Finite Group Databasebeta

← #13247370

Subgroups — #13247370

Order 1792.

422 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 107 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#11731602C211yesmaximal, characteristic
2#11728540C211yesmaximal, characteristic
3#11728282C211yesmaximal, characteristic
4C7×#1286C211yesmaximal, characteristic
5#11725552C211yesmaximal, characteristic
6#11725521C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C2×#6450—21no—
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C7×#382C2211yescharacteristic
13C2×#6398—21no—
14#89745—21no—
15C4×C4⋊Dic7C2211yescharacteristic
16#89697C2211yescharacteristic
17#89696C2211yescharacteristic
18#89422C412yes—
19#89421C412yes—
20#89419—22no—
21#89349—21no—
22#89264—22no—
23#8766—71nomaximal, Sylow
24C22×Dic14D411yescharacteristic
25C2×C7×M16—21no—
26C2×C7×C4⋊C4—21no—
27C2×C42×C7C2311yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7D411yescharacteristic
30C2×Dic7⋊C4—22no—
31C2×Dic7⋊C4—22no—
32C2×Dic7⋊C4D412yes—
33C2×C4×Dic7—21no—
34C2×C4×Dic7—22no—
35C2×C4×Dic7C2×C412yes—
36C2×#1136—21no—
37#6450—42no—
38#6450—42no—
39C4×Dic14—21no—
40C4×Dic14—21no—
41C4×Dic14—21no—
42C4×Dic14—21no—
43C7×C4⋊C8—21no—
44C7×C4⋊C8—21no—
45C7×C4⋊C8—21no—
46C7×C22⋊C8—21no—
47C4×C7×C8—21no—
48C7×#95—21no—
49C7×#95C2×C412yes—
50C7⋊#95—22no—
51C7⋊#95C2×C411yescharacteristic
52C7⋊#95C2×C411yescharacteristic
53C22⋊(C7⋊C8)—21no—
54#6398—42no—
55C4⋊(C7⋊C8)—21no—
56C4⋊(C7⋊C8)—21no—
57C4⋊(C7⋊C8)—21no—
58C4×C7⋊C8—21no—
59#2351—71no—
60#1801—71no—
61#1461—71no—
62#1303—71no—
63#1303—71no—
64#1286—71no—
65#1286D711yescharacteristic
66C22×C4×C7—22no—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—21no—
71C22×Dic7—21no—
72C22×Dic7C22⋊C412yes—
73C22×Dic7C22⋊C412yes—
74C2×Dic14—21no—
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×Dic14—21no—
78C2×Dic14D812yes—
79C2×Dic14SD1612yes—
80C7×M16—41no—
81C7×M16—41no—
82C2×C7×C8—41no—
83C2×C7×C8—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C42×C7#4111yescharacteristic
87C42×C7#4111yescharacteristic
88C42×C7#4111yescharacteristic
89C42×C7#4111yescharacteristic
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—42no—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—44no—
96Dic7⋊C4—44no—
97C4×Dic7—22no—
98C4×Dic7—22no—
99C4×Dic7—41no—
100C4×Dic7—41no—
101C4×Dic7—44no—
102#1136—41no—
103#1136—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106C2×C4×Q8—71no—
107C2×C4×Q8—141no—
108#431—71no—
109#431C2×D711yescharacteristic
110#423—71no—
111#423C2×D711yescharacteristic
112C2×#101—141no—
113#389—142no—
114#389—141no—
115#382—72no—
116#382C2×D711yescharacteristic
117C4×C4⋊C4—71no—
118C4×C4⋊C4—72no—
119#343—141no—
120#324—142no—
121C23×C7C2×C22⋊C411yescharacteristic
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7#11711yescharacteristic
128C2×C4×C7#11711yescharacteristic
129C2×C4×C7#12311yescharacteristic
130C2×C4×C7#12311yescharacteristic
131C2×C4×C7#12311yescharacteristic
132C2×C4×C7#12311yescharacteristic
133C2×C4×C7C2×D811yescharacteristic
134C2×C4×C7C2×SD1611yescharacteristic
135C2×C4×C7#13711yescharacteristic
136C2×C4×C7#13711yescharacteristic
137C2×Dic7—22no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7—22no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143C2×Dic7—42no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7#10214yes—
148Dic14—22no—
149Dic14—22no—
150Dic14—22no—
151Dic14—22no—
152C7×C8—81no—
153C7×C8—81no—
154C7⋊C8—81no—
155C7⋊C8—81no—
156C22×Q8—71no—
157C2×M16—21no—
158C2×M16—141no—
159C4×Q8—141no—
160C4×Q8—141no—
161C4×Q8—141no—
162C4×Q8—141no—
163C4×Q8—282no—
164C4×Q8—282no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—72no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—142no—
169C2×C4⋊C4—142no—
170C2×C4⋊C4—141no—
171C2×C42—72no—
172C2×C42—142no—
173C2×C42—141no—
174C2×C42C22×D711yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—141no—
179C4⋊C8—141no—
180C4⋊C8—141no—
181#101—282no—
182C22⋊C8—21no—
183C22⋊C8—141no—
184C4×C8—21no—
185C4×C8—141no—
186#95—21no—
187#95—71no—
188#95—71no—
189#95—142no—
190#95C4×D712yes—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7#38111yescharacteristic
194C22×C7C2×#10211yescharacteristic
195C22×C7#41911yescharacteristic
196C4×C7—21no—
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—82no—
201C4×C7#48011yescharacteristic
202C4×C7#48011yescharacteristic
203C4×C7#48311yescharacteristic
204C4×C7#48311yescharacteristic
205Dic7—24no—
206Dic7—42no—
207Dic7—42no—
208Dic7—81no—
209C2×Q8—72no—
210C2×Q8—72no—
211C2×Q8—141no—
212C2×Q8—141no—
213C2×Q8—141no—
214C2×Q8—141no—
215C22×C4—22no—
216C22×C4—72no—
217C22×C4—72no—
218C22×C4—141no—
219C22×C4—141no—
220C22×C4C2×C4×D711yescharacteristic
221C22×C4D4×D711yescharacteristic
222C22×C4D4×D711yescharacteristic
223M16—41no—
224M16—41no—
225M16—281no—
226M16—281no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—281no—
230C2×C8—281no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—142no—
234C4⋊C4—141no—
235C4⋊C4—142no—
236C4⋊C4—141no—
237C4⋊C4—284no—
238C4⋊C4—284no—
239C4⋊C4—282no—
240C42—142no—
241C42—142no—
242C42—284no—
243C42—281no—
244C42—281no—
245C42#115711yescharacteristic
246C42#115711yescharacteristic
247C42#115911yescharacteristic
248C42#116111yescharacteristic
249C2×C7—21no—
250C2×C7—21no—
251C2×C7#182311yescharacteristic
252C2×C7#182711yescharacteristic
253C2×C7#183011yescharacteristic
254C23D7×C22⋊C411yescharacteristic
255Q8—142no—
256Q8—142no—
257Q8—142no—
258Q8—142no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—74no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—281no—
270C2×C4—281no—
271C2×C4—282no—
272C2×C4—281no—
273C2×C4—281no—
274C2×C4—281no—
275C2×C4#643511yescharacteristic
276C2×C4#644611yescharacteristic
277C2×C4#644611yescharacteristic
278C2×C4#645511yescharacteristic
279C2×C4#645711yescharacteristic
280C2×C4#645711yescharacteristic
281C2×C4D7×D811yescharacteristic
282C2×C4D7×SD1611yescharacteristic
283C2×C4#647811yescharacteristic
284C2×C4#648111yescharacteristic
285C8—81no—
286C8—81no—
287C8—561no—
288C8—561no—
289C7#876611yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22#8943611yescharacteristic
293C22D7×#10211yescharacteristic
294C22#8957411yescharacteristic
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—82no—
300C4—144no—
301C4—282no—
302C4—282no—
303C4—561no—
304C4#8954611yescharacteristic
305C4#8957711yescharacteristic
306C4#8963411yescharacteristic
307C4#8965411yescharacteristic
308C2—21no—
309C2—21no—
310C2#1172646511yescharacteristic
311C2#1172662511yescharacteristic
312C2#1172678311yescharacteristic
subgroupG/H
centerC22#89436
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1286C2
socleC22×C7#381
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1286D7
hypocenterC7#8766
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
289. C7#8766semidirect

Maximal subgroups

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

subgroupindex
#117316022
#117285402
#117282822
C7×#12862
#117255522
#117255212
#117254372
#87667