Finite Group Databasebeta

← #354×D107 #28338120

Subgroups — #354×D107 #28338120

Order 13696.

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×D107C211yesmaximal, characteristic
2#99×D107C212yesmaximal
3#20292422C211yesmaximal, characteristic, complemented
4#20292421C212yesmaximal, complemented
5#354×C107C211yesmaximal, characteristic, complemented
6C2×Q8×D107C411yescharacteristic, complemented
7C2×D4×D107C2211yescharacteristic
8#16812098C411yescharacteristic, complemented
9#16812097C411yescharacteristic, complemented
10#16812096C2211yescharacteristic
11#16812095—22no—
12C22⋊C4×D107—22no—
13C22⋊C4×D107—22no—
14C42×D107C411yescharacteristic, complemented
15#16812024C2212yes—
16#16812023—22no—
17#124×C107C2211yescharacteristic
18#99×C107C2212yes—
19C23×D107D412yes—
20C2×Dic214—21no—
21C2×Dic214C2×C411yescharacteristic, complemented
22Q8×D107—41no—
23Q8×D107—41no—
24C2×D428C2×C411yescharacteristic
25C2×C22⋊D107D412yes—
26D4×D107—41no—
27D4×D107—41no—
28C2×C4×D107—21no—
29C2×C4×D107—21no—
30C2×C4×D107—42no—
31C2×C4×D107C2×C411yescharacteristic
32C22⋊Dic107—22no—
33C22⋊Dic107—22no—
34(C2×D107)⋊C4—22no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—42no—
37C4×Dic107—21no—
38C4×Dic107C2×C411yescharacteristic, complemented
39C2×Q8×C107C2×C411yescharacteristic, complemented
40C2×D4×C107C2311yescharacteristic
41C22⋊C4×C107—22no—
42C22⋊C4×C107—22no—
43C42×C107C2×C411yescharacteristic, complemented
44C23×C107C2×D412yes—
45C22×D107—22no—
46C22×D107—22no—
47C22×D107—44no—
48C22×D107C22⋊C411yescharacteristic
49C22×D107C22⋊C412yes—
50Q8×C107—41no—
51D4×C107—41no—
52C2×C4×C107—21no—
53C2×C4×C107—21no—
54C2×C4×C107—42no—
55C2×C4×C107C22×C411yescharacteristic
56C22⋊D107—42no—
57C2×Dic107—21no—
58C2×Dic107—21no—
59C2×Dic107—42no—
60C2×Dic107C22⋊C411yescharacteristic
61D428—41no—
62C4×D107—21no—
63C4×D107—21no—
64C4×D107—42no—
65C4×D107—41no—
66C4×D107—41no—
67C4×D107—84no—
68Dic214—42no—
69Dic214—41no—
70C22×C107—22no—
71C22×C107—42no—
72C22×C107C2×C22⋊C411yescharacteristic
73C2×D107—22no—
74C2×D107—22no—
75C2×D107—44no—
76C2×D107—42no—
77C2×D107—82no—
78C2×D107#9911yescharacteristic
79C2×D107#9911yescharacteristic
80C4×C107—21no—
81C4×C107—41no—
82C4×C107—41no—
83C4×C107—82no—
84Dic107—21no—
85Dic107—41no—
86Dic107—41no—
87Dic107—82no—
88C2×C107—21no—
89C2×C107—42no—
90C2×C107C2×#9911yescharacteristic
91D107—21no—
92D107—42no—
93D107#35412yesdirect product factor
94C2×#354—1071nomaximal, Sylow
95C107C2×#35411yescharacteristic, complemented, Sylow
96C2×#124—1071no—
97C2×#99—1072no—
98#354—1071no—
99#354—1072no—
100#354D10711yescharacteristic, direct product factor
101C22×Q8—1071no—
102C22×D4—1071no—
103#124—1071no—
104#124—1071no—
105#124—1071no—
106#124—2142no—
107#124C2×D10711yescharacteristic
108C2×C22⋊C4—2142no—
109C2×C22⋊C4—2142no—
110C2×C42—1071no—
111#99—1072no—
112#99—2142no—
113#99C2×D10712yes—
114C24—1072no—
115C2×Q8—1071no—
116C2×Q8—4281no—
117C2×Q8—4281no—
118C2×Q8—2141no—
119C2×Q8C4×D10711yescharacteristic, complemented
120C2×D4—1072no—
121C2×D4—1071no—
122C2×D4—4281no—
123C2×D4—4281no—
124C2×D4C22×D10711yescharacteristic
125C22×C4—1071no—
126C22×C4—4282no—
127C22×C4—2141no—
128C22×C4—2141no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—4282no—
132C22⋊C4—2142no—
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C22⋊C4—2142no—
136C42—1071no—
137C42—2141no—
138C42C4×D10711yescharacteristic, complemented
139C23—1071no—
140C23—1072no—
141C23—4284no—
142C23—2142no—
143C23—2142no—
144C23D4×D10712yes—
145Q8—41no—
146Q8—4281no—
147Q8—4282no—
148D4—41no—
149D4—4281no—
150D4—4282no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—8564no—
155C2×C4—1071no—
156C2×C4—4281no—
157C2×C4—4281no—
158C2×C4—4282no—
159C2×C4—4282no—
160C2×C4—2141no—
161C2×C4—2141no—
162C2×C4—2141no—
163C2×C4—2141no—
164C2×C4C2×C4×D10711yescharacteristic
165C22—22no—
166C22—42no—
167C22—8562no—
168C22—1071no—
169C22—1071no—
170C22—4282no—
171C22—4284no—
172C22—2142no—
173C22—2142no—
174C22C22⋊C4×D10711yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—8562no—
180C4—4281no—
181C4—4281no—
182C4—2141no—
183C2—21no—
184C2—42no—
185C2—1072no—
186C2—4282no—
187C2—2141no—
188C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC2×C4×C107C22×C4
FrattiniC2×D4C22×D107
Fitting#354×C107C2
socleC2×C107C2×#99
prime radicalC2×D4×D107C22
cosocleC2×D4×C107C23
hypercenter#354D107
hypocenterC107C2×#354
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #20292422C2semidirect
4. #20292421C2semidirect
5. #354×C107C2semidirect
6. C2×Q8×D107C4semidirect
8. #16812098C4semidirect
9. #16812097C4semidirect
14. C42×D107C4semidirect
21. C2×Dic214C2×C4semidirect
38. C4×Dic107C2×C4semidirect
39. C2×Q8×C107C2×C4semidirect
43. C42×C107C2×C4semidirect
93. D107#354direct (N × H)
95. C107C2×#354semidirect
100. #354D107direct (N × H)
119. C2×Q8C4×D107semidirect
138. C42C4×D107semidirect

Maximal subgroups

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

subgroupindex
#124×D1072
#99×D1072
#202924222
#202924212
#354×C1072
C2×#354107