Finite Group Databasebeta

← #34013502

Subgroups — #34013502

Order 18688.

470 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 112 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#22626898C211yesmaximal, characteristic, complemented
2#22626167C211yesmaximal, characteristic, complemented
3C2×#17989484C212yesmaximal, complemented
4#22624982C212yesmaximal, complemented
5#22618637C211yesmaximal, characteristic
6#17989645—24no—
7C2×#14556665—22no—
8C2×#14556665C2212yes—
9C2×#14556665C2212yes—
10#17989484—24no—
11#17989483—22no—
12C2×#14556598C2211yescharacteristic
13C22×#11924011C2211yescharacteristic
14#17989094C411yescharacteristic
15#17988662—22no—
16#17988655—22no—
17#17988613C411yescharacteristic
18#17988550C411yescharacteristic
19#17988545C2211yescharacteristic
20#408×C73C411yescharacteristic
21#14556665—24no—
22#14556665—22no—
23#14556665—22no—
24#14556665—22no—
25#14556665D414yescomplemented
26C2×#11924031—24no—
27C2×#11924031—22no—
28C2×#11924031—22no—
29C22×C4×D73C2×C411yescharacteristic
30#14556598—22no—
31#14556598—22no—
32C2×#11924012C2×C411yescharacteristic
33C2×#11924012D412yes—
34C2×#11924011—21no—
35C2×#11924011—21no—
36C2×#11924011—22no—
37C2×#11924011—22no—
38C22×#10596831D412yes—
39C22×#10596831C2311yescharacteristic
40#14556572C812yes—
41#14556567—21no—
42#14556562—21no—
43#14556559C812yes—
44C4×C22⋊D73—22no—
45#14556550C2×C411yescharacteristic
46#14556520—22no—
47#14556503—22no—
48#14556501—21no—
49C8×Dic73—21no—
50C2×#41×C73C2×C411yescharacteristic
51C2×M16×C73C2×C411yescharacteristic
52C22×C8×C73C2×C411yescharacteristic
53C22⋊C8×C73—22no—
54#11924031—24no—
55#11924031—42no—
56C23×D73C2×C811yescharacteristic
57C2×C22⋊D73—22no—
58C22×Dic73C2×C811yescharacteristic
59C2×C4×D73—21no—
60C2×C4×D73—22no—
61C2×C4×D73—21no—
62C2×C4×D73—22no—
63#11924012—22no—
64#11924012—24no—
65#11924011—41no—
66#11924011—42no—
67#11924011—41no—
68C2×#10596831—22no—
69C2×#10596831—22no—
70C2×#10596831C22⋊C412yes—
71C2×#10596831C2×D412yes—
72C2×#10596831C2×D411yescharacteristic
73C2×#10596831C2×D411yescharacteristic
74C2×#10596831#4112yes—
75C4⋊Dic73—21no—
76#11924007—22no—
77C22⋊Dic73—22no—
78C22⋊Dic73C2×C812yes—
79#11924005—22no—
80C4×Dic73—21no—
81#41×C73—22no—
82#41×C73—22no—
83C2×Q8×C73M1611yescharacteristic
84C2×D4×C73C2×C812yes—
85C2×D4×C73M1611yescharacteristic
86C22×C4×C73C22⋊C412yes—
87C22×C4×C73C22×C411yescharacteristic
88M16×C73—22no—
89C2×C8×C73—21no—
90C2×C8×C73—21no—
91C2×C8×C73—22no—
92C23×C73C22⋊C812yes—
93C23×C73C22×C811yescharacteristic
94C22×D73—21no—
95C22×D73—22no—
96C22×D73—81no—
97Q8×C73—22no—
98D4×C73—24no—
99D4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73C22⋊C812yes—
103C2×C4×C73C2×C22⋊C411yescharacteristic
104C2×C4×C73C4×D412yes—
105C2×C4×C73C2×M1611yescharacteristic
106C2×C4×C73#13112yes—
107C22⋊D73—82no—
108C2×Dic73—22no—
109C2×Dic73—21no—
110C2×Dic73—22no—
111C4×D73—41no—
112C4×D73—41no—
113C4×D73—42no—
114C8×C73—41no—
115C8×C73—41no—
116#10596831—22no—
117#10596831#12011yescharacteristic
118#10596831#12112yes—
119#10596831#12311yescharacteristic
120C22×C73—22no—
121C22×C73—41no—
122C22×C73—42no—
123C22×C73C2×C22⋊C811yescharacteristic
124C22×C73C8×D412yes—
125C2×D73—41no—
126C2×D73—41no—
127C2×D73—41no—
128C2×D73—82no—
129C4×C73—22no—
130C4×C73#38611yescharacteristic
131C4×C73#40811yescharacteristic
132C4×C73#43512yes—
133Dic73—42no—
134Dic73—41no—
135#12989—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—22no—
138C2×C73—42no—
139C2×C73#186911yescharacteristic
140D73—81no—
141#3407—731no—
142#2930—731no—
143#2115—731no—
144C2×#349—732no—
145#1378—732no—
146C73#1298911yescharacteristic, complemented, Sylow
147C2×#131—731no—
148C22×M16—731no—
149#525—731no—
150#504—1464no—
151C2×M32—732no—
152C2×M32—732no—
153C2×M32—1462no—
154#436—1462no—
155#435—1462no—
156#424—731no—
157#408#6440811yescharacteristic
158#407—731no—
159#405—731no—
160#349—1464no—
161#349—1462no—
162C2×#41C2×#6440811yescharacteristic
163C23×C4—731no—
164#131—1462no—
165#131—1462no—
166C2×M16—732no—
167C2×M16—731no—
168C2×M16—1462no—
169C2×M16—1461no—
170C2×M16—1462no—
171C2×M16—1461no—
172C2×M16C2×#6440811yescharacteristic
173C22×C8—731no—
174C22×C8—732no—
175C22×C8C2×#6440811yescharacteristic
176#123—732no—
177C22⋊Q8—1461no—
178#121—1461no—
179#120—732no—
180C4×D4—1462no—
181#117—731no—
182M32—734no—
183M32—1462no—
184M32—1462no—
185M32—1462no—
186M32—1464no—
187C2×C16—1462no—
188C2×C16—1462no—
189C2×C16—1464no—
190C4⋊C8—1462no—
191C22⋊C8—22no—
192C22⋊C8—1462no—
193#97—1461no—
194C4×C8—1461no—
195C24—731no—
196#41—22no—
197#41—22no—
198C2×Q8#1192403811yescharacteristic
199C2×D4—1462no—
200C2×D4#1192403612yes—
201C2×D4#1192403811yescharacteristic
202C22×C4—731no—
203C22×C4—1461no—
204C22×C4—1462no—
205C22×C4—1461no—
206C22×C4—1462no—
207C22×C4C22⋊#6440812yes—
208C22×C4C22×#6440811yescharacteristic
209M16—22no—
210M16—2922no—
211M16—2921no—
212M16—2921no—
213M16—1464no—
214M16—1462no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—731no—
219C2×C8—731no—
220C2×C8—732no—
221C2×C8—732no—
222C2×C8—732no—
223C2×C8—1462no—
224C2×C8—1462no—
225C4⋊C4—1461no—
226C4⋊C4—1462no—
227C22⋊C4—732no—
228C22⋊C4—1462no—
229C22⋊C4—1462no—
230C42—1461no—
231C16—2922no—
232C16—1464no—
233C23—5841no—
234C23—1461no—
235C23—1462no—
236C23#1455664812yes—
237C23C2×#1192403611yescharacteristic
238Q8—22no—
239D4—24no—
240D4—22no—
241D4—5842no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—2921no—
245C2×C4—2921no—
246C2×C4—2922no—
247C2×C4—1462no—
248C2×C4—1461no—
249C2×C4—1462no—
250C2×C4#1455664812yes—
251C2×C4C2×C22⋊#6440811yescharacteristic
252C2×C4D4×#6440812yes—
253C2×C4C2×#1192403811yescharacteristic
254C2×C4#1455667812yes—
255C8—41no—
256C8—41no—
257C8—732no—
258C8—731no—
259C8—731no—
260C8—1462no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—2921no—
265C22—2921no—
266C22—2921no—
267C22—5842no—
268C22C2×#1455664811yescharacteristic
269C22#1798961412yes—
270C4—22no—
271C4—2922no—
272C4—2921no—
273C4#1798953511yescharacteristic
274C4#1798956011yescharacteristic
275C4#1798961612yes—
276C2—21no—
277C2—22no—
278C2—42no—
279C2—5841no—
280C2#2262590611yescharacteristic
subgroupG/H
centerC4#17989535
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#408×C73C4
socleC2×C73#1869
prime radical#17989094C4
cosocleC22×#10596831C23
hypercenter#408#64408
hypocenterC73#12989
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #22626898C2semidirect
2. #22626167C2semidirect
3. C2×#17989484C2semidirect
4. #22624982C2semidirect
25. #14556665D4semidirect
146. C73#12989semidirect

Maximal subgroups

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

subgroupindex
#226268982
#226261672
C2×#179894842
#226249822
#226186372
#1298973