Finite Group Databasebeta

← #354×D173 #36355222

Subgroups — #354×D173 #36355222

Order 22144.

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

normal subgroups only

This group has 284 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#124×D173C211yesmaximal, characteristic
2#99×D173C212yesmaximal
3#26011286C211yesmaximal, characteristic, complemented
4#26011285C212yesmaximal, complemented
5#354×C173C211yesmaximal, characteristic, complemented
6C2×Q8×D173C411yescharacteristic, complemented
7C2×D4×D173C2211yescharacteristic
8#19149668C411yescharacteristic, complemented
9#19149667C411yescharacteristic, complemented
10#19149666C2211yescharacteristic
11#19149665—22no—
12C22⋊C4×D173—22no—
13C22⋊C4×D173—22no—
14C42×D173C411yescharacteristic, complemented
15#19149594C2212yes—
16#19149593—22no—
17#124×C173C2211yescharacteristic
18#99×C173C2212yes—
19C23×D173D412yes—
20C2×Dic346—21no—
21C2×Dic346C2×C411yescharacteristic, complemented
22Q8×D173—41no—
23Q8×D173—41no—
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173D412yes—
26D4×D173—41no—
27D4×D173—41no—
28C2×C4×D173—21no—
29C2×C4×D173—21no—
30C2×C4×D173—42no—
31C2×C4×D173C2×C411yescharacteristic
32C22⋊Dic173—22no—
33C22⋊Dic173—22no—
34#14583803—22no—
35#14583803—22no—
36#14583803—42no—
37C4×Dic173—21no—
38C4×Dic173C2×C411yescharacteristic, complemented
39C2×Q8×C173C2×C411yescharacteristic, complemented
40C2×D4×C173C2311yescharacteristic
41C22⋊C4×C173—22no—
42C22⋊C4×C173—22no—
43C42×C173C2×C411yescharacteristic, complemented
44C22×D173—22no—
45C22×D173—22no—
46C22×D173—44no—
47C22×D173C22⋊C411yescharacteristic
48C22×D173C22⋊C412yes—
49Dic346—42no—
50Dic346—41no—
51D692—41no—
52C22⋊D173—42no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—42no—
56C2×Dic173C22⋊C411yescharacteristic
57C4×D173—21no—
58C4×D173—21no—
59C4×D173—42no—
60C4×D173—41no—
61C4×D173—41no—
62C4×D173—84no—
63C23×C173C2×D412yes—
64Q8×C173—41no—
65D4×C173—41no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—42no—
69C2×C4×C173C22×C411yescharacteristic
70C22×C173—22no—
71C22×C173—42no—
72C22×C173C2×C22⋊C411yescharacteristic
73C2×D173—22no—
74C2×D173—22no—
75C2×D173—44no—
76C2×D173—42no—
77C2×D173—82no—
78C2×D173#9911yescharacteristic
79C2×D173#9911yescharacteristic
80C4×C173—21no—
81C4×C173—41no—
82C4×C173—41no—
83C4×C173—82no—
84Dic173—21no—
85Dic173—41no—
86Dic173—41no—
87Dic173—82no—
88C2×C173—21no—
89C2×C173—42no—
90C2×C173C2×#9911yescharacteristic
91D173—21no—
92D173—42no—
93D173#35412yesdirect product factor
94C173C2×#35411yescharacteristic, complemented, Sylow
95C2×#354—1731nomaximal, Sylow
96C2×#124—1731no—
97C2×#99—1732no—
98#354—1731no—
99#354—1732no—
100#354D17311yescharacteristic, direct product factor
101C22×Q8—1731no—
102C22×D4—1731no—
103#124—3462no—
104#124—1731no—
105#124—1731no—
106#124—1731no—
107#124C2×D17311yescharacteristic
108C2×C22⋊C4—3462no—
109C2×C22⋊C4—3462no—
110C2×C42—1731no—
111#99—3462no—
112#99—1732no—
113#99C2×D17312yes—
114C24—1732no—
115C2×Q8—3461no—
116C2×Q8—1731no—
117C2×Q8—6921no—
118C2×Q8—6921no—
119C2×Q8C4×D17311yescharacteristic, complemented
120C2×D4—1732no—
121C2×D4—1731no—
122C2×D4—6921no—
123C2×D4—6921no—
124C2×D4C22×D17311yescharacteristic
125C22×C4—3461no—
126C22×C4—3461no—
127C22×C4—1731no—
128C22×C4—6922no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—3462no—
132C22⋊C4—3462no—
133C22⋊C4—3462no—
134C22⋊C4—3462no—
135C22⋊C4—6922no—
136C42—3461no—
137C42—1731no—
138C42C4×D17311yescharacteristic, complemented
139C23—3462no—
140C23—3462no—
141C23—1731no—
142C23—1732no—
143C23—6924no—
144C23D4×D17312yes—
145Q8—41no—
146Q8—6921no—
147Q8—6922no—
148D4—41no—
149D4—6921no—
150D4—6922no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—3461no—
158C2×C4—13844no—
159C2×C4—1731no—
160C2×C4—6922no—
161C2×C4—6921no—
162C2×C4—6921no—
163C2×C4—6922no—
164C2×C4C2×C4×D17311yescharacteristic
165C22—22no—
166C22—42no—
167C22—3462no—
168C22—3462no—
169C22—13842no—
170C22—1731no—
171C22—1731no—
172C22—6922no—
173C22—6924no—
174C22C22⋊C4×D17311yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—3461no—
180C4—13842no—
181C4—6921no—
182C4—6921no—
183C2—21no—
184C2—42no—
185C2—3461no—
186C2—1732no—
187C2—6922no—
188C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#354×C173C2
socleC2×C173C2×#99
prime radicalC2×D4×D173C22
cosocleC2×D4×C173C23
hypercenter#354D173
hypocenterC173C2×#354
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #26011286C2semidirect
4. #26011285C2semidirect
5. #354×C173C2semidirect
6. C2×Q8×D173C4semidirect
8. #19149668C4semidirect
9. #19149667C4semidirect
14. C42×D173C4semidirect
21. C2×Dic346C2×C4semidirect
38. C4×Dic173C2×C4semidirect
39. C2×Q8×C173C2×C4semidirect
43. C42×C173C2×C4semidirect
93. D173#354direct (N × H)
94. C173C2×#354semidirect
100. #354D173direct (N × H)
119. C2×Q8C4×D173semidirect
138. C42C4×D173semidirect

Maximal subgroups

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

subgroupindex
#124×D1732
#99×D1732
#260112862
#260112852
#354×C1732
C2×#354173