Finite Group Databasebeta

← #34012531

Subgroups — #34012531

Order 18688.

464 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#17989498C212yesmaximal
2#22625684C212yesmaximal
3#22625029C212yesmaximal
4#22612648C211yesmaximal, characteristic
5C2×#14556652C2214yes—
6C4×#11924037—22no—
7#17989498C418yescomplemented
8#17989467—22no—
9C2×#14556591C2211yescharacteristic
10C2×#14556556C411yescharacteristic
11C2×#14556556C2211yescharacteristic
12#17988492C411yescharacteristic
13#17988492C2211yescharacteristic
14#17988470—22no—
15#17988470—22no—
16#17988433C411yescharacteristic
17#382×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556652—24no—
20#14556652C2×C418yes—
21#14556646—42no—
22C4×#10596833—42no—
23C22×Dic146C2×C411yescharacteristic
24C22×Dic146C2311yescharacteristic
25#14556591—22no—
26#14556591C2×C414yes—
27#14556556—42no—
28#14556556—42no—
29C2×C4⋊Dic73—22no—
30C2×#11924007—21no—
31C2×#11924007D412yes—
32C2×C4×Dic73—21no—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73D412yes—
36#14556498—21no—
37#14556498—21no—
38#14556498—22no—
39#14556498—22no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C42×C73C2×C411yescharacteristic
43#95×C73—22no—
44#11924037—28no—
45#11924037—44no—
46C2×#10596833—44no—
47C2×Dic146—21no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C2×Dic146C4212yes—
53C2×Dic146C22×C412yes—
54C2×Dic146#4112yes—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73C22⋊C411yescharacteristic
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C2×D412yes—
60C4⋊Dic73—28no—
61#11924007—42no—
62#11924007—42no—
63C4×Dic73—22no—
64C4×Dic73—22no—
65C4×Dic73—22no—
66C4×Dic73—42no—
67C4×Dic73—41no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73C22⋊C412yes—
71C22×C4×C73C22×C411yescharacteristic
72C4⋊C4×C73—41no—
73C4⋊C4×C73—41no—
74C42×C73—21no—
75C42×C73C4212yes—
76C23×C73C2×C22⋊C411yescharacteristic
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—22no—
80C2×C4×C73—22no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73—41no—
86C2×C4×C73C2×C4211yescharacteristic
87C2×C4×C73#11711yescharacteristic
88C2×Dic73—22no—
89C2×Dic73—22no—
90C2×Dic73—22no—
91C2×Dic73—21no—
92C2×Dic73—41no—
93C2×Dic73—41no—
94C2×Dic73—41no—
95C2×Dic73—41no—
96C2×Dic73—41no—
97C2×Dic73—41no—
98C2×Dic73—41no—
99C2×Dic73#9912yes—
100C2×Dic73C4×D414yes—
101Dic146—42no—
102Dic146—42no—
103Dic146—42no—
104Dic146—42no—
105#10596833—48no—
106C22×C73—21no—
107C22×C73—21no—
108C22×C73C4×C22⋊C411yescharacteristic
109C22×C73C2×#9911yescharacteristic
110C22×C73#41011yescharacteristic
111C4×C73—21no—
112C4×C73—21no—
113C4×C73—42no—
114C4×C73—42no—
115C4×C73—81no—
116Dic73—22no—
117Dic73—24no—
118Dic73—41no—
119Dic73—42no—
120Dic73—81no—
121#12631—731nomaximal, Sylow
122C2×C73—21no—
123C2×C73—21no—
124C2×C73C4×#9911yescharacteristic
125C2×C73#213311yescharacteristic
126C2×C73#213511yescharacteristic
127#2327—731no—
128C2×#356—732no—
129C4×#101—732no—
130#1394—732no—
131C73#1263111yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—731no—
133C2×C4×Q8—731no—
134C2×C4×Q8—731no—
135C2×#101—734no—
136C4×M16—1462no—
137#391—731no—
138#391—731no—
139#387—1462no—
140#387—1462no—
141#382#6440811yescharacteristic
142#376—731no—
143#356—738no—
144#343—1462no—
145C22×Q8—731no—
146C22×Q8—731no—
147C2×M16—1464no—
148C4⋊Q8—734no—
149C4⋊Q8—1462no—
150C4×Q8—2922no—
151C4×Q8—2922no—
152C2×C4⋊C4—732no—
153C2×C4⋊C4—1461no—
154C2×C4⋊C4—1462no—
155C2×C4⋊C4C2×#6440811yescharacteristic
156C2×C4⋊C4C2×#6440811yescharacteristic
157C2×C42—731no—
158C2×C42—732no—
159C2×C42—731no—
160C2×C42—1461no—
161C2×C42C2×#6440811yescharacteristic
162#101—738no—
163#101—1464no—
164#97—2922no—
165C4×C8—2922no—
166#95—22no—
167#95—1461no—
168#95—1461no—
169#95—1462no—
170#95—1462no—
171C2×Q8—2921no—
172C2×Q8—2921no—
173C2×Q8—2921no—
174C2×Q8—2921no—
175C2×Q8—732no—
176C2×Q8—732no—
177C2×Q8—732no—
178C2×Q8—1461no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—731no—
182C22×C4—731no—
183C22×C4—732no—
184C22×C4—1461no—
185C22×C4—1461no—
186C22×C4C22⋊#6440812yes—
187C22×C4C22×#6440811yescharacteristic
188M16—2924no—
189M16—1468no—
190C2×C8—2924no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—2922no—
194C4⋊C4—2922no—
195C4⋊C4—1468no—
196C42—21no—
197C42—2921no—
198C42—2922no—
199C42—1462no—
200C42—1462no—
201C42—1462no—
202C42C4×#6440812yes—
203C23C2×C22⋊#6440811yescharacteristic
204Q8—2922no—
205Q8—2922no—
206Q8—2922no—
207Q8—2922no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—2921no—
218C2×C4—2921no—
219C2×C4—2921no—
220C2×C4—2921no—
221C2×C4—2921no—
222C2×C4—2921no—
223C2×C4—2921no—
224C2×C4—734no—
225C2×C4—732no—
226C2×C4—1462no—
227C2×C4—1461no—
228C2×C4—1462no—
229C2×C4—1462no—
230C2×C4C2×C4×#6440811yescharacteristic
231C2×C4#1455667011yescharacteristic
232C8—2928no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊#6440811yescharacteristic
236C22D73⋊#9911yescharacteristic
237C22#1798956811yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—2922no—
244C4—2921no—
245C4—5841no—
246C4—1464no—
247C4—1462no—
248C2—21no—
249C2—21no—
250C2#2262567611yescharacteristic
251C2#2262622711yescharacteristic
252C2#2262623411yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#382×C73C4
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Dic146C23
hypercenter#382#64408
hypocenterC73#12631
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #382

Complements

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

NH ≅ G/Ntype
7. #17989498C4semidirect
131. C73#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#179894982
#226256842
#226250292
#226126482
#1263173