Finite Group Databasebeta

← #34009998

Subgroups — #34009998

Order 18688.

368 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 86 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#22626167C212yesmaximal, complemented
2#22624880C212yesmaximal, complemented
3#22624864C212yesmaximal, complemented
4#22618637C211yesmaximal, characteristic
5C2×#14556665—22no—
6#17989483—22no—
7#17989483C2214yes—
8C2×#14556598C2211yescharacteristic
9C22×#11924011C411yescharacteristic
10#17989094C411yescharacteristic
11#17988662—22no—
12#17988655—22no—
13#17988613C2211yescharacteristic
14#17988550C2211yescharacteristic
15#17988545C411yescharacteristic
16#408×C73C411yescharacteristic
17#14556665—24no—
18#14556665—42no—
19C2×#11924031—42no—
20C22×C4×D73C2×C411yescharacteristic
21#14556598—42no—
22C2×#11924012D412yes—
23C2×#11924012C2311yescharacteristic
24C2×#11924011—21no—
25C2×#11924011—21no—
26C2×#11924011—42no—
27C22×#10596831C2×C411yescharacteristic
28C22×#10596831D412yes—
29#14556572C812yes—
30#14556567—21no—
31#14556562—21no—
32#14556559C812yes—
33C4×C22⋊D73—22no—
34#14556550C2×C411yescharacteristic
35#14556520—22no—
36#14556503—22no—
37#14556501—21no—
38C8×Dic73—21no—
39C2×#41×C73C2×C411yescharacteristic
40C2×M16×C73C2×C411yescharacteristic
41C22×C8×C73C2×C411yescharacteristic
42C22⋊C8×C73—22no—
43#11924031—44no—
44C23×D73C2×C811yescharacteristic
45C2×C22⋊D73—22no—
46C22×Dic73C2×C811yescharacteristic
47C2×C4×D73—21no—
48C2×C4×D73—21no—
49C2×C4×D73—41no—
50C2×C4×D73—41no—
51#11924012—42no—
52#11924012—41no—
53#11924011—42no—
54#11924011—42no—
55C2×#10596831—22no—
56C2×#10596831—22no—
57C2×#10596831—22no—
58C2×#10596831C22⋊C411yescharacteristic
59C2×#10596831C22⋊C411yescharacteristic
60C2×#10596831C2×D412yes—
61C4⋊Dic73—21no—
62#11924007—22no—
63C22⋊Dic73—22no—
64C22⋊Dic73C2×C812yes—
65#11924005—22no—
66C4×Dic73—21no—
67#41×C73—41no—
68#41×C73—41no—
69C2×Q8×C73M1611yescharacteristic
70C2×D4×C73C2×C812yes—
71C2×D4×C73M1611yescharacteristic
72C22×C4×C73C22⋊C412yes—
73C22×C4×C73C22×C411yescharacteristic
74M16×C73—41no—
75C2×C8×C73—21no—
76C2×C8×C73—21no—
77C2×C8×C73—41no—
78C23×C73C22⋊C812yes—
79C23×C73C22×C811yescharacteristic
80C22×D73—21no—
81C22×D73—41no—
82C22×D73—81no—
83Q8×C73—41no—
84D4×C73—42no—
85D4×C73—41no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
90C2×C4×C73C22⋊C812yes—
91C2×C4×C73C2×C22⋊C411yescharacteristic
92C2×C4×C73C2×M1611yescharacteristic
93C22⋊D73—82no—
94C2×Dic73—22no—
95C2×Dic73—21no—
96C2×Dic73—41no—
97C4×D73—42no—
98C4×D73—41no—
99C4×D73—41no—
100C8×C73—41no—
101C8×C73—41no—
102#10596831—21no—
103#10596831—22no—
104#10596831#9912yes—
105C22×C73—21no—
106C22×C73—22no—
107C22×C73—42no—
108C22×C73—41no—
109C22×C73C2×C22⋊C811yescharacteristic
110C2×D73—41no—
111C2×D73—41no—
112C2×D73—41no—
113C2×D73—82no—
114C4×C73—21no—
115C4×C73—22no—
116C4×C73C2×#9911yescharacteristic
117C4×C73#41311yescharacteristic
118Dic73—42no—
119Dic73—41no—
120#11484—731nomaximal, Sylow
121C2×C73—21no—
122C2×C73—41no—
123C2×C73—42no—
124C2×C73#145911yescharacteristic
125D73—81no—
126#2930—731no—
127#2115—732no—
128#1328—732no—
129#1320—732no—
130C73#1148411yescharacteristic, complemented, Sylow
131C2×#131—731no—
132C22×M16—731no—
133#525—731no—
134C2×M32—1462no—
135#436—1462no—
136#435—1462no—
137#424—731no—
138#408#6440811yescharacteristic
139#407—731no—
140#405—731no—
141#349—734no—
142#349—1462no—
143C2×#41C2×#6440811yescharacteristic
144C23×C4—731no—
145#131—2922no—
146C2×M16—2922no—
147C2×M16—732no—
148C2×M16—731no—
149C2×M16—1461no—
150C2×M16—1461no—
151C2×M16C2×#6440811yescharacteristic
152C22×C8—732no—
153C22×C8—731no—
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—22no—
166C22⋊C8—1462no—
167#97—1461no—
168C4×C8—1461no—
169C24—731no—
170#41—41no—
171#41—41no—
172C2×Q8#1192403811yescharacteristic
173C2×D4—1462no—
174C2×D4#1192403612yes—
175C2×D4#1192403811yescharacteristic
176C22×C4—2921no—
177C22×C4—2921no—
178C22×C4—731no—
179C22×C4—1461no—
180C22×C4—1461no—
181C22×C4C22⋊#6440812yes—
182C22×C4C22×#6440811yescharacteristic
183M16—41no—
184M16—2922no—
185M16—2922no—
186M16—2921no—
187M16—2922no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—731no—
192C2×C8—731no—
193C2×C8—732no—
194C2×C8—1462no—
195C2×C8—1462no—
196C2×C8—1462no—
197C4⋊C4—1461no—
198C4⋊C4—1462no—
199C22⋊C4—732no—
200C22⋊C4—1462no—
201C22⋊C4—1462no—
202C42—1461no—
203C16—2924no—
204C23—2921no—
205C23—5841no—
206C23—1461no—
207C23#1455664812yes—
208C23C2×#1192403611yescharacteristic
209Q8—41no—
210D4—42no—
211D4—41no—
212D4—5842no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—2922no—
218C2×C4—2921no—
219C2×C4—2921no—
220C2×C4—2921no—
221C2×C4—1462no—
222C2×C4—1461no—
223C2×C4#1455664812yes—
224C2×C4C2×C22⋊#6440811yescharacteristic
225C2×C4C2×#1192403811yescharacteristic
226C8—41no—
227C8—41no—
228C8—732no—
229C8—1462no—
230C8—1461no—
231C22—21no—
232C22—22no—
233C22—42no—
234C22—41no—
235C22—2921no—
236C22—2921no—
237C22—2921no—
238C22—5842no—
239C22C2×#1455664811yescharacteristic
240C4—21no—
241C4—22no—
242C4—2922no—
243C4—2921no—
244C4D73⋊#9911yescharacteristic
245C4#1798957411yescharacteristic
246C2—21no—
247C2—41no—
248C2—42no—
249C2—5841no—
250C2#2262516911yescharacteristic
subgroupG/H
centerC4D73⋊#99
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#408×C73C4
socleC2×C73#1459
prime radical#17989094C4
cosocleC2×#11924012C23
hypercenter#408#64408
hypocenterC73#11484
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #22626167C2semidirect
2. #22624880C2semidirect
3. #22624864C2semidirect
130. C73#11484semidirect

Maximal subgroups

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

subgroupindex
#226261672
#226248802
#226248642
#226186372
#1148473