Finite Group Databasebeta

← #36355256

Subgroups — #36355256

Order 22144.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012066C211yesmaximal, characteristic
2#26011474C212yesmaximal
3#26011290C211yesmaximal, characteristic, complemented
4#26011289C212yesmaximal, complemented
5#356×C173C211yesmaximal, characteristic, complemented
6C2×#14583827C2211yescharacteristic
7C2×Q8×D173C411yescharacteristic
8M16×D173—22no—
9#19149677C2211yescharacteristic
10#19149668C411yescharacteristic
11#19149667C411yescharacteristic
12#19149664—22no—
13#19149657—22no—
14#19149636C411yescharacteristic
15#19149598C2212yes—
16#19149595—22no—
17C4⋊Q8×C173C2211yescharacteristic
18#101×C173C2212yes—
19#14583827—42no—
20C2×Dic346—21no—
21C2×Dic346C2×C411yescharacteristic
22Q8×D173—42no—
23C2×D692C2×C411yescharacteristic
24C2×D692D412yes—
25C2×C4×D173—21no—
26C2×C4×D173C2×C411yescharacteristic
27C2×C4×D173D412yes—
28#14583810—22no—
29#14583809—42no—
30C8×D173—42no—
31#14583805—21no—
32#14583805—22no—
33#14583803—21no—
34#14583803—22no—
35#14583803—22no—
36C4×Dic173C2×C411yescharacteristic
37C2×Q8×C173C2×C411yescharacteristic
38C2×Q8×C173C2311yescharacteristic
39M16×C173—22no—
40C4⋊C4×C173—22no—
41C42×C173C2×C411yescharacteristic
42C22×D173C22⋊C411yescharacteristic
43C22×D173C22⋊C412yes—
44Dic346—42no—
45Dic346—41no—
46D692—41no—
47D692—42no—
48C2×Dic173—21no—
49C2×Dic173—21no—
50C2×Dic173C22⋊C411yescharacteristic
51C4×D173—22no—
52C4×D173—22no—
53C4×D173—22no—
54C4×D173—42no—
55#13063436—42no—
56Q8×C173—41no—
57Q8×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173—21no—
60C2×C4×C173C22×C411yescharacteristic
61C2×C4×C173C2×D412yes—
62C8×C173—42no—
63C22×C173C2×C22⋊C411yescharacteristic
64C2×D173—21no—
65C2×D173—22no—
66C2×D173—41no—
67C2×D173—42no—
68C4×C173—22no—
69C4×C173—21no—
70C4×C173—41no—
71C4×C173—41no—
72Dic173—41no—
73Dic173—41no—
74Dic173#9912yes—
75C2×C173—21no—
76C2×C173C2×#9911yescharacteristic
77D173—41no—
78D173—42no—
79C173#213511yescharacteristic, complemented, Sylow
80#2135—1731nomaximal, Sylow
81#536—1731no—
82#413—1732no—
83#356—1731no—
84#356—1732no—
85#356D17311yescharacteristic, complemented
86C2×#41—1731no—
87C22×Q8—1731no—
88C2×M16—3462no—
89C4⋊Q8—1731no—
90C4⋊Q8C2×D17311yescharacteristic
91#124—1731no—
92#124—1731no—
93#123—3462no—
94C22⋊Q8—3462no—
95#117—1731no—
96#101—1732no—
97#101C2×D17312yes—
98#100—3462no—
99#41—6922no—
100C2×Q8—3461no—
101C2×Q8—1731no—
102C2×Q8—6922no—
103C2×Q8C4×D17311yescharacteristic
104C2×Q8C22×D17311yescharacteristic
105C2×D4—1731no—
106C2×D4—1732no—
107C22×C4—3461no—
108C22×C4—1731no—
109C22×C4—1732no—
110M16—22no—
111M16—3462no—
112M16—6922no—
113C2×C8—6922no—
114C4⋊C4—22no—
115C4⋊C4—3461no—
116C4⋊C4—3462no—
117C22⋊C4—3461no—
118C22⋊C4—3462no—
119C22⋊C4—3462no—
120C42—1731no—
121C42C4×D17311yescharacteristic
122C23—1732no—
123C23—1731no—
124Q8—41no—
125Q8—41no—
126Q8—6921no—
127Q8—6922no—
128D4—6922no—
129D4—6921no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—3462no—
133C2×C4—3462no—
134C2×C4—3462no—
135C2×C4—3461no—
136C2×C4—3461no—
137C2×C4—1731no—
138C2×C4—6922no—
139C2×C4C2×C4×D17311yescharacteristic
140C2×C4D4×D17312yes—
141C8—42no—
142C8—6922no—
143C22—3461no—
144C22—3462no—
145C22—6922no—
146C22—6921no—
147C22C22⋊C4×D17311yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—1732no—
153C4—6921no—
154C4—6921no—
155C2—21no—
156C2—6921no—
157C2—6922no—
158C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×Q8C22×D173
Fitting#356×C173C2
socleC2×C173C2×#99
prime radicalC2×#14583827C22
cosocleC2×Q8×C173C23
hypercenter#356D173
hypocenterC173#2135
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #26011290C2semidirect
4. #26011289C2semidirect
5. #356×C173C2semidirect
79. C173#2135semidirect
85. #356D173semidirect

Maximal subgroups

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

subgroupindex
#260120662
#260114742
#260112902
#260112892
#356×C1732
#2135173