Finite Group Databasebeta

← #13263293

Subgroups — #13263293

Order 1792.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#11736071C211yesmaximal, characteristic
2#11726869C212yesmaximal, complemented
3C7×#1320C211yesmaximal, characteristic, complemented
4#11725487C211yesmaximal, characteristic, complemented
5#11725337C212yesmaximal, complemented
6#90482C411yescharacteristic
7C23×C7⋊C8C411yescharacteristic
8C2×#6470C411yescharacteristic
9C7×#407C2211yescharacteristic
10#89901C2211yescharacteristic
11#89894C2211yescharacteristic
12#89891C411yescharacteristic
13#89502—22no—
14(C22⋊D7)⋊C8—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11537—71nomaximal, Sylow
20C23×C4×C7C2×C411yescharacteristic
21C2×#1157C2×C411yescharacteristic
22C2×C7×M16C2×C411yescharacteristic
23C2×C7×M16C2311yescharacteristic
24#6516—21no—
25#6493—21no—
26#6492—21no—
27C4×C22⋊D7—22no—
28#6489C2×C411yescharacteristic
29#6487—21no—
30C22×C7⋊C8—21no—
31C22×C7⋊C8—21no—
32C22×C7⋊C8—42no—
33C22×C7⋊C8C2×C411yescharacteristic
34C22×C7⋊C8C2×C411yescharacteristic
35#6470—42no—
36C2×#1131—21no—
37C2×C8×D7—21no—
38C7×M32—22no—
39C7×C22⋊C8—22no—
40(C2×D7)⋊C8—22no—
41#6389D412yes—
42Dic7⋊C8—22no—
43C8×Dic7D412yes—
44#6386—22no—
45#2929—71no—
46#2166—72no—
47#1321—72no—
48#1320—71no—
49#1320D711yescharacteristic, complemented
50C24×C7M1611yescharacteristic
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7—41no—
54C22×C4×C7—41no—
55C22×C4×C7M1611yescharacteristic
56C22×C4×C7C22×C411yescharacteristic
57C2×C22⋊D7—22no—
58C2×C22⋊D7—21no—
59#1157—41no—
60#1157—41no—
61C2×D28M1611yescharacteristic
62C2×C4×D7—21no—
63C2×Dic14M1611yescharacteristic
64C7×M16—41no—
65C7×M16—41no—
66C2×C7×C8—21no—
67C2×C7×C8C2×D412yes—
68C22⋊Dic7—22no—
69C22⋊Dic7—21no—
70(C2×D7)⋊C4—22no—
71C4⋊Dic7—21no—
72Dic7⋊C4—22no—
73C4×Dic7C22⋊C412yes—
74C2×C7⋊C8—22no—
75C2×C7⋊C8—21no—
76C2×C7⋊C8—42no—
77C2×C7⋊C8—42no—
78C2×C7⋊C8—41no—
79C2×C7⋊C8C22⋊C411yescharacteristic
80C2×C7⋊C8C22⋊C411yescharacteristic
81#1131—42no—
82C8×D7—42no—
83C7×C16—42no—
84C7⋊C16—42no—
85C2×#131—71no—
86C23×C8—71no—
87#525—71no—
88#435—142no—
89C8×D4—142no—
90#424—71no—
91#408—71no—
92#407C2×D711yescharacteristic
93#405—71no—
94#364—142no—
95#349—72no—
96#349C2×D712yes—
97#347—142no—
98C23×C7—21no—
99C23×C7—41no—
100C23×C7—42no—
101C23×C7C2×M1611yescharacteristic
102C22×D7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—42no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—41no—
110C2×C4×C7C2×C22⋊C411yescharacteristic
111C2×C4×C7C2×M1611yescharacteristic
112C22⋊D7—42no—
113C22⋊D7—44no—
114C2×Dic7—22no—
115C2×Dic7—21no—
116D28—41no—
117C4×D7—41no—
118C4×D7—41no—
119Dic14—41no—
120C7×C8—22no—
121C7×C8—41no—
122C7⋊C8—21no—
123C7⋊C8—41no—
124C7⋊C8#10012yes—
125C2×#41—71no—
126C23×C4C4×D711yescharacteristic
127#131—282no—
128C2×M16—141no—
129C2×M16C4×D711yescharacteristic
130C2×M16C22×D711yescharacteristic
131C22×C8—71no—
132C22×C8—71no—
133C22×C8—141no—
134C22×C8—141no—
135C22×C8—141no—
136C22×C8—282no—
137#123—141no—
138C22⋊Q8—141no—
139#121—141no—
140#120—141no—
141C4×D4—142no—
142#117—71no—
143M32—22no—
144M32—142no—
145C4⋊C8—142no—
146C22⋊C8—22no—
147C22⋊C8—142no—
148#97—72no—
149C4×C8—72no—
150C22×C7—22no—
151C22×C7—21no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—41no—
157C22×C7#40711yescharacteristic
158C2×D7—42no—
159C2×D7—41no—
160C4×C7—21no—
161C4×C7—41no—
162C4×C7#41011yescharacteristic
163C4×C7C2×#10011yescharacteristic
164Dic7—42no—
165Dic7—41no—
166C24#113111yescharacteristic
167#41—281no—
168#41—281no—
169C2×Q8—71no—
170C2×D4—71no—
171C2×D4—142no—
172C2×D4—141no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—141no—
178C22×C4#113111yescharacteristic
179C22×C4C2×C4×D711yescharacteristic
180M16—41no—
181M16—41no—
182M16—282no—
183C2×C8—21no—
184C2×C8—71no—
185C2×C8—71no—
186C2×C8—142no—
187C2×C8—141no—
188C2×C8—282no—
189C2×C8—282no—
190C2×C8—282no—
191C2×C8—281no—
192C2×C8D4×D712yes—
193C4⋊C4—142no—
194C4⋊C4—141no—
195C22⋊C4—142no—
196C22⋊C4—142no—
197C22⋊C4—141no—
198C42—72no—
199C16—42no—
200C16—282no—
201C2×C7—21no—
202C2×C7—42no—
203C2×C7—41no—
204C2×C7#146611yescharacteristic
205D7—81no—
206C23—21no—
207C23—41no—
208C23—42no—
209C23—141no—
210C23C2×#113111yescharacteristic
211Q8—281no—
212D4—281no—
213D4—284no—
214D4—282no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—142no—
223C2×C4—141no—
224C2×C4—281no—
225C2×C4—281no—
226C2×C4D7×C22⋊C411yescharacteristic
227C2×C4D7×M1611yescharacteristic
228C8—22no—
229C8—41no—
230C8—72no—
231C8—141no—
232C8—281no—
233C7#1153711yescharacteristic, complemented, Sylow
234C22—22no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—282no—
242C22—281no—
243C22#8949811yescharacteristic
244C4—21no—
245C4—41no—
246C4—282no—
247C4—281no—
248C4#8951211yescharacteristic
249C4D7×#10011yescharacteristic
250C2—21no—
251C2—42no—
252C2—41no—
253C2—561no—
254C2#1172585711yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1320C2
socleC2×C7#1466
prime radical#90482C4
cosocleC2×C7×M16C23
hypercenter#1320D7
hypocenterC7#11537
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #11726869C2semidirect
3. C7×#1320C2semidirect
4. #11725487C2semidirect
5. #11725337C2semidirect
49. #1320D7semidirect
233. C7#11537semidirect

Maximal subgroups

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

subgroupindex
#117360712
#117268692
C7×#13202
#117254872
#117253372
#115377