Finite Group Databasebeta

← #13258295

Subgroups — #13258295

Order 1792.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C7×#2844C211yesmaximal, characteristic
2#11727304C211yesmaximal, characteristic
3#11727302C212yesmaximal
4#11725436C211yesmaximal, characteristic
5#11725436C212yesmaximal
6C7×#400—21no—
7C7×#400—22no—
8C7×#395—21no—
9C7×#395—22no—
10C7×#382C2212yes—
11C7×#382C2211yescharacteristic
12C2×#6382—21no—
13C2×#6381—21no—
14#89707C2211yescharacteristic
15C2×#6377C2212yes—
16C2×#6377C2211yescharacteristic
17#89324—22no—
18#89324—22no—
19#10584—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×C7×C4⋊C4—21no—
22C2×C7×C4⋊C4—22no—
23C2×C7×C4⋊C4D411yescharacteristic
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C7×C4⋊C4D412yes—
26C2×C7×C4⋊C4D412yes—
27C2×C42×C7C2311yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7—22no—
30C22×C7⋊C8—21no—
31C22×C7⋊C8—22no—
32C7×#95—22no—
33C7×#95—21no—
34C7×#95—21no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38#6382—42no—
39#6381—42no—
40#6377—21no—
41#6377—22no—
42#6377Q812yescomplemented
43#6377Q814yescomplemented
44#2844D711yescharacteristic
45#1847—71no—
46#1836—72no—
47#1295—72no—
48#1295—71no—
49C2×C7×Q8—81no—
50C2×C7×Q8—82no—
51C22×C4×C7—21no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7C2×D411yescharacteristic
56C22×C4×C7C2×D412yes—
57C22×Dic7—41no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—42no—
60C7×C4⋊C4—41no—
61C7×C4⋊C4—41no—
62C7×C4⋊C4—81no—
63C7×C4⋊C4—82no—
64C42×C7C2×Q811yescharacteristic
65C42×C7C2×Q812yes—
66C42×C7#4111yescharacteristic
67C4⋊Dic7—44no—
68C4⋊Dic7—41no—
69C4⋊Dic7—41no—
70C2×C7⋊C8—22no—
71C2×C7⋊C8—22no—
72C2×C7⋊C8—22no—
73C2×C7⋊C8—24no—
74C2×C7⋊C8—42no—
75C2×C7⋊C8—42no—
76C2×#107—141no—
77C2×#106—141no—
78C2×#97—72no—
79C2×#97—71no—
80#400—22no—
81#400—21no—
82#395—22no—
83#395—21no—
84#384—71no—
85#382C2×D712yes—
86#382C2×D711yescharacteristic
87#340—142no—
88#340—142no—
89C23×C7#12011yescharacteristic
90C7×Q8—84no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—41no—
96C2×C4×C7—41no—
97C2×C4×C7—42no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7C22⋊Q811yescharacteristic
106C2×C4×C7C22⋊Q812yes—
107C2×C4×C7C4⋊Q811yescharacteristic
108C2×C4×C7C4⋊Q812yes—
109C2×Dic7—81no—
110C2×Dic7—82no—
111C7⋊C8—24no—
112C7⋊C8—44no—
113C22×Q8—21no—
114C22×C8—141no—
115C22×C8—142no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—142no—
119C2×C4⋊C4—141no—
120C2×C4⋊C4C22⋊D712yes—
121C2×C4⋊C4C22⋊D712yes—
122C2×C4⋊C4C22⋊D711yescharacteristic
123C2×C4⋊C4C22⋊D711yescharacteristic
124C2×C42C22×D711yescharacteristic
125#107—282no—
126#106—282no—
127#97—72no—
128#97—74no—
129#97—141no—
130#97—142no—
131#95—22no—
132#95—21no—
133#95—21no—
134#95—22no—
135#95—22no—
136#95—22no—
137C22×C7#39511yescharacteristic
138C22×C7#45311yescharacteristic
139C22×C7#45412yes—
140C22×C7#45512yes—
141C22×C7#45611yescharacteristic
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—22no—
145C4×C7—22no—
146C4×C7—81no—
147C4×C7—81no—
148C4×C7—82no—
149C4×C7—82no—
150Dic7—161no—
151C2×Q8—81no—
152C2×Q8—82no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—281no—
158C22×C4C2×C22⋊D711yescharacteristic
159C22×C4C2×C22⋊D712yes—
160C2×C8—142no—
161C2×C8—142no—
162C2×C8—142no—
163C2×C8—144no—
164C2×C8—282no—
165C2×C8—282no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—81no—
171C4⋊C4—82no—
172C4⋊C4—281no—
173C4⋊C4—281no—
174C4⋊C4—284no—
175C42Q8×D712yes—
176C42Q8×D711yescharacteristic
177C42#116111yescharacteristic
178C2×C7#165711yescharacteristic
179C2×C7#165911yescharacteristic
180C2×C7#166412yes—
181C2×C7#206011yescharacteristic
182C2×C7#206112yes—
183C23#651611yescharacteristic
184Q8—84no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—561no—
200C2×C4—562no—
201C2×C4#650711yescharacteristic
202C2×C4#650712yes—
203C2×C4#650911yescharacteristic
204C2×C4#650912yes—
205C8—144no—
206C8—284no—
207C7#1058411yescharacteristic, complemented, Sylow
208C22#9000111yescharacteristic
209C22#9001311yescharacteristic
210C22#9001412yes—
211C22#9001712yes—
212C22#9001811yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—82no—
219C4—81no—
220C4—82no—
221C4—1121no—
222C2#1172965411yescharacteristic
223C2#1172965512yes—
224C2#1172970311yescharacteristic
225C2#1172970911yescharacteristic
226C2#1172971112yes—
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2844C2
socle, prime radicalC23×C7#120
hypercenter#2844D7
hypocenterC7#10584
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
42. #6377Q8semidirect
43. #6377Q8semidirect
207. C7#10584semidirect

Maximal subgroups

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

subgroupindex
C7×#28442
#117273042
#117273022
#117254362
#105847