Finite Group Databasebeta

← #33000201

Subgroups — #33000201

Order 18688.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#22612187C211yesmaximal, characteristic, complemented
2C16×D292C211yesmaximal, characteristic, complemented
3#22609297C211yesmaximal, characteristic, complemented
4#22609287C211yesmaximal, characteristic
5#22607979C211yesmaximal, characteristic, complemented
6#22607976C211yesmaximal, characteristic, complemented
7C73×#1372C211yesmaximal, characteristic, complemented
8C2×#14556536—21no—
9C2×#14556536—21no—
10C2×C16×D73—21no—
11#17988967C2211yescharacteristic
12#17988964C2211yescharacteristic
13#17988868—21no—
14#17988783—21no—
15C8×D292C2211yescharacteristic
16C4⋊#11924003C2211yescharacteristic
17#17988366—21no—
18#17988332—21no—
19#17988330—21no—
20C4×C16×C73C2211yescharacteristic
21#335×C73C2211yescharacteristic
22#335×C73C2211yescharacteristic
23C2×Dic292—21no—
24C2×Dic292—21no—
25C2×D584—21no—
26C2×D584—21no—
27C2×D584—41no—
28C2×C8×D73—21no—
29#14556591—21no—
30#14556588—21no—
31C4×D292D411yescharacteristic
32#14556536—42no—
33#14556536—42no—
34C2×#11924003—21no—
35C16×D73—42no—
36#14556525—41no—
37C4⋊#10596831D411yescharacteristic
38#14556513—41no—
39#14556513—41no—
40#14556503—21no—
41C2×C16×C73D411yescharacteristic
42C2×C16×C73D411yescharacteristic
43C4⋊C8×C73—21no—
44C4⋊C8×C73—21no—
45C4×C8×C73C2311yescharacteristic
46C2×Dic146—41no—
47C2×D292—41no—
48C2×D292—41no—
49C2×D292—41no—
50C2×D292D811yescharacteristic
51C2×C4×D73—21no—
52Dic292—41no—
53Dic292—41no—
54Dic292—81no—
55D584—41no—
56D584—41no—
57D584—81no—
58D584—82no—
59C2×#10596831—21no—
60C8×D73—41no—
61C8×D73—41no—
62C4⋊Dic73—41no—
63C4⋊Dic73—41no—
64C4⋊Dic73D811yescharacteristic
65#11924005—21no—
66#11924003—41no—
67C2×C8×C73—41no—
68C2×C8×C73—41no—
69C2×C8×C73C2×D411yescharacteristic
70C2×C8×C73#4111yescharacteristic
71C42×C73C2×D411yescharacteristic
72C16×C73—21no—
73C16×C73D812yescomplemented
74C22×D73—21no—
75C22×D73—81no—
76C2×C4×C73#12111yescharacteristic
77C2×C4×C73C2×D811yescharacteristic
78C2×C4×C73#13511yescharacteristic
79C2×Dic73—21no—
80C2×Dic73—81no—
81D292—21no—
82D292—81no—
83D292—81no—
84D292—81no—
85D292—81no—
86D292—161no—
87D292SD3212yescomplemented
88C4×D73—41no—
89C4×D73—41no—
90Dic146—81no—
91Dic146—81no—
92C8×C73—21no—
93C8×C73—81no—
94C8×C73—81no—
95C8×C73C2×D811yescharacteristic
96C8×C73AGL(1,ℤ/8)11yescharacteristic
97#10596831—41no—
98C22×C73#46611yescharacteristic
99C2×D73—22no—
100C2×D73—41no—
101C2×D73—41no—
102C2×D73—161no—
103C2×D73—161no—
104C2×D73—161no—
105C4×C73—21no—
106C4×C73#45911yescharacteristic
107C4×C73#47211yescharacteristic
108C4×C73C2×SD3211yescharacteristic
109C4×C73#50811yescharacteristic
110Dic73—41no—
111Dic73—161no—
112#12432—731nomaximal, Sylow
113C2×C73#166711yescharacteristic
114C2×C73#221711yescharacteristic
115C2×C73#222311yescharacteristic
116D73—42no—
117D73—321no—
118#2248—731no—
119D4×C16—731no—
120#1669—731no—
121#1667—731no—
122#1372D7311yescharacteristic, complemented
123#1362—731no—
124#1361—731no—
125C73#1243211yescharacteristic, complemented, Sylow
126C2×SD32—1461no—
127C2×SD32—1461no—
128C22×C16—1461no—
129#494—731no—
130#493—731no—
131#474—1461no—
132#459—1461no—
133C8×D4—731no—
134C4⋊C16—731no—
135C22⋊C16—1461no—
136C4×C16C2×D7311yescharacteristic
137#335C2×D7311yescharacteristic
138#335C2×D7311yescharacteristic
139#332—1461no—
140#331—1461no—
141C2×Q16—1461no—
142C2×Q16—1461no—
143C2×D8—2921no—
144C2×D8—1461no—
145C2×D8—1461no—
146C22×C8—1461no—
147C4⋊Q8—1461no—
148#127—1461no—
149C4×D4—731no—
150SD32—2922no—
151SD32—2922no—
152C2×C16—2922no—
153C2×C16—1461no—
154C2×C16D29211yescharacteristic
155C2×C16D29211yescharacteristic
156#107—2921no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—731no—
160#102—2921no—
161#102—2921no—
162C22⋊C8—1461no—
163C4×C8C22×D7311yescharacteristic
164C2×Q8—2921no—
165C2×D4—2921no—
166C2×D4—2921no—
167C2×D4—2921no—
168C2×D4—731no—
169C22×C4—1461no—
170Q16—2921no—
171Q16—2921no—
172Q16—5841no—
173D8—2921no—
174D8—2921no—
175D8—5841no—
176D8—5842no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—2921no—
180C2×C8—2921no—
181C2×C8—1461no—
182C2×C8C2×D29211yescharacteristic
183C2×C8#1192402911yescharacteristic
184C4⋊C4—2921no—
185C4⋊C4—2921no—
186C4⋊C4—731no—
187C22⋊C4—1461no—
188C42D4×D7311yescharacteristic
189C16—21no—
190C16—2921no—
191C16D58412yescomplemented
192C23—5841no—
193C23—1461no—
194Q8—5841no—
195Q8—5841no—
196D4—5841no—
197D4—5841no—
198D4—5841no—
199D4—5841no—
200D4—732no—
201D4—11681no—
202D4—1461no—
203C2×C4—2921no—
204C2×C4—2921no—
205C2×C4—5841no—
206C2×C4—1461no—
207C2×C4#1455656611yescharacteristic
208C2×C4D8×D7311yescharacteristic
209C2×C4#1455661411yescharacteristic
210C8—21no—
211C8—81no—
212C8—81no—
213C8—2921no—
214C8C2×D58411yescharacteristic
215C8#1455661911yescharacteristic
216C22—2921no—
217C22—2921no—
218C22—11681no—
219C22—11681no—
220C22—11681no—
221C22—1462no—
222C22#1798882411yescharacteristic
223C4—21no—
224C4—2921no—
225C4—11681no—
226C4#1798878311yescharacteristic
227C4#1798885811yescharacteristic
228C4SD32×D7311yescharacteristic
229C4#1798901811yescharacteristic
230C2—23361no—
231C2—2922no—
232C2#2260928711yescharacteristic
233C2#2261202811yescharacteristic
234C2#2261206211yescharacteristic
subgroupG/H
centerC22#17988824
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1372C2
socleC22×C73#466
prime radical#22609287C2
hypercenter#1372D73
hypocenterC73#12432
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #22612187C2semidirect
2. C16×D292C2semidirect
3. #22609297C2semidirect
5. #22607979C2semidirect
6. #22607976C2semidirect
7. C73×#1372C2semidirect
73. C16×C73D8semidirect
87. D292SD32semidirect
122. #1372D73semidirect
125. C73#12432semidirect
191. C16D584semidirect

Maximal subgroups

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

subgroupindex
#226121872
C16×D2922
#226092972
#226092872
#226079792
#226079762
C73×#13722
#1243273