Finite Group Databasebeta

← #37510348

Subgroups — #37510348

Order 22912.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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#26039067C211yesmaximal, characteristic
2#26038479C211yesmaximal, characteristic
3#26038304C211yesmaximal, characteristic
4#26038207C211yesmaximal, characteristic
5#26038200C211yesmaximal, characteristic
6#26038198C211yesmaximal, characteristic
7#428×C179C211yesmaximal, characteristic
8C2×#15668243C2211yescharacteristic
9#19153113—21no—
10#19153112—21no—
11#19153105—21no—
12#19153104—21no—
13#19153090—22no—
14#19153072C2211yescharacteristic
15C2×C4⋊Dic179C2211yescharacteristic
16C2×C4×Dic179C2211yescharacteristic
17#19153046—21no—
18#19153044—21no—
19C179⋊#95—22no—
20C2×M16×C179C2211yescharacteristic
21#117×C179C2211yescharacteristic
22C2×C4⋊C4×C179C2211yescharacteristic
23#107×C179—21no—
24#106×C179—21no—
25C2×Dic358—22no—
26C22×Dic179—21no—
27C22×Dic179D412yes—
28#15668243—21no—
29#15668243Q812yescomplemented
30C2×C179⋊C8—21no—
31C4⋊Dic179—22no—
32C4⋊Dic179—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179D411yescharacteristic
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—22no—
37Dic179⋊C4—22no—
38C22⋊Dic179—21no—
39C4×Dic179—21no—
40C4×Dic179—21no—
41C4×Dic179Q812yescomplemented
42C22×C4×C179—21no—
43C22×C4×C179C2311yescharacteristic
44M16×C179—21no—
45M16×C179Q812yescomplemented
46C2×C8×C179—21no—
47C4⋊C4×C179—21no—
48C4⋊C4×C179—21no—
49C4⋊C4×C179D411yescharacteristic
50C4⋊C4×C179D411yescharacteristic
51C22⋊C4×C179—21no—
52C42×C179—21no—
53Dic358—82no—
54C2×Dic179—21no—
55C2×Dic179—21no—
56C2×Dic179—22no—
57C2×Dic179—22no—
58C2×Dic179—41no—
59C2×Dic179—41no—
60C2×Dic179—42no—
61C179⋊C8—22no—
62C23×C179C2×D411yescharacteristic
63C2×C4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179—42no—
66C2×C4×C179—41no—
67C2×C4×C179C2×D411yescharacteristic
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×Q811yescharacteristic
70C2×C4×C179C2×Q811yescharacteristic
71C2×C4×C179C2×Q811yescharacteristic
72C2×C4×C179#4111yescharacteristic
73C8×C179—22no—
74C22×C179—21no—
75C22×C179—21no—
76C22×C179#12011yescharacteristic
77C22×C179C22⋊Q811yescharacteristic
78C22×C179C4⋊Q811yescharacteristic
79C4×C179—41no—
80C4×C179—41no—
81C4×C179—81no—
82C4×C179C22⋊Q811yescharacteristic
83C4×C179C22⋊Q811yescharacteristic
84C4×C179C4⋊Q811yescharacteristic
85C4×C179C4⋊Q811yescharacteristic
86Dic179—42no—
87Dic179—41no—
88Dic179—42no—
89Dic179—41no—
90C2×C179—21no—
91C2×C179—21no—
92C2×C179#39511yescharacteristic
93C2×C179#45411yescharacteristic
94C2×C179#45511yescharacteristic
95C179#206111yescharacteristic, complemented, Sylow
96#2061—1791nomaximal, Sylow
97#544—1791no—
98#428—1791no—
99#428D17911yescharacteristic
100#382—1791no—
101#343—1791no—
102#339—1791no—
103#339—1791no—
104C2×M16—1791no—
105C2×M16C2×D17911yescharacteristic
106C4⋊Q8—3581no—
107C4⋊Q8—3581no—
108#125—3581no—
109#125—3581no—
110C22⋊Q8—3582no—
111#117—1791no—
112#117C2×D17911yescharacteristic
113C2×C4⋊C4—1791no—
114C2×C4⋊C4C2×D17911yescharacteristic
115C2×C42—1791no—
116#107—21no—
117#107—3581no—
118#106—21no—
119#106—3581no—
120#95—3582no—
121C2×Q8—3582no—
122C22×C4—21no—
123C22×C4—3581no—
124C22×C4—1792no—
125C22×C4C22×D17911yescharacteristic
126M16—21no—
127M16—3581no—
128M16—1792no—
129M16Dic35812yescomplemented
130C2×C8—21no—
131C2×C8—3581no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3582no—
135C4⋊C4—3582no—
136C4⋊C4—3582no—
137C4⋊C4—3581no—
138C4⋊C4—3581no—
139C4⋊C4—1791no—
140C4⋊C4—1791no—
141C4⋊C4C22⋊D17911yescharacteristic
142C4⋊C4C22⋊D17911yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—3581no—
145C42—21no—
146C42—3581no—
147C42—3581no—
148C42—1792no—
149C23D4×D17911yescharacteristic
150Q8—14322no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—3581no—
156C2×C4—3581no—
157C2×C4—3582no—
158C2×C4—3582no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4—7162no—
162C2×C4D4×D17911yescharacteristic
163C2×C4C2×C22⋊D17911yescharacteristic
164C2×C4Q8×D17911yescharacteristic
165C2×C4Q8×D17911yescharacteristic
166C2×C4C2×Dic35811yescharacteristic
167C2×C4#1566825911yescharacteristic
168C8—22no—
169C8—3582no—
170C22—21no—
171C22—21no—
172C22#1915308111yescharacteristic
173C22#1915309111yescharacteristic
174C22#1915311211yescharacteristic
175C4—41no—
176C4—41no—
177C4—81no—
178C4—7162no—
179C4—7161no—
180C4—7162no—
181C4—7161no—
182C4#1915309111yescharacteristic
183C4#1915309211yescharacteristic
184C4#1915311111yescharacteristic
185C4#1915311211yescharacteristic
186C2—21no—
187C2—21no—
188C2#2603835911yescharacteristic
189C2#2603861011yescharacteristic
190C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#428×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#428D179
hypocenterC179#2061
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
29. #15668243Q8semidirect
41. C4×Dic179Q8semidirect
45. M16×C179Q8semidirect
95. C179#2061semidirect
129. M16Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390672
#260384792
#260383042
#260382072
#260382002
#260381982
#428×C1792
#2061179