Finite Group Databasebeta

← #37575688

Subgroups — #37575688

Order 24448.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#20259301C211yesmaximal, characteristic
2#27132567C212yesmaximal, complemented
3#27132560C212yesmaximal, complemented
4#27132557C212yesmaximal
5#27132551C212yesmaximal
6#27132550C212yesmaximal, complemented
7#27132548C212yesmaximal, complemented
8#27132523C211yesmaximal, characteristic
9#381×C191C211yesmaximal, characteristic
10C22×Dic382C2211yescharacteristic
11#20259301C418yescomplemented
12C2×C4⋊Dic191C2212yes—
13C2×C4⋊Dic191C2211yescharacteristic
14C2×Dic191⋊C4C2212yes—
15C2×Dic191⋊C4C2212yes—
16C2×Dic191⋊C4C2212yes—
17C2×C22⋊Dic191C2211yescharacteristic
18C2×C22⋊Dic191C2211yescharacteristic
19C2×C22⋊Dic191C2212yes—
20C2×C4×Dic191C2212yes—
21C2×C4×Dic191C2212yes—
22C191⋊#95C2212yes—
23C191⋊#95C2212yes—
24C191⋊#95C2212yes—
25C191⋊#95C2212yes—
26C191⋊#95C2212yes—
27C23×C4×C191C2211yescharacteristic
28C2×C22⋊C4×C191C2212yes—
29#95×C191C2212yes—
30#95×C191C2211yescharacteristic
31#95×C191C2211yescharacteristic
32C2×Dic382—42no—
33C2×Dic382C2×C414yes—
34C22×Dic191C2312yes—
35C22×Dic191C2312yes—
36C22×Dic191C2312yes—
37C22×Dic191C2312yes—
38C4⋊Dic191—42no—
39C4⋊Dic191C2×C414yes—
40Dic191⋊C4—42no—
41Dic191⋊C4—42no—
42Dic191⋊C4C2×C418yes—
43C22⋊Dic191—41no—
44C22⋊Dic191—41no—
45C22⋊Dic191C2×C418yes—
46C4×Dic191—42no—
47C4×Dic191—42no—
48C24×C191C2311yescharacteristic
49C22×C4×C191—42no—
50C22×C4×C191C2×C414yes—
51C22×C4×C191C2312yes—
52C22×C4×C191C2312yes—
53C22×C4×C191C2311yescharacteristic
54C22×C4×C191C2311yescharacteristic
55C22⋊C4×C191—42no—
56Dic382—44no—
57C2×Dic191—22no—
58C2×Dic191—22no—
59C2×Dic191—22no—
60C2×Dic191—22no—
61C2×Dic191—22no—
62C2×Dic191—22no—
63C2×Dic191—22no—
64C2×Dic191—22no—
65C2×Dic191—22no—
66C2×Dic191—22no—
67C2×Dic191C22×C414yes—
68C2×Dic191C22×C414yes—
69C23×C191—22no—
70C23×C191—21no—
71C23×C191—22no—
72C23×C191—21no—
73C23×C191C22×C412yes—
74C23×C191C2411yescharacteristic
75C2×C4×C191—22no—
76C2×C4×C191—22no—
77C2×C4×C191—22no—
78C2×C4×C191—22no—
79C2×C4×C191—22no—
80C2×C4×C191—22no—
81C2×C4×C191—22no—
82C2×C4×C191—22no—
83C2×C4×C191—44no—
84C2×C4×C191C22×C412yes—
85C2×C4×C191C22×C412yes—
86C22×C191—22no—
87C22×C191—22no—
88C22×C191—22no—
89C22×C191—42no—
90C22×C191—42no—
91C22×C191C23×C411yescharacteristic
92C22×C191H(5,𝔽2)12yes—
93C22×C191H(5,𝔽2)11yescharacteristic
94C22×C1912−1+412yes—
95C22×C1912−1+411yescharacteristic
96C4×C191—22no—
97C4×C191—22no—
98C4×C191—42no—
99C4×C191—42no—
100Dic191—24no—
101Dic191—24no—
102Dic191—42no—
103Dic191—42no—
104C2×C191—42no—
105C2×C191#51811yescharacteristic
106C2×C191#51911yescharacteristic
107C2×C191#52011yescharacteristic
108C2×C191#56212yes—
109C2×C191#56312yes—
110C191#237811yescharacteristic, complemented, Sylow
111#2378—1911nomaximal, Sylow
112C2×C22⋊Q8—1911no—
113#391—1912no—
114#389—1912no—
115#388—1912no—
116#388—1912no—
117#387—1912no—
118#387—1912no—
119#381D19111yescharacteristic
120#380—1911no—
121C22×Q8—1911no—
122C23×C4C2×D19111yescharacteristic
123C22⋊Q8—1918no—
124C2×C4⋊C4—1911no—
125C2×C4⋊C4—1912no—
126C2×C4⋊C4—1912no—
127C2×C4⋊C4—1912no—
128C2×C4⋊C4—1912no—
129C2×C22⋊C4—1912no—
130C2×C22⋊C4—1911no—
131C2×C22⋊C4—1911no—
132C2×C22⋊C4C2×D19112yes—
133C2×C42—1912no—
134C2×C42—1912no—
135#95—1912no—
136#95—1912no—
137#95—1912no—
138#95—1912no—
139#95—1912no—
140#95C2×D19111yescharacteristic
141#95C2×D19112yes—
142#95C2×D19111yescharacteristic
143C24C22×D19111yescharacteristic
144C2×Q8—1914no—
145C2×Q8—7642no—
146C22×C4—42no—
147C22×C4—1912no—
148C22×C4—1912no—
149C22×C4—1912no—
150C22×C4—1912no—
151C22×C4C4×D19114yes—
152C22×C4C22×D19112yes—
153C22×C4C22×D19112yes—
154C22×C4C22×D19111yescharacteristic
155C22×C4C22×D19111yescharacteristic
156C4⋊C4—1918no—
157C4⋊C4—1914no—
158C4⋊C4—7642no—
159C4⋊C4—7642no—
160C4⋊C4—7642no—
161C22⋊C4—42no—
162C22⋊C4—1918no—
163C22⋊C4—7641no—
164C22⋊C4—7641no—
165C42—7642no—
166C42—7642no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23C2×C4×D19112yes—
172C23C23×D19111yescharacteristic
173Q8—7644no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—3822no—
184C2×C4—3822no—
185C2×C4—3822no—
186C2×C4—3822no—
187C2×C4—3822no—
188C2×C4—3822no—
189C2×C4—3822no—
190C2×C4—3822no—
191C2×C4—3822no—
192C2×C4—3822no—
193C2×C4—1914no—
194C2×C4—1914no—
195C2×C4C2×C4×D19112yes—
196C2×C4C2×C4×D19112yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D19111yescharacteristic
203C22#2025936712yes—
204C22#2025936811yescharacteristic
205C22#2025936912yes—
206C22#2025936911yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—3824no—
212C4—3824no—
213C4—7642no—
214C4—7642no—
215C2—42no—
216C2#2713313611yescharacteristic
217C2#2713314011yescharacteristic
218C2#2713314311yescharacteristic
219C2#2713341012yes—
220C2#2713341612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutatorC22×C191C23×C4
Fitting#381×C191C2
socle, cosocleC23×C191C24
prime radicalC24×C191C23
hypercenter#381D191
hypocenterC191#2378
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #27132567C2semidirect
3. #27132560C2semidirect
6. #27132550C2semidirect
7. #27132548C2semidirect
11. #20259301C4semidirect
110. C191#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#202593012
#271325672
#271325602
#271325572
#271325512
#271325502
#271325482
#271325232
#381×C1912
#2378191