1798863911yescharacteristic
269C22#1798866011yescharacteristic
270C22#1798866211yescharacteristic
271C4—81no—
272C4—2922no—
273C4—2921no—
274C4#1798847911yescharacteristic
275C4#1798864711yescharacteristic
276C4#1798865011yescharacteristic
277C4C8×D29211yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—5841no—
282C2#2261053811yescharacteristic
subgroupG/H
centerC8#14556566
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1379C2
socleC2×C73#1923
prime radicalC2×#11924028C2×C4
cosocleC22×C8×C73C23
hypercenter#1379D73
hypocenterC73#13123
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1379

Complements

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

NH ≅ G/Ntype
1. #22611697C2semidirect
4. #22611527C2semidirect
6. #22608018C2semidirect
7. C73×#1379C2semidirect
147. #1379D73semidirect
149. C73#13123semidirect

Maximal subgroups

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

subgroupindex
#226116972
#226115582
C2×#179883702
#226115272
#226097322
#226080182
C73×#13792
#1312373