Finite Group Databasebeta

← #13263124

Subgroups — #13263124

Order 1792.

356 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 87 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#11736102C211yesmaximal, characteristic
2#11726847C212yesmaximal, complemented
3C7×#1315C211yesmaximal, characteristic, complemented
4#11725495C211yesmaximal, characteristic, complemented
5#11725329C212yesmaximal, complemented
6#90523C411yescharacteristic
7C2×#6511C411yescharacteristic
8C2×#6470C411yescharacteristic
9C7×#408C2211yescharacteristic
10#89890C411yescharacteristic
11#89882C2211yescharacteristic
12C2×C8×Dic7C2211yescharacteristic
13#89501—22no—
14(C22⋊D7)⋊C8—22no—
15C7×C22⋊C16C2212yes—
16C22⋊(C7⋊C16)C2212yes—
17#89297—22no—
18Dic7⋊C16—22no—
19#11491—71nomaximal, Sylow
20C2×C7×#41C2×C411yescharacteristic
21C2×#1157C2×C411yescharacteristic
22C2×C7×M16C2×C411yescharacteristic
23C22×C7×C8C2311yescharacteristic
24#6511—42no—
25#6510C811yescharacteristic
26#6507C811yescharacteristic
27#6503C811yescharacteristic
28#6502—22no—
29#6499C811yescharacteristic
30C4×C22⋊D7—22no—
31C2×C4×Dic7C2×C411yescharacteristic
32C2×#1136—21no—
33C2×#1136C2×C411yescharacteristic
34C22×C7⋊C8—21no—
35C22×C7⋊C8C2×C411yescharacteristic
36#6470—42no—
37C2×#1131—21no—
38C2×C8×D7—21no—
39C2×C7×C16—22no—
40C7×C22⋊C8—22no—
41(C2×D7)⋊C8—22no—
42Dic7⋊C8—22no—
43C8×Dic7D412yes—
44C8×Dic7D412yes—
45C2×C7⋊C16—22no—
46#2931—71no—
47#2163—72no—
48#1315—71no—
49#1315—72no—
50#1315D711yescharacteristic, complemented
51C7×#41—41no—
52C7×#41—41no—
53C2×C7×Q8C2×C811yescharacteristic
54C2×C7×D4—21no—
55C2×C7×D4C2×C811yescharacteristic
56C22×C4×C7—21no—
57C22×C4×C7C22×C411yescharacteristic
58C2×C22⋊D7—21no—
59C2×C22⋊D7—22no—
60C22×Dic7M1612yes—
61#1157—41no—
62#1157—41no—
63C2×D28C2×C811yescharacteristic
64C2×C4×D7—21no—
65C2×Dic14C2×C811yescharacteristic
66C7×M16—41no—
67C2×C7×C8—21no—
68C2×C7×C8—41no—
69C2×C7×C8C2×D412yes—
70C22⋊Dic7—22no—
71(C2×D7)⋊C4—22no—
72Dic7⋊C4—22no—
73C4×Dic7C22⋊C412yes—
74C4×Dic7C2×C811yescharacteristic
75C4×Dic7C2×C811yescharacteristic
76#1136—42no—
77#1136—41no—
78C2×C7⋊C8—22no—
79C2×C7⋊C8—21no—
80C2×C7⋊C8—42no—
81C2×C7⋊C8C22⋊C411yescharacteristic
82C2×C7⋊C8C22⋊C411yescharacteristic
83#1131—42no—
84C8×D7—42no—
85C7×C16—42no—
86C7⋊C16—42no—
87C2×#131—71no—
88C2×#131—71no—
89#532—71no—
90#436—142no—
91C8×D4—142no—
92#423—71no—
93#408—71no—
94#408C2×D711yescharacteristic
95C2×C4×C8—71no—
96C4⋊C16—142no—
97C22⋊C16—72no—
98C22⋊C16C2×D712yes—
99#346—142no—
100C23×C7—21no—
101C23×C7C2×M1611yescharacteristic
102C22×D7—21no—
103C7×Q8—41no—
104C7×D4—41no—
105C7×D4—42no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—41no—
110C2×C4×C7—41no—
111C2×C4×C7C2×C22⋊C411yescharacteristic
112C2×C4×C7C22×C811yescharacteristic
113C22⋊D7—44no—
114C22⋊D7—42no—
115C2×Dic7—21no—
116C2×Dic7—42no—
117C2×Dic7C22⋊C812yes—
118C2×Dic7C22⋊C812yes—
119D28—41no—
120C4×D7—41no—
121C4×D7—41no—
122Dic14—41no—
123C7×C8—22no—
124C7×C8—41no—
125C7⋊C8—21no—
126C7⋊C8—41no—
127C7⋊C8#10012yes—
128C2×#41—71no—
129C2×#41C4×D711yescharacteristic
130#131—282no—
131#131—282no—
132C2×M16—71no—
133C2×M16—141no—
134C2×M16—141no—
135C2×M16C4×D711yescharacteristic
136C22×C8—71no—
137C22×C8—141no—
138C22×C8—141no—
139C22×C8C22×D711yescharacteristic
140C4⋊Q8—71no—
141#127—71no—
142#124—71no—
143#124—71no—
144#121—142no—
145C4×D4—142no—
146C2×C42—71no—
147C2×C16—22no—
148C2×C16—142no—
149C4⋊C8—142no—
150C22⋊C8—22no—
151C22⋊C8—142no—
152C4×C8—72no—
153C4×C8—72no—
154C22×C7—22no—
155C22×C7—21no—
156C22×C7—42no—
157C22×C7—41no—
158C22×C7C2×C22⋊C811yescharacteristic
159C2×D7—42no—
160C2×D7—41no—
161C4×C7—21no—
162C4×C7—41no—
163C4×C7#41011yescharacteristic
164C4×C7C2×#10011yescharacteristic
165Dic7—22no—
166Dic7—22no—
167Dic7—41no—
168#41—41no—
169#41—41no—
170#41—281no—
171#41—281no—
172C2×Q8—71no—
173C2×Q8C8×D711yescharacteristic
174C2×D4—21no—
175C2×D4—71no—
176C2×D4—142no—
177C2×D4—141no—
178C2×D4C8×D711yescharacteristic
179C22×C4—21no—
180C22×C4—72no—
181C22×C4—141no—
182C22×C4C2×C4×D711yescharacteristic
183M16—41no—
184M16—282no—
185M16—282no—
186M16—281no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—71no—
190C2×C8—71no—
191C2×C8—142no—
192C2×C8—141no—
193C2×C8—282no—
194C2×C8—282no—
195C2×C8D4×D712yes—
196C4⋊C4—142no—
197C22⋊C4—142no—
198C22⋊C4—142no—
199C42—72no—
200C42—71no—
201C42—71no—
202C16—42no—
203C16—282no—
204C2×C7—41no—
205C2×C7—81no—
206C2×C7#146011yescharacteristic
207C2×C7#211712yes—
208D7—81no—
209C23—21no—
210C23—141no—
211C23D7×M1611yescharacteristic
212Q8—41no—
213Q8—281no—
214D4—41no—
215D4—42no—
216D4—281no—
217D4—284no—
218D4—282no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—72no—
225C2×C4—72no—
226C2×C4—141no—
227C2×C4—282no—
228C2×C4—281no—
229C2×C4—281no—
230C2×C4D7×C22⋊C411yescharacteristic
231C2×C4C2×C8×D711yescharacteristic
232C8—22no—
233C8—41no—
234C8—72no—
235C8—141no—
236C8—281no—
237C7#1149111yescharacteristic, complemented, Sylow
238C22—21no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—281no—
243C22—282no—
244C22D7×C22⋊C811yescharacteristic
245C4—21no—
246C4—41no—
247C4—142no—
248C4—142no—
249C4—281no—
250C4#8951211yescharacteristic
251C4D7×#10011yescharacteristic
252C2—41no—
253C2—81no—
254C2—561no—
255C2#1172585511yescharacteristic
256C2#1172687912yes—
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC22×C8C22×D7
FittingC7×#1315C2
socleC22×C7C2×C22⋊C8
prime radical#90523C4
cosocleC22×C7×C8C23
hypercenter#1315D7
hypocenterC7#11491
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #11726847C2semidirect
3. C7×#1315C2semidirect
4. #11725495C2semidirect
5. #11725329C2semidirect
50. #1315D7semidirect
237. C7#11491semidirect

Maximal subgroups

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

subgroupindex
#117361022
#117268472
C7×#13152
#117254952
#117253292
#114917