Finite Group Databasebeta

← #37510354

Subgroups — #37510354

Order 22912.

314 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 75 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#26039065C211yesmaximal, characteristic
2#26038314C212yesmaximal
3#26038311C211yesmaximal, characteristic
4#26038205C212yesmaximal
5#342×C179C211yesmaximal, characteristic
6C23×Dic179C2211yescharacteristic
7#19153092—21no—
8#19153092—21no—
9#19153090—22no—
10#19153090—22no—
11C2×C4⋊Dic179—22no—
12C2×Dic179⋊C4—21no—
13C2×C22⋊Dic179C2212yes—
14C2×C22⋊Dic179C2211yescharacteristic
15C179⋊#95—22no—
16C179⋊#95—21no—
17C2×C22⋊C4×C179C2212yes—
18C2×C22⋊C4×C179C2211yescharacteristic
19C2×Dic358—21no—
20C2×Dic358—21no—
21C22×Dic179—21no—
22C22×Dic179—21no—
23C22×Dic179—22no—
24C22×Dic179—42no—
25C22×Dic179D412yes—
26C22×Dic179D412yes—
27C4⋊Dic179—21no—
28C4⋊Dic179—21no—
29C4⋊Dic179—44no—
30Dic179⋊C4—22no—
31Dic179⋊C4—22no—
32Dic179⋊C4—42no—
33C22⋊Dic179—22no—
34C22⋊Dic179—22no—
35C22⋊Dic179—21no—
36C22⋊Dic179Q812yescomplemented
37C24×C179C2311yescharacteristic
38C22×C4×C179—22no—
39C22×C4×C179D411yescharacteristic
40C22×C4×C179D411yescharacteristic
41C22⋊C4×C179—22no—
42C22⋊C4×C179—22no—
43C22⋊C4×C179Q814yescomplemented
44Dic358—81no—
45Dic358—81no—
46C2×Dic179—22no—
47C2×Dic179—24no—
48C2×Dic179—22no—
49C2×Dic179—22no—
50C2×Dic179—22no—
51C2×Dic179—41no—
52C2×Dic179—41no—
53C2×Dic179—41no—
54C2×Dic179—42no—
55C2×Dic179—41no—
56C2×Dic179—42no—
57C2×Dic179—42no—
58C23×C179—41no—
59C23×C179—41no—
60C23×C179C2×D412yes—
61C23×C179C2×D411yescharacteristic
62C23×C179C2×Q811yescharacteristic
63C23×C179C2×Q812yes—
64C23×C179#4111yescharacteristic
65C2×C4×C179—21no—
66C2×C4×C179—21no—
67C2×C4×C179—24no—
68C2×C4×C179—42no—
69C2×C4×C179—42no—
70C2×C4×C179—41no—
71C2×C4×C179—41no—
72C22×C179—22no—
73C22×C179—21no—
74C22×C179—22no—
75C22×C179—21no—
76C22×C179—41no—
77C22×C179—42no—
78C22×C179—41no—
79C22×C179#12011yescharacteristic
80C22×C179C22⋊Q812yes—
81C22×C179C22⋊Q811yescharacteristic
82C22×C179C4⋊Q812yes—
83C22×C179C4⋊Q811yescharacteristic
84C4×C179—44no—
85C4×C179—41no—
86C4×C179—41no—
87Dic179—42no—
88Dic179—42no—
89Dic179—42no—
90Dic179—42no—
91C2×C179—21no—
92C2×C179—22no—
93C2×C179—22no—
94C2×C179—21no—
95C2×C179#39511yescharacteristic
96C2×C179B(4,𝔽2)11yescharacteristic
97C2×C179#45811yescharacteristic
98C179#206211yescharacteristic, complemented, Sylow
99#2062—1791nomaximal, Sylow
100#543—1791no—
101#385—1791no—
102#385—1792no—
103#342—1792no—
104#342D17911yescharacteristic
105C23×C4—1791no—
106C22⋊Q8—3582no—
107C22⋊Q8—3582no—
108C22⋊Q8—3581no—
109C22⋊Q8—3581no—
110C2×C4⋊C4—3582no—
111C2×C4⋊C4—3581no—
112C2×C22⋊C4—1792no—
113C2×C22⋊C4—1791no—
114C2×C22⋊C4C2×D17911yescharacteristic
115C2×C22⋊C4C2×D17912yes—
116#95—3582no—
117#95—3581no—
118C24C22×D17911yescharacteristic
119C2×Q8—3581no—
120C2×Q8—3581no—
121C22×C4—22no—
122C22×C4—3582no—
123C22×C4—3581no—
124C22×C4—3581no—
125C22×C4—1792no—
126C22×C4—1792no—
127C22×C4—7162no—
128C22×C4D71611yescharacteristic
129C22×C4D71611yescharacteristic
130C4⋊C4—3582no—
131C4⋊C4—3582no—
132C4⋊C4—3581no—
133C4⋊C4—3581no—
134C4⋊C4—7162no—
135C4⋊C4—7164no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—3582no—
139C22⋊C4—3582no—
140C22⋊C4—3581no—
141C22⋊C4—1792no—
142C22⋊C4Dic35814yescomplemented
143C23—41no—
144C23—41no—
145C23D4×D17912yes—
146C23C2×D71611yescharacteristic
147C23Q8×D17911yescharacteristic
148C23C2×Dic35812yes—
149C23#1566825811yescharacteristic
150Q8—14321no—
151Q8—14321no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3584no—
160C2×C4—3582no—
161C2×C4—3582no—
162C2×C4—3582no—
163C2×C4—3582no—
164C2×C4—7161no—
165C2×C4—7161no—
166C2×C4—7162no—
167C2×C4—7161no—
168C2×C4—7161no—
169C2×C4—7162no—
170C2×C4—7162no—
171C22—22no—
172C22—21no—
173C22—22no—
174C22—21no—
175C22—41no—
176C22—42no—
177C22—41no—
178C22#1915308311yescharacteristic
179C22#1915309012yes—
180C22#1915309311yescharacteristic
181C22#1915311212yes—
182C22#1915311311yescharacteristic
183C4—44no—
184C4—41no—
185C4—41no—
186C4—7162no—
187C4—7162no—
188C4—7162no—
189C4—7162no—
190C2—22no—
191C2—21no—
192C2—22no—
193C2—21no—
194C2#2603836011yescharacteristic
195C2#2603862711yescharacteristic
196C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, prime radical, cosocleC24×C179C23
FrattiniC24C22×D179
Fitting#342×C179C2
socleC22×C179#120
hypercenter#342D179
hypocenterC179#2062
LCSG ▷ C24×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
36. C22⋊Dic179Q8semidirect
43. C22⋊C4×C179Q8semidirect
98. C179#2062semidirect
142. C22⋊C4Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390652
#260383142
#260383112
#260382052
#342×C1792
#2062179