Finite Group Databasebeta

← #13254755

Subgroups — #13254755

Order 1792.

468 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3249C211yesmaximal, characteristic
2#11740814C212yesmaximal
3#11729576C212yesmaximal
4#11728227C211yesmaximal, characteristic
5#11725514C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C4×Dic14C2211yescharacteristic
8C7×#477—22no—
9C7×#467—22no—
10C7×#462—22no—
11C2×C7×C4⋊C8C2211yescharacteristic
12#89866—22no—
13#89862—22no—
14#89822—22no—
15#89815—22no—
16#89802—24no—
17#89698C2212yes—
18C2×#6377C2211yescharacteristic
19#89345—22no—
20#89338—24no—
21#10031—71nomaximal, Sylow
22C22×Dic14D411yescharacteristic
23C2×C7×Q16—22no—
24C22×C7×C8D412yes—
25C7×C4⋊Q8D412yes—
26C7×C4⋊Q8D412yes—
27C7×#125—22no—
28C7×C22⋊Q8—22no—
29C7×C22⋊Q8—22no—
30C4×C7×Q8—22no—
31C4×C7×Q8—22no—
32C7×#117—22no—
33C2×C42×C7C2311yescharacteristic
34C2×#1144—24no—
35C2×C4⋊Dic7D411yescharacteristic
36C2×Dic7⋊C4—21no—
37C2×C4×Dic7—21no—
38C22×C7⋊C8—21no—
39C4×Dic14—22no—
40C4×Dic14—22no—
41C7×#107—22no—
42C7×C4⋊C8—22no—
43C7×#103—24no—
44#6409—24no—
45C22⋊(C7⋊C8)—22no—
46#6384—24no—
47#6382—24no—
48C4⋊(C7⋊C8)—22no—
49#6377—22no—
50#6377D414yes—
51#3249D711yescharacteristic
52#3113—72no—
53#1985—71no—
54#1531—72no—
55#1290—71no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7#4111yescharacteristic
61C22×Dic7—21no—
62C22×Dic7—21no—
63C2×Dic14—22no—
64C2×Dic14—42no—
65C7×Q16—82no—
66C2×C7×C8—22no—
67C2×C7×C8—42no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C22⋊C4—42no—
75C42×C7—42no—
76C42×C7C2×D412yes—
77C42×C7C2×D412yes—
78#1144—48no—
79C4⋊Dic7—22no—
80Dic7⋊C4—42no—
81C4×Dic7—41no—
82C4×Dic7—41no—
83C2×C7⋊C8—22no—
84C2×C7⋊C8—24no—
85C2×C7⋊C8—41no—
86C2×C7⋊C8—41no—
87#533C2×D712yes—
88C2×C4×Q8—71no—
89#483—142no—
90#477—22no—
91#475—142no—
92#467—22no—
93#462—22no—
94#462—142no—
95#451—142no—
96#441—144no—
97#432—72no—
98C2×C4⋊C8C2×D711yescharacteristic
99C2×#97—71no—
100#336—142no—
101#330—144no—
102C23×C7#12111yescharacteristic
103C7×Q8—44no—
104C7×Q8—42no—
105C7×Q8—42no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—41no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7#12012yes—
113C2×C4×C7#12011yescharacteristic
114C2×C4×C7#12112yes—
115C2×C4×C7#12711yescharacteristic
116C2×Dic7—22no—
117C2×Dic7—41no—
118C2×Dic7—41no—
119C2×Dic7—41no—
120C2×Dic7—42no—
121Dic14—24no—
122Dic14—42no—
123C7×C8—42no—
124C7⋊C8—42no—
125C7⋊C8—44no—
126C22×Q8—71no—
127C2×Q16—22no—
128C2×Q16—144no—
129C22×C8—141no—
130C22×C8C22⋊D712yes—
131C4⋊Q8C22⋊D712yes—
132C4⋊Q8C22⋊D712yes—
133#125—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—142no—
139C4×Q8—142no—
140#117—22no—
141C2×C4⋊C4—71no—
142C2×C4⋊C4—141no—
143C2×C42—141no—
144C2×C42C22×D711yescharacteristic
145#107—22no—
146#106—144no—
147C4⋊C8—22no—
148C4⋊C8—142no—
149#103—24no—
150#103—144no—
151#103—144no—
152C22⋊C8—142no—
153#97—74no—
154#97—142no—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7#39211yescharacteristic
158C22×C7#47312yes—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—82no—
163C4×C7—82no—
164C4×C7#45511yescharacteristic
165C4×C7#45611yescharacteristic
166Dic7—42no—
167Dic7—81no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—142no—
171C2×Q8—282no—
172C22×C4—141no—
173C22×C4—141no—
174C22×C4D4×D711yescharacteristic
175C22×C4#115911yescharacteristic
176C22×C4C2×C22⋊D711yescharacteristic
177Q16—82no—
178Q16—288no—
179C2×C8—22no—
180C2×C8—42no—
181C2×C8—142no—
182C2×C8—144no—
183C2×C8—281no—
184C2×C8—281no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—142no—
192C4⋊C4—282no—
193C22⋊C4—42no—
194C42—42no—
195C42—281no—
196C42—281no—
197C42D4×D712yes—
198C42C2×C22⋊D712yes—
199C2×C7—22no—
200C2×C7#200711yescharacteristic
201C2×C7#201011yescharacteristic
202C2×C7#202011yescharacteristic
203C23#650111yescharacteristic
204Q8—44no—
205Q8—42no—
206Q8—42no—
207Q8—144no—
208Q8—282no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—142no—
216C2×C4—281no—
217C2×C4—281no—
218C2×C4—282no—
219C2×C4—281no—
220C2×C4#650012yes—
221C2×C4#650212yes—
222C2×C4#650311yescharacteristic
223C2×C4#651611yescharacteristic
224C8—42no—
225C8—282no—
226C8—284no—
227C7#1003111yescharacteristic, complemented, Sylow
228C22—21no—
229C22—21no—
230C22#8996412yes—
231C22#8999511yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—82no—
236C4—82no—
237C4—282no—
238C4—561no—
239C4#9001411yescharacteristic
240C4#9001811yescharacteristic
241C2—22no—
242C2#1172963511yescharacteristic
243C2#1172982811yescharacteristic
244C2#1172987211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3249C2
socleC22×C7#392
prime radicalC23×C7#121
hypercenter#3249D7
hypocenterC7#10031
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
227. C7#10031semidirect

Maximal subgroups

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

subgroupindex
C7×#32492
#117408142
#117295762
#117282272
#117255142
#100317