Finite Group Databasebeta

← #353×D179 #37510656

Subgroups — #353×D179 #37510656

Order 22912.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D179C211yesmaximal, characteristic
2#99×D179C212yesmaximal, complemented
3#26038232C211yesmaximal, characteristic, complemented
4#26038231C212yesmaximal, complemented
5#353×C179C211yesmaximal, characteristic, complemented
6C2×D4×D179—22no—
7C2×D4×D179C411yescharacteristic, complemented
8C2×D4×D179C2211yescharacteristic
9#19153110C411yescharacteristic, complemented
10#19153109—22no—
11#19153109C411yescharacteristic, complemented
12#19153109C2211yescharacteristic
13C22⋊C4×D179—22no—
14C42×D179C411yescharacteristic, complemented
15#19153028C2212yescomplemented
16#19153027—22no—
17#127×C179C2211yescharacteristic
18#99×C179C2212yescomplemented
19C23×D179—21no—
20C23×D179D412yes—
21C2×D716—22no—
22C2×D716C2×C411yescharacteristic
23C2×D716C2×C411yescharacteristic, complemented
24C2×C22⋊D179—22no—
25C2×C22⋊D179—21no—
26C2×C22⋊D179—22no—
27C2×C22⋊D179D412yes—
28D4×D179—41no—
29D4×D179—41no—
30D4×D179—48no—
31D4×D179—41no—
32D4×D179—41no—
33C2×C4×D179—21no—
34C2×C4×D179—42no—
35C2×C4×D179C2×C411yescharacteristic
36C22⋊Dic179—22no—
37(C2×D179)⋊C4—42no—
38C4×Dic179—21no—
39C4×Dic179C2×C411yescharacteristic, complemented
40C2×D4×C179—22no—
41C2×D4×C179C2×C411yescharacteristic, complemented
42C2×D4×C179C2311yescharacteristic
43C22⋊C4×C179—22no—
44C42×C179C2×C411yescharacteristic, complemented
45C22×D179—22no—
46C22×D179—22no—
47C22×D179—21no—
48C22×D179—44no—
49C22×D179—44no—
50C22×D179—41no—
51C22×D179—41no—
52C22×D179C22⋊C411yescharacteristic
53C22×D179C22⋊C412yes—
54D716—41no—
55D716—41no—
56D716—44no—
57C22⋊D179—42no—
58C22⋊D179—44no—
59C22⋊D179—44no—
60C22⋊D179—42no—
61C2×Dic179—21no—
62C2×Dic179—42no—
63C2×Dic179C22⋊C411yescharacteristic
64C4×D179—24no—
65C4×D179—21no—
66C4×D179—21no—
67C4×D179—84no—
68C23×C179—21no—
69C23×C179C2×D412yes—
70D4×C179—41no—
71D4×C179—41no—
72D4×C179—44no—
73C2×C4×C179—21no—
74C2×C4×C179—42no—
75C2×C4×C179C22×C411yescharacteristic
76C22×C179—22no—
77C22×C179—42no—
78C22×C179—42no—
79C22×C179—41no—
80C22×C179C2×C22⋊C411yescharacteristic
81C2×D179—22no—
82C2×D179—22no—
83C2×D179—44no—
84C2×D179—42no—
85C2×D179—42no—
86C2×D179—41no—
87C2×D179—82no—
88C2×D179—82no—
89C2×D179—82no—
90C2×D179#9911yescharacteristic
91C2×D179#9911yescharacteristic
92C4×C179—22no—
93C4×C179—21no—
94C4×C179—82no—
95Dic179—22no—
96Dic179—21no—
97Dic179—82no—
98C2×C179—21no—
99C2×C179—42no—
100C2×C179—81no—
101C2×C179C2×#9911yescharacteristic
102D179—21no—
103D179—42no—
104D179—81no—
105D179#35312yesdirect product factor
106C179C2×#35311yescharacteristic, complemented, Sylow
107C2×#353—1791nomaximal, Sylow
108C2×#127—1791no—
109C2×#99—1792no—
110#353—1791no—
111#353—1792no—
112#353D17911yescharacteristic, direct product factor
113C22×D4—3582no—
114C22×D4—1791no—
115C22×D4—1791no—
116#127—3582no—
117#127—1791no—
118#127—1791no—
119#127—1791no—
120#127C2×D17911yescharacteristic
121C2×C22⋊C4—3582no—
122C2×C42—1791no—
123#99—3582no—
124#99—1792no—
125#99C2×D17912yescomplemented
126C24—3581no—
127C24—1792no—
128C2×D4—22no—
129C2×D4—3582no—
130C2×D4—3582no—
131C2×D4—3582no—
132C2×D4—3581no—
133C2×D4—1792no—
134C2×D4—1791no—
135C2×D4—1791no—
136C2×D4—7161no—
137C2×D4—7161no—
138C2×D4—7161no—
139C2×D4—7161no—
140C2×D4—7168no—
141C2×D4C4×D17911yescharacteristic, complemented
142C2×D4C22×D17911yescharacteristic
143C22×C4—3581no—
144C22×C4—1791no—
145C22×C4—7162no—
146C22⋊C4—22no—
147C22⋊C4—3582no—
148C22⋊C4—7162no—
149C42—3581no—
150C42—1791no—
151C42C4×D17911yescharacteristic, complemented
152C23—21no—
153C23—3582no—
154C23—3582no—
155C23—3581no—
156C23—1791no—
157C23—1792no—
158C23—7164no—
159C23—7164no—
160C23—7161no—
161C23—7161no—
162C23D4×D17912yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—7161no—
167D4—7161no—
168D4—7164no—
169D4—7162no—
170D4—7164no—
171D4—7164no—
172D4—7162no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—3584no—
176C2×C4—3581no—
177C2×C4—3581no—
178C2×C4—3581no—
179C2×C4—14324no—
180C2×C4—1791no—
181C2×C4—7162no—
182C2×C4C2×C4×D17911yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—3582no—
188C22—3582no—
189C22—14322no—
190C22—14322no—
191C22—14322no—
192C22—1791no—
193C22—1791no—
194C22—7162no—
195C22—7162no—
196C22—7164no—
197C22—7161no—
198C22C22⋊C4×D17911yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—3582no—
203C4—3581no—
204C4—14322no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—3581no—
209C2—14321no—
210C2—1792no—
211C2—7162no—
212C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC2×C4×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#353×C179C2
socleC2×C179C2×#99
prime radical#127×D179C2
cosocleC2×D4×C179C23
hypercenter#353D179
hypocenterC179C2×#353
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D179C2semidirect
3. #26038232C2semidirect
4. #26038231C2semidirect
5. #353×C179C2semidirect
7. C2×D4×D179C4semidirect
9. #19153110C4semidirect
11. #19153109C4semidirect
14. C42×D179C4semidirect
15. #19153028C22semidirect
18. #99×C179C22semidirect
23. C2×D716C2×C4semidirect
39. C4×Dic179C2×C4semidirect
41. C2×D4×C179C2×C4semidirect
44. C42×C179C2×C4semidirect
105. D179#353direct (N × H)
106. C179C2×#353semidirect
112. #353D179direct (N × H)
125. #99C2×D179semidirect
141. C2×D4C4×D179semidirect
151. C42C4×D179semidirect

Maximal subgroups

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

subgroupindex
#127×D1792
#99×D1792
#260382322
#260382312
#353×C1792
C2×#353179