Finite Group Databasebeta

← #13251663

Subgroups — #13251663

Order 1792.

602 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 111 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#11739861C211yesmaximal, characteristic
2#11739749C211yesmaximal, characteristic, complemented
3C7×#1800C211yesmaximal, characteristic, complemented
4#11728250C212yesmaximal, complemented
5#11725519C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6501—21no—
8C2×C4×D28C2211yescharacteristic
9C7×#423C2212yes—
10C2×C7×#100—21no—
11#89985—22no—
12#89979—22no—
13#89784—24no—
14#89759—21no—
15#89732C412yes—
16#89725C412yes—
17#89708C2211yescharacteristic
18#89697C2212yes—
19C7×#343—21no—
20#89278—24no—
21#9489—71nomaximal, Sylow
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D412yes—
24C22×C22⋊D7—22no—
25C22×D28C2×C411yescharacteristic
26C22×C4×D7—21no—
27C2×C7×M16—22no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C22⋊C4C2×C412yes—
32C2×C42×C7C2311yescharacteristic
33C2×C22⋊Dic7—22no—
34#6501—44no—
35C2×(C2×D7)⋊C4—21no—
36C2×(C2×D7)⋊C4—22no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×#1136—24no—
40C4×D28—22no—
41C4×D28—22no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8—22no—
44C7×#100—42no—
45C7⋊#95—22no—
46C7⋊#95—21no—
47C7⋊#95C2×C412yes—
48C22⋊(C7⋊C8)—24no—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51C4×C7⋊C8D414yescomplemented
52#2975—71no—
53#2535—71no—
54#1800D711yescharacteristic, complemented
55#1486—72no—
56#1286—72no—
57C24×C7C22⋊C412yes—
58C23×D7—21no—
59C2×C7×D4—22no—
60C2×C7×D4—21no—
61C2×C7×D4—22no—
62C2×C7×D4—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—82no—
66C22×C4×C7C22⋊C412yes—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C2×C22⋊D7—44no—
71C2×C22⋊D7—44no—
72C2×C22⋊D7—82no—
73C22×Dic7—22no—
74C22×Dic7—21no—
75C2×D28—22no—
76C2×D28—42no—
77C2×C4×D7—42no—
78C2×C4×D7—42no—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C7×M16—42no—
82C7×M16—42no—
83C2×C7×C8—42no—
84C7×C4⋊C4—22no—
85C7×C22⋊C4—24no—
86C42×C7C2×D412yes—
87C42×C7#4112yes—
88C22⋊Dic7—44no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—82no—
92C4⋊Dic7—22no—
93C4⋊Dic7—42no—
94#1136—84no—
95C2×C7⋊C8—24no—
96C2×#121—141no—
97C2×C4×D4—71no—
98C2×C4×D4C2×D711yescharacteristic
99#436—144no—
100#431—72no—
101#423C2×D712yes—
102C2×#100—21no—
103#407—142no—
104#397—72no—
105#392—72no—
106#386—142no—
107#382—71no—
108#380—141no—
109#343—21no—
110D4⋊C8—144no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7C2×C22⋊C411yescharacteristic
117C22×D7—22no—
118C22×D7—41no—
119C22×D7—41no—
120C22×D7—41no—
121C22×D7—41no—
122C22×D7—41no—
123C22×D7—41no—
124C7×D4—42no—
125C7×D4—42no—
126C2×C4×C7—22no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—44no—
130C2×C4×C7—42no—
131C2×C4×C7—82no—
132C2×C4×C7C4×D412yes—
133C2×C4×C7#12011yescharacteristic
134C2×C4×C7#12112yes—
135C2×C4×C7#12311yescharacteristic
136C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C7AGL(1,ℤ/8)12yes—
138C2×C4×C7#13711yescharacteristic
139C22⋊D7—88no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142C2×Dic7—41no—
143C2×Dic7—41no—
144C2×Dic7—42no—
145C2×Dic7—41no—
146D28—24no—
147D28—42no—
148C4×D7—82no—
149C4×D7—82no—
150C7×C8—82no—
151C7⋊C8—44no—
152C22×D4—71no—
153C22×D4—142no—
154C22×D4C22⋊D711yescharacteristic
155C23×C4—141no—
156C23×C4D2812yes—
157C2×M16—22no—
158C2×M16—144no—
159#121—284no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—142no—
163C4×D4—142no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—141no—
166C2×C4⋊C4C22⋊D711yescharacteristic
167C2×C22⋊C4—142no—
168C2×C22⋊C4—141no—
169C2×C22⋊C4—142no—
170C2×C22⋊C4C4×D712yes—
171C2×C42C22×D711yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—142no—
175#100—42no—
176C22⋊C8—144no—
177C22⋊C8—142no—
178C4×C8—74no—
179#95—72no—
180#95—142no—
181#95—141no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—44no—
185C22×C7—42no—
186C22×C7—42no—
187C22×C7—42no—
188C22×C7—82no—
189C22×C7#38611yescharacteristic
190C22×C7#41711yescharacteristic
191C22×C7#41811yescharacteristic
192C2×D7—42no—
193C2×D7—42no—
194C2×D7—42no—
195C2×D7—81no—
196C2×D7—81no—
197C2×D7—81no—
198C2×D7—81no—
199C4×C7—22no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—42no—
203C4×C7#46812yes—
204C4×C7#46912yes—
205Dic7—82no—
206Dic7—81no—
207C24—141no—
208C24(C2×D7)⋊C412yes—
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—142no—
214C2×D4—282no—
215C2×D4—284no—
216C2×D4—284no—
217C2×D4—562no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—82no—
221C22×C4—142no—
222C22×C4—141no—
223C22×C4—281no—
224C22×C4—282no—
225C22×C4—281no—
226C22×C4—282no—
227C22×C4(C2×D7)⋊C412yes—
228C22×C4C2×C4×D711yescharacteristic
229C22×C4C2×D2811yescharacteristic
230C22×C4C2×C22⋊D711yescharacteristic
231M16—42no—
232M16—42no—
233M16—564no—
234C2×C8—42no—
235C2×C8—144no—
236C4⋊C4—22no—
237C4⋊C4—142no—
238C4⋊C4—282no—
239C22⋊C4—24no—
240C22⋊C4—284no—
241C22⋊C4—281no—
242C22⋊C4—281no—
243C22⋊C4—562no—
244C42D4×D712yes—
245C42#115912yes—
246C2×C7—21no—
247C2×C7—21no—
248C2×C7—82no—
249C2×C7#189111yescharacteristic
250C2×C7#189511yescharacteristic
251C2×C7#191111yescharacteristic
252D7—82no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—142no—
259C23—281no—
260C23—281no—
261C23—281no—
262C23—281no—
263C23—281no—
264C23—281no—
265C23C2×(C2×D7)⋊C411yescharacteristic
266D4—42no—
267D4—42no—
268D4—144no—
269D4—282no—
270D4—568no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—282no—
278C2×C4—282no—
279C2×C4—281no—
280C2×C4—281no—
281C2×C4—281no—
282C2×C4—282no—
283C2×C4—562no—
284C2×C4—562no—
285C2×C4(C22⋊D7)⋊C412yes—
286C2×C4#644511yescharacteristic
287C2×C4#644911yescharacteristic
288C2×C4#649511yescharacteristic
289C2×C4#650212yes—
290C2×C4#650511yescharacteristic
291C2×C4#651212yes—
292C8—82no—
293C8—284no—
294C7#948911yescharacteristic, complemented, Sylow
295C22—21no—
296C22—21no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—282no—
303C22—282no—
304C22—282no—
305C22—561no—
306C22—561no—
307C22—561no—
308C22—561no—
309C22#8973011yescharacteristic
310C22#8973811yescharacteristic
311C22#8977311yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—562no—
317C4—561no—
318C4#8981112yes—
319C4#8982112yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—562no—
324C2#1172727111yescharacteristic
325C2#1172782511yescharacteristic
326C2#1172789011yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1800C2
socleC22×C7#386
prime radical#11739861C2
cosocleC2×C42×C7C23
hypercenter#1800D7
hypocenterC7#9489
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #11739749C2semidirect
3. C7×#1800C2semidirect
4. #11728250C2semidirect
5. #11725519C2semidirect
51. C4×C7⋊C8D4semidirect
54. #1800D7semidirect
294. C7#9489semidirect

Maximal subgroups

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

subgroupindex
#117398612
#117397492
C7×#18002
#117282502
#117255192
#94897