Finite Group Databasebeta

← #36352504

Subgroups — #36352504

Order 22144.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#26012120C211yesmaximal, characteristic, complemented
2#26011644C211yesmaximal, characteristic
3#26011641C211yesmaximal, characteristic
4#26011409C211yesmaximal, characteristic
5#26011227C211yesmaximal, characteristic, complemented
6#26011216C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8C22×D692C2211yescharacteristic
9C2×#14583815—21no—
10C2×#14583813—21no—
11#19149678—21no—
12#19149677—21no—
13#19149669—21no—
14#19149669—21no—
15#19149658C2211yescharacteristic
16#19149656—21no—
17#19149635—21no—
18C2×#14583805C2211yescharacteristic
19C2×#14583803—21no—
20C2×#14583803—21no—
21#19149601—21no—
22#19149599—21no—
23C22⋊#13063436C2211yescharacteristic
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173—21no—
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—42no—
33C22×Dic173D411yescharacteristic
34C22×Dic173D411yescharacteristic
35#14583815—42no—
36#14583813—42no—
37C2×#13063436—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45#14583805—21no—
46#14583805—21no—
47C22⋊Dic173—21no—
48#14583803—42no—
49#14583803—42no—
50C4×Dic173—21no—
51C4×Dic173—21no—
52C2×Q8×C173D411yescharacteristic
53C22×C4×C173—21no—
54C22×C4×C173C2311yescharacteristic
55C2×C8×C173—21no—
56C4⋊C4×C173—21no—
57C4⋊C4×C173D411yescharacteristic
58C22⋊C4×C173—21no—
59C22×D173—22no—
60C22×D173—41no—
61C22×D173—41no—
62C22×D173—41no—
63C22×D173—41no—
64C22×D173—41no—
65C22×D173—41no—
66Dic346—41no—
67Dic346—81no—
68D692—22no—
69D692—22no—
70D692—41no—
71D692—41no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—41no—
78C2×Dic173SD1612yescomplemented
79#13063436—41no—
80C23×C173C2×D411yescharacteristic
81Q8×C173—41no—
82C2×C4×C173—21no—
83C2×C4×C173—21no—
84C2×C4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2×D411yescharacteristic
90C8×C173—41no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173#12011yescharacteristic
94C22×C173C2×SD1611yescharacteristic
95C22×C173#13711yescharacteristic
96C2×D173—42no—
97C2×D173—42no—
98C2×D173—42no—
99C2×D173—81no—
100C2×D173—81no—
101C2×D173—81no—
102C2×D173—81no—
103C4×C173—21no—
104C4×C173—21no—
105C4×C173—41no—
106C4×C173—41no—
107C4×C173—81no—
108Dic173—22no—
109Dic173—41no—
110Dic173—41no—
111Dic173—41no—
112C2×C173—21no—
113C2×C173—21no—
114C2×C173#44811yescharacteristic
115C2×C173#45411yescharacteristic
116C2×C173B(4,𝔽2)11yescharacteristic
117D173—82no—
118C173#159911yescharacteristic, complemented, Sylow
119#1599—1731nomaximal, Sylow
120#544—1731no—
121#450—1731no—
122#450—1731no—
123#394—1731no—
124#328—1731no—
125#328D17311yescharacteristic, complemented
126#324—1731no—
127C22×D4—1731no—
128C2×SD16—3461no—
129C2×SD16—3461no—
130C4⋊Q8—3461no—
131C4⋊Q8—3461no—
132#125—3461no—
133#125—3461no—
134C22⋊Q8—3461no—
135C22⋊Q8—1731no—
136C22⋊Q8C2×D17311yescharacteristic
137#117—3461no—
138C2×C4⋊C4—1731no—
139C2×C22⋊C4—3461no—
140C2×C22⋊C4—3461no—
141#102—3461no—
142#102—3461no—
143C22⋊C8—1731no—
144C22⋊C8C2×D17311yescharacteristic
145#95C2×D17311yescharacteristic
146C24—3461no—
147C2×Q8—3461no—
148C2×Q8—1731no—
149C2×Q8D69211yescharacteristic
150C2×D4—3461no—
151C2×D4—3461no—
152C2×D4—6922no—
153C22×C4—21no—
154C22×C4—1731no—
155C22×C4—1731no—
156C22×C4C22×D17311yescharacteristic
157SD16—6922no—
158SD16—6922no—
159C2×C8—21no—
160C2×C8—3461no—
161C4⋊C4—21no—
162C4⋊C4—3461no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—3461no—
170C4⋊C4—1731no—
171C4⋊C4D69211yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—3461no—
174C22⋊C4—6922no—
175C22⋊C4—6922no—
176C42—3461no—
177C42—3461no—
178C23—3462no—
179C23—6921no—
180C23—6921no—
181C23—6921no—
182C23—6921no—
183C23—6921no—
184C23—6921no—
185C23D4×D17311yescharacteristic
186Q8—41no—
187Q8—13841no—
188Q8—6921no—
189D4—3462no—
190D4—3462no—
191D4—6921no—
192D4—6921no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—1732no—
205C2×C4—6921no—
206C2×C4D4×D17311yescharacteristic
207C2×C4C2×D69211yescharacteristic
208C8—41no—
209C8—6921no—
210C22—21no—
211C22—21no—
212C22—13841no—
213C22—13841no—
214C22—13841no—
215C22—13841no—
216C22—6922no—
217C22—6922no—
218C22—6922no—
219C22#1914964911yescharacteristic
220C22SD16×D17311yescharacteristic
221C22#1914971111yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—3462no—
228C4—6921no—
229C4—6921no—
230C4—6921no—
231C2—21no—
232C2—21no—
233C2—13842no—
234C2#2601163111yescharacteristic
235C2#2601166511yescharacteristic
236C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radicalC22×D692C22
hypercenter#328D173
hypocenterC173#1599
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #26012120C2semidirect
5. #26011227C2semidirect
6. #26011216C2semidirect
7. #328×C173C2semidirect
78. C2×Dic173SD16semidirect
118. C173#1599semidirect
125. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260121202
#260116442
#260116412
#260114092
#260112272
#260112162
#328×C1732
#1599173