Finite Group Databasebeta

← #28336995

Subgroups — #28336995

Order 13696.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293198C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic, complemented
3#20292644C211yesmaximal, characteristic
4C2×#16812027C211yesmaximal, characteristic, complemented
5#20292384C211yesmaximal, characteristic, complemented
6#20292383C211yesmaximal, characteristic, complemented
7#421×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10#16812107C412yescomplemented
11#16812096C412yescomplemented
12#16812090—22no—
13#16812088—22no—
14C4×Dic214—22no—
15#16812074—22no—
16#16812068C2211yescharacteristic
17#16812068C2211yescharacteristic
18#16812065—22no—
19#16812036—22no—
20#16812027—22no—
21C2×#41×C107C2211yescharacteristic
22C2×M16×C107C2211yescharacteristic
23#117×C107C2211yescharacteristic
24#104×C107—22no—
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214C2×C412yes—
30C2×Dic214D412yes—
31C22×Dic107D412yes—
32#13133802—21no—
33#13133802—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37Dic107⋊C4—24no—
38Dic107⋊C4—24no—
39C22⋊Dic107—21no—
40C22⋊Dic107—21no—
41C22⋊Dic107—24no—
42C4×Dic107—24no—
43C4×Dic107C2×C412yes—
44C4×Dic107D412yes—
45#41×C107—21no—
46#41×C107—21no—
47#41×C107—21no—
48#41×C107—21no—
49C2×Q8×C107C2×C411yescharacteristic
50C2×D4×C107—21no—
51C2×D4×C107C2×C411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54M16×C107—21no—
55M16×C107—21no—
56C2×C8×C107—21no—
57C4⋊C4×C107—21no—
58C22⋊C4×C107—21no—
59C42×C107D412yes—
60C23×C107—21no—
61C23×C107C2×D411yescharacteristic
62Q8×C107—21no—
63Q8×C107—21no—
64D4×C107—21no—
65D4×C107—21no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107C22×C411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107#4111yescharacteristic
77C2×C4×C107#4111yescharacteristic
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107C22⋊C414yes—
83Dic214—22no—
84Dic214—22no—
85Dic214—22no—
86Dic214—22no—
87C8×C107—41no—
88C107⋊C8—41no—
89C22×C107—41no—
90C22×C107—42no—
91C22×C107—41no—
92C22×C107C2×C22⋊C411yescharacteristic
93C22×C107#12011yescharacteristic
94C22×C107#12311yescharacteristic
95C4×C107—42no—
96C4×C107—41no—
97C4×C107C4×D411yescharacteristic
98C4×C107C4×D411yescharacteristic
99C4×C107#12111yescharacteristic
100C4×C107#12111yescharacteristic
101Dic107—24no—
102Dic107—42no—
103Dic107—42no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107—81no—
108C2×C107#38611yescharacteristic
109#1912—1071nomaximal, Sylow
110C107#191211yescharacteristic, complemented, Sylow
111#536—1071no—
112#519—1071no—
113#421—1071no—
114#421D10711yescharacteristic
115C2×#101—1071no—
116H(3,ℤ/4)—1071no—
117H(3,ℤ/4)—1071no—
118C2×#41C2×D10711yescharacteristic
119C22×Q8—1071no—
120C2×M16—1071no—
121C2×M16C2×D10711yescharacteristic
122C4⋊Q8—1072no—
123#124—1072no—
124#123—2142no—
125C22⋊Q8—2142no—
126C4×Q8—2142no—
127C4×Q8—2142no—
128#117—1071no—
129#117—1071no—
130#117—2142no—
131#117C2×D10711yescharacteristic
132#104—22no—
133#104—2142no—
134#101—2142no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—1072no—
140C2×Q8—1072no—
141C2×Q8—2141no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8C4×D10711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D10711yescharacteristic
148C22×C4—21no—
149C22×C4—1072no—
150C22×C4C22×D10711yescharacteristic
151M16—21no—
152M16—21no—
153M16—2141no—
154M16—2141no—
155C2×C8—21no—
156C2×C8—2141no—
157C4⋊C4—21no—
158C4⋊C4—2144no—
159C4⋊C4—2144no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C22⋊C4—21no—
163C22⋊C4—2144no—
164C22⋊C4—2141no—
165C22⋊C4—2141no—
166C42—1072no—
167C42—1072no—
168C42—2144no—
169C42C22⋊D10712yes—
170C23—21no—
171C23D4×D10711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—2142no—
175Q8—2142no—
176Q8—2142no—
177Q8—2142no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1074no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4C2×C22⋊D10711yescharacteristic
195C2×C4#1313381711yescharacteristic
196C2×C4#1313381811yescharacteristic
197C8—41no—
198C8—4281no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D10711yescharacteristic
203C22#1681207711yescharacteristic
204C22#1681209111yescharacteristic
205C4—42no—
206C4—41no—
207C4—4282no—
208C4—4282no—
209C4—2144no—
210C4(C22⋊D107)⋊C411yescharacteristic
211C4C4×C22⋊D10711yescharacteristic
212C4#1681208011yescharacteristic
213C4#1681208211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2029250311yescharacteristic
subgroupG/H
centerC2#20292503
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#421×C107C2
socleC2×C107#386
prime radicalC2×#41×C107C22
cosocleC22×C4×C107C23
hypercenter#421D107
hypocenterC107#1912
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20293104C2semidirect
4. C2×#16812027C2semidirect
5. #20292384C2semidirect
6. #20292383C2semidirect
10. #16812107C4semidirect
11. #16812096C4semidirect
110. C107#1912semidirect

Maximal subgroups

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

subgroupindex
#202931982
#202931042
#202926442
C2×#168120272
#202923842
#202923832
#421×C1072
#1912107