Finite Group Databasebeta

← #32995297

Subgroups — #32995297

Order 18688.

364 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 83 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#22618648C211yesmaximal, characteristic
2#22611550C212yesmaximal, complemented
3#22607882C211yesmaximal, characteristic, complemented
4#22607868C212yesmaximal, complemented
5C73×#1327C211yesmaximal, characteristic, complemented
6C2×#14556599C411yescharacteristic
7C2×#14556599C2211yescharacteristic
8#17989137C411yescharacteristic
9M32×D73—22no—
10#17988657—22no—
11#17988657—22no—
12#17988555C411yescharacteristic
13#17988555C2211yescharacteristic
14C2×#14556502C411yescharacteristic
15#17988369C2212yes—
16#17988368—22no—
17#423×C73C2211yescharacteristic
18#349×C73C2212yes—
19C2×#11924028C2×C411yescharacteristic
20C2×#11924028C2×C411yescharacteristic
21#14556599—42no—
22#14556599—42no—
23C2×#11924011—21no—
24C2×#11924011D412yes—
25C22×#10596831C2×C411yescharacteristic
26C22×#10596831C2×C411yescharacteristic
27C2×C8×D73—21no—
28C2×C8×D73D412yes—
29#14556591C811yescharacteristic
30#14556588C811yescharacteristic
31#14556579C811yescharacteristic
32#14556579C811yescharacteristic
33#14556565—22no—
34C4×D292—22no—
35#14556531—22no—
36#14556530—42no—
37C16×D73—42no—
38#14556503—22no—
39#14556503—22no—
40#14556502—21no—
41#14556502—21no—
42C2×M16×C73C2×C411yescharacteristic
43C2×M16×C73C2311yescharacteristic
44C2×C42×C73C2×C411yescharacteristic
45M32×C73—22no—
46C4⋊C8×C73—22no—
47#11924028—41no—
48#11924028—41no—
49#11924028—41no—
50#11924028—41no—
51C2×Dic146C2×C811yescharacteristic
52C2×Dic146C2×C811yescharacteristic
53C2×D292—22no—
54C2×D292C2×C811yescharacteristic
55C2×D292C2×C811yescharacteristic
56C2×C22⋊D73—21no—
57C2×C22⋊D73M1612yes—
58C2×C4×D73—21no—
59C2×C4×D73C22⋊C412yes—
60#11924011—42no—
61#11924011—42no—
62C2×#10596831—22no—
63C2×#10596831—21no—
64C2×#10596831—41no—
65C2×#10596831C22⋊C411yescharacteristic
66C2×#10596831C22⋊C411yescharacteristic
67C8×D73—22no—
68C8×D73—22no—
69C8×D73—42no—
70C4⋊Dic73—22no—
71#11924005—22no—
72#11924005—22no—
73#11924003—42no—
74C22×C4×C73—21no—
75C22×C4×C73C22×C411yescharacteristic
76M16×C73—41no—
77M16×C73—41no—
78C2×C8×C73—21no—
79C2×C8×C73C2×D412yes—
80C42×C73—21no—
81C42×C73C2×C811yescharacteristic
82C42×C73C2×C811yescharacteristic
83C16×C73—42no—
84C23×C73C2×M1611yescharacteristic
85C22×D73—21no—
86C22×D73C22⋊C812yes—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—81no—
92C2×C4×C73C2×C22⋊C411yescharacteristic
93C2×C4×C73C22×C811yescharacteristic
94C22⋊D73—42no—
95C22⋊D73—42no—
96C2×Dic73—21no—
97C2×Dic73C22⋊C812yes—
98D292—41no—
99D292—41no—
100D292—82no—
101C4×D73—22no—
102C4×D73—22no—
103C4×D73—41no—
104C4×D73—41no—
105Dic146—41no—
106Dic146—41no—
107C8×C73—22no—
108C8×C73—41no—
109#10596831—21no—
110#10596831—41no—
111#10596831#9912yes—
112C22×C73—21no—
113C22×C73—41no—
114C22×C73C2×C22⋊C811yescharacteristic
115C2×D73—22no—
116C2×D73—42no—
117C2×D73—42no—
118C2×D73—41no—
119C4×C73—21no—
120C4×C73—41no—
121C4×C73—41no—
122C4×C73C2×#9911yescharacteristic
123C4×C73#41311yescharacteristic
124Dic73—22no—
125Dic73—41no—
126#11486—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—41no—
129C2×C73#145911yescharacteristic
130D73—42no—
131D73—81no—
132#2932—731no—
133#2115—732no—
134#1327—731no—
135#1327D7311yescharacteristic, complemented
136#1321—732no—
137C73#1148611yescharacteristic, complemented, Sylow
138C2×#131—731no—
139C2×#131—731no—
140#532—731no—
141C2×M32—1462no—
142#435—1462no—
143#435—1462no—
144#423C2×D7311yescharacteristic
145#408—731no—
146#408—731no—
147C2×#97—731no—
148#349—732no—
149#349—1462no—
150#349C2×D7312yes—
151C2×#41—731no—
152C2×#41—731no—
153#131—2922no—
154#131—2922no—
155C2×M16—732no—
156C2×M16—1461no—
157C2×M16C4×D7311yescharacteristic
158C2×M16C22×D7311yescharacteristic
159C22×C8—731no—
160C22×C8—731no—
161C22×C8—732no—
162C22×C8—1461no—
163C4⋊Q8—731no—
164#127—731no—
165#124—731no—
166#124—731no—
167#121—1462no—
168C4×D4—1462no—
169C2×C42C4×D7311yescharacteristic
170M32—22no—
171M32—2922no—
172M32—1462no—
173C2×C16—2922no—
174C4⋊C8—22no—
175C22⋊C8—1462no—
176C22⋊C8—1462no—
177#97—1461no—
178#97—1461no—
179#41—2921no—
180#41—2921no—
181#41—2921no—
182#41—2921no—
183C2×Q8—731no—
184C2×Q8—731no—
185C2×D4—732no—
186C2×D4—731no—
187C2×D4—731no—
188C2×D4—1462no—
189C2×D4—1461no—
190C22×C4—21no—
191C22×C4—732no—
192C22×C4—1461no—
193C22×C4C2×C4×D7311yescharacteristic
194M16—41no—
195M16—41no—
196M16—2922no—
197M16—2922no—
198C2×C8—21no—
199C2×C8—2922no—
200C2×C8—2921no—
201C2×C8—731no—
202C2×C8—731no—
203C2×C8—1462no—
204C2×C8—1462no—
205C2×C8—1461no—
206C2×C8—1462no—
207C2×C8D4×D7312yes—
208C4⋊C4—1462no—
209C22⋊C4—1462no—
210C22⋊C4—1462no—
211C42—21no—
212C42C8×D7311yescharacteristic
213C42C8×D7311yescharacteristic
214C16—42no—
215C16—2922no—
216C23—732no—
217C23—1461no—
218C23M16×D7311yescharacteristic
219Q8—2921no—
220Q8—2921no—
221D4—2921no—
222D4—2921no—
223D4—2922no—
224D4—2922no—
225D4—5842no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—81no—
231C2×C4—2921no—
232C2×C4—2921no—
233C2×C4—732no—
234C2×C4—1461no—
235C2×C4—1462no—
236C2×C4—1462no—
237C2×C4C22⋊C4×D7311yescharacteristic
238C2×C4C2×C8×D7311yescharacteristic
239C8—22no—
240C8—41no—
241C8—2921no—
242C8—732no—
243C8—1461no—
244C22—21no—
245C22—41no—
246C22—2922no—
247C22—2922no—
248C22—2921no—
249C22—1462no—
250C22C22⋊C8×D7311yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—2921no—
255C4—1462no—
256C4#99×D7311yescharacteristic
257C4#1798857011yescharacteristic
258C2—21no—
259C2—41no—
260C2—2922no—
261C2—5841no—
262C2#2260824811yescharacteristic
subgroupG/H
centerC4#99×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×M16C22×D73
FittingC73×#1327C2
socleC2×C73#1459
prime radical#17989137C4
cosocleC2×M16×C73C23
hypercenter#1327D73
hypocenterC73#11486
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #22611550C2semidirect
3. #22607882C2semidirect
4. #22607868C2semidirect
5. C73×#1327C2semidirect
135. #1327D73semidirect
137. C73#11486semidirect

Maximal subgroups

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

subgroupindex
#226186482
#226115502
#226078822
#226078682
C73×#13272
#1148673