Finite Group Databasebeta

← C2×#20292809 #28345802

Subgroups — C2×#20292809 #28345802

Order 13696.

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812157C211yesmaximal, characteristic
2C2×#16812140C212yesmaximal, complemented
3#20292809C218yesmaximal, direct product factor
4C2×#16812037C212yesmaximal, complemented
5C2×#16812028C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
7#16812157—24no—
8#16812157—22no—
9#16812157C2214yes—
10C2×#13133819—22no—
11C2×#13133818—22no—
12C2×#13133818C2212yes—
13C2×Q8×D107—22no—
14#16812140—24no—
15#16812140C2218yescomplemented
16C2×#13133808—22no—
17C2×#13133805—22no—
18C2×#13133802C2212yes—
19#16812037C2218yescomplemented
20#16812028C2214yescomplemented
21C22×Q8×C107C2211yescharacteristic
22C22×Q8×C107C2211yescharacteristic
23C4⋊Q8×C107—22no—
24C4⋊Q8×C107C2214yescomplemented
25C2×C4⋊C4×C107—22no—
26C2×C42×C107C2211yescharacteristic
27#13133819—28no—
28#13133819—28no—
29#13133818—28no—
30#13133818—28no—
31#13133818—22no—
32#13133818—22no—
33#13133818D414yes—
34#13133818C2314yes—
35C2×Dic214—22no—
36C2×Dic214D412yes—
37Q8×D107—28no—
38Q8×D107—28no—
39C2×D428—22no—
40C2×D428D412yes—
41C2×C22⋊D107—22no—
42C2×C4×D107—24no—
43C2×C4×D107—22no—
44#13133808—28no—
45#13133805—28no—
46#13133802—22no—
47#13133802C2314yes—
48C2×C107⋊C8—22no—
49C2×Q8×C107—22no—
50C2×Q8×C107—22no—
51C2×Q8×C107—21no—
52C2×Q8×C107—22no—
53C2×Q8×C107—22no—
54C2×Q8×C107D412yes—
55C2×Q8×C107C2312yes—
56C2×Q8×C107C2312yes—
57C22×C4×C107—21no—
58C22×C4×C107—21no—
59C22×C4×C107D412yes—
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—28no—
62C42×C107—21no—
63C42×C107C2312yes—
64C23×C107C2×D411yescharacteristic
65C22×D107—22no—
66Q8×C107—24no—
67Q8×C107—24no—
68Q8×C107—24no—
69Q8×C107—42no—
70C2×C4×C107—24no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D414yes—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2411yescharacteristic
82C22⋊D107—24no—
83C22⋊D107—24no—
84C2×Dic107—22no—
85D428—28no—
86D428—22no—
87D428C2×D414yes—
88C4×D107—216no—
89C4×D107—24no—
90C4×D107—24no—
91Dic214—28no—
92Dic214—22no—
93Dic214C2×D414yes—
94C107⋊C8—24no—
95C22×C107—22no—
96C22×C107#12011yescharacteristic
97C22×C107#12011yescharacteristic
98C22×C107C22×D411yescharacteristic
99C2×D107—24no—
100C2×D107—44no—
101C4×C107—24no—
102C4×C107—24no—
103C4×C107—42no—
104C4×C107#12012yes—
105C4×C107C22×D412yes—
106Dic107—24no—
107C2×C107—21no—
108C2×C107—21no—
109C2×C107#45612yescomplemented
110C2×C107C2×#12011yescharacteristic
111D107—44no—
112C2×#456—1071nomaximal, Sylow
113C107C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4—1071no—
115C2×#137—1072no—
116C2×C4⋊Q8D10711yescharacteristic, complemented
117#456—1078no—
118C2×#104—1072no—
119C2×#101—1071no—
1202−1+4—1074no—
1212−1+4—2144no—
1222−1+4—2142no—
123C2×#41—1072no—
124C2×#41—2142no—
125C2×#41—2142no—
126C22×Q8—2142no—
127C22×Q8C2×D10711yescharacteristic
128C22×Q8C2×D10711yescharacteristic
129#137—1078no—
130#137—2144no—
131C2×Q16—2142no—
132C2×SD16—2142no—
133C2×M16—1072no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D10714yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D10711yescharacteristic
138#104—1078no—
139#101—1074no—
140#41—1074no—
141#41—1074no—
142#41—2148no—
143#41—2148no—
144#41—2148no—
145#41—2142no—
146#41—2142no—
147#41—2148no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1072no—
154C2×Q8—2142no—
155C2×Q8—2148no—
156C2×Q8—2148no—
157C2×Q8C22⋊D10712yes—
158C2×Q8C22×D10712yes—
159C2×Q8C22×D10712yes—
160C2×D4—1072no—
161C2×D4—2142no—
162C2×D4—2142no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—2144no—
166C22×C4—2142no—
167C22×C4C22⋊D10712yes—
168C22×C4C22×D10711yescharacteristic
169Q16—2148no—
170SD16—2148no—
171M16—1074no—
172M16—2142no—
173C2×C8—2142no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D10712yes—
177C23—2142no—
178C23C2×C22⋊D10711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—1074no—
184Q8—2148no—
185Q8—2142no—
186D4—1074no—
187D4—2148no—
188D4—2144no—
189D4—2144no—
190D4—2142no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—21416no—
199C2×C4—2142no—
200C2×C4—2144no—
201C2×C4—2144no—
202C2×C4D4×D10712yes—
203C2×C4D4×D10712yes—
204C2×C4C2×C22⋊D10714yes—
205C2×C4C2×C22⋊D10711yescharacteristic
206C2×C4C23×D10711yescharacteristic
207C8—2144no—
208C22—22no—
209C22—4284no—
210C22—2144no—
211C22#1681207711yescharacteristic
212C22#1681207711yescharacteristic
213C22C22×C22⋊D10711yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—2144no—
218C4#1681207712yes—
219C4C2×D4×D10712yes—
220C2—21no—
221C2—21no—
222C2—4284no—
223C2#2029280912yesdirect product factor
224C2C2×#1681207711yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4⋊Q8×C107C2
socleC22×C107#120
prime radicalC2×#16812157C2
hypercenterC2×C4⋊Q8D107
hypocenterC107C2×#456
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20292809C2direct (N × H)
4. C2×#16812037C2semidirect
5. C2×#16812028C2semidirect
6. C2×C4⋊Q8×C107C2semidirect
15. #16812140C22semidirect
19. #16812037C22semidirect
20. #16812028C22semidirect
24. C4⋊Q8×C107C22semidirect
109. C2×C107#456semidirect
113. C107C2×#456semidirect
116. C2×C4⋊Q8D107semidirect
135. C4⋊Q8C2×D107semidirect
223. C2#20292809direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121572
C2×#168121402
#202928092
C2×#168120372
C2×#168120282
C2×C4⋊Q8×C1072
C2×#456107