Finite Group Databasebeta

← #28344832

Subgroups — #28344832

Order 13696.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#16812100C211yesmaximal, characteristic, complemented
2C2×#16812096C211yesmaximal, characteristic
3#20292533C211yesmaximal, characteristic, complemented
4#20292532C211yesmaximal, characteristic, complemented
5#20292530C211yesmaximal, characteristic
6#20292520C214yesmaximal, complemented
7#20292515C214yesmaximal, complemented
8C42×Dic107C211yesmaximal, characteristic, complemented
9#390×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11#16812100—24no—
12#16812096—24no—
13C2×C4⋊Dic107C2212yes—
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4C2212yes—
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2214yes—
19C2×C4×Dic107C2212yes—
20C2×C4×Dic107C2212yes—
21C2×C4×Dic107C2211yescharacteristic
22C2×C4×Dic107C2211yescharacteristic
23C107⋊#95C2214yes—
24C107⋊#95C2214yes—
25C22×D4×C107C2211yescharacteristic
26C2×C4⋊C4×C107C2211yescharacteristic
27C2×C22⋊C4×C107C2214yes—
28C2×C42×C107C2211yescharacteristic
29C2×Dic214—22no—
30C2×Dic214—22no—
31C2×Dic214—22no—
32C22×Dic107C2312yes—
33C22×Dic107C2312yes—
34C22×Dic107C2311yescharacteristic
35C22×Dic107C2311yescharacteristic
36C22×Dic107C2312yes—
37C4⋊Dic107—22no—
38C4⋊Dic107—24no—
39Dic107⋊C4—24no—
40C22⋊Dic107—24no—
41C22⋊Dic107—24no—
42C22⋊Dic107—28no—
43C4×Dic107—24no—
44C4×Dic107—22no—
45C4×Dic107—24no—
46C4×Dic107D414yes—
47C24×C107C2312yes—
48C2×D4×C107—22no—
49C2×D4×C107—22no—
50C2×D4×C107—22no—
51C22×C4×C107C2312yes—
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—22no—
55C22⋊C4×C107—28no—
56C42×C107—22no—
57C23×C107—22no—
58C23×C107—24no—
59C23×C107—22no—
60C23×C107—22no—
61C23×C107—24no—
62C23×C107C2411yescharacteristic
63D4×C107—44no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107—24no—
68C2×C4×C107C2×D412yes—
69C2×C4×C107#4114yes—
70C2×C4×C107#4112yes—
71C2×C4×C107#4112yes—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—24no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107C2×D412yes—
82C2×Dic107C2×D412yes—
83C2×Dic107#4114yes—
84Dic214—44no—
85C22×C107—42no—
86C22×C107—44no—
87C22×C107—44no—
88C22×C107—42no—
89C22×C107—42no—
90C22×C107C22×D411yescharacteristic
91C22×C107C2×#4112yes—
92C22×C107C2×#4111yescharacteristic
93C22×C107C2×#4112yes—
94C22×C107C2×#4111yescharacteristic
95C4×C107—24no—
96C4×C107—42no—
97C4×C107#12414yes—
98Dic107—24no—
99Dic107—22no—
100Dic107—22no—
101Dic107—42no—
102Dic107—42no—
103C2×C107—82no—
104C2×C107C2×#12411yescharacteristic
105C2×C107#52912yes—
106C2×C107#52912yes—
107C2×C107#53211yescharacteristic
108C2×C107#53211yescharacteristic
109#2861—1071nomaximal, Sylow
110C107#286111yescharacteristic, complemented, Sylow
111C2×#125—1071no—
112C2×#124—1071no—
113#391—1071no—
114#391—1071no—
115#390—1071no—
116#390D10711yescharacteristic
117#388—1074no—
118#388—1074no—
119C43—1071no—
120C22×Q8—1071no—
121C22×D4C2×D10711yescharacteristic
122#125—2144no—
123#124—2144no—
124C2×C4⋊C4—1072no—
125C2×C4⋊C4—1072no—
126C2×C4⋊C4—1071no—
127C2×C4⋊C4C2×D10711yescharacteristic
128C2×C22⋊C4—1072no—
129C2×C22⋊C4—1072no—
130C2×C22⋊C4—1074no—
131C2×C22⋊C4C2×D10714yes—
132C2×C42—1071no—
133C2×C42—1072no—
134C2×C42—1072no—
135C2×C42—1071no—
136C2×C42C2×D10711yescharacteristic
137#95—1074no—
138#95—1074no—
139C24C22×D10712yes—
140C2×Q8—2142no—
141C2×Q8—2142no—
142C2×Q8—2142no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C22×C4—1072no—
147C22×C4—1071no—
148C22×C4—1071no—
149C22×C4—1072no—
150C22×C4—1072no—
151C22×C4C22×D10712yes—
152C22×C4C22×D10712yes—
153C22×C4C22×D10711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—2144no—
156C4⋊C4—2144no—
157C4⋊C4—2142no—
158C22⋊C4—28no—
159C22⋊C4—2144no—
160C22⋊C4—2144no—
161C22⋊C4—2148no—
162C42—22no—
163C42—1074no—
164C42—2144no—
165C42—2144no—
166C42—2142no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23C23×D10711yescharacteristic
173Q8—4284no—
174D4—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—1072no—
180C2×C4—1072no—
181C2×C4—1074no—
182C2×C4—2142no—
183C2×C4—2142no—
184C2×C4—2144no—
185C2×C4—2142no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4D4×D10712yes—
192C2×C4#1313381714yes—
193C2×C4#1313381712yes—
194C2×C4#1313381812yes—
195C22—42no—
196C22—44no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22C2×D4×D10711yescharacteristic
201C22#41×D10712yes—
202C22C2×#1313381711yescharacteristic
203C22C2×#1313381712yes—
204C22C2×#1313381811yescharacteristic
205C4—24no—
206C4—42no—
207C4—4282no—
208C4—4282no—
209C4—2144no—
210C4—2142no—
211C4—2142no—
212C4#1681209514yes—
213C2—82no—
214C2C2×#1681209511yescharacteristic
215C2#2029315812yes—
216C2#2029315912yes—
217C2#2029317511yescharacteristic
218C2#2029317611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
prime radicalC22×D4×C107C22
hypercenter#390D107
hypocenterC107#2861
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812100C2semidirect
3. #20292533C2semidirect
4. #20292532C2semidirect
6. #20292520C2semidirect
7. #20292515C2semidirect
8. C42×Dic107C2semidirect
110. C107#2861semidirect

Maximal subgroups

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

subgroupindex
C2×#168121002
C2×#168120962
#202925332
#202925322
#202925302
#202925202
#202925152
C42×Dic1072
#390×C1072
#2861107