Finite Group Databasebeta

← #13249556

Subgroups — #13249556

Order 1792.

630 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×C7×#382C211yesmaximal, characteristic
2#11739520C211yesmaximal, characteristic
3#11729285C211yesmaximal, characteristic
4C2×#89247C212yesmaximal
5#11725418C212yesmaximal
6C22×C7×C4⋊C4—21no—
7C22×C42×C7C2211yescharacteristic
8C2×#6487C2211yescharacteristic
9C7×#382—24no—
10C2×C7×#95—21no—
11C2×C7×#95C2212yes—
12#89984C2211yescharacteristic
13#89983—21no—
14#89982—21no—
15C2×C22⋊(C7⋊C8)C2212yes—
16#89705—22no—
17#89701C412yescomplemented
18#89700C412yescomplemented
19#89319—22no—
20#89247—24no—
21#9152—71nomaximal, Sylow
22C23×C4×C7—22no—
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×Dic14C2×C411yescharacteristic
27C2×C7×C4⋊C4—24no—
28C2×C7×C4⋊C4—42no—
29C2×C7×C4⋊C4—42no—
30C2×C42×C7—22no—
31C2×C42×C7—22no—
32C2×C42×C7C2×C412yes—
33C2×C42×C7D412yes—
34C2×C22⋊Dic7—21no—
35C2×C22⋊Dic7—21no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×C4⋊Dic7C2×C412yes—
38#6487—22no—
39#6487—22no—
40C2×Dic7⋊C4—22no—
41C2×Dic7⋊C4—21no—
42C22×C7⋊C8—22no—
43C7×#95—24no—
44C7×#95—22no—
45C7×#95D414yes—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#2855—71no—
53C2×#382D711yescharacteristic
54#1817—71no—
55C2×#328—72no—
56#1294—72no—
57C24×C7C2×D411yescharacteristic
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—24no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—44no—
68C22×C4×C7—44no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7SD1612yes—
74C22×C4×C7Q1612yes—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D412yes—
78C22×C4×C7#4112yes—
79C22×Dic7—22no—
80C22×Dic7—21no—
81C22×Dic7—21no—
82C2×Dic14—22no—
83C2×Dic14—42no—
84C7×C4⋊C4—48no—
85C42×C7—24no—
86C42×C7—24no—
87C22⋊Dic7—41no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C22⋊Dic7—41no—
91C4⋊Dic7—22no—
92C4⋊Dic7—42no—
93Dic7⋊C4—44no—
94Dic7⋊C4—42no—
95C2×C7⋊C8—44no—
96C2×C7⋊C8—42no—
97C2×C22⋊Q8—71no—
98C22×C4⋊C4—21no—
99C22×C42C2×D711yescharacteristic
100C2×C22⋊C8—72no—
101#391—72no—
102#389—72no—
103#387—142no—
104#385—141no—
105#382—24no—
106#381—141no—
107#380—71no—
108C2×#95—21no—
109C2×#95C2×D712yes—
110#336—142no—
111#328—144no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7C2×C22⋊C411yescharacteristic
115C23×C7#12011yescharacteristic
116C23×C7#12311yescharacteristic
117C23×C7C2×SD1611yescharacteristic
118C23×C7C2×Q1611yescharacteristic
119C23×C7AGL(1,ℤ/8)11yescharacteristic
120C23×C7#13711yescharacteristic
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7—44no—
138C2×C4×C7—44no—
139C2×C4×C7—84no—
140C2×C4×C7—84no—
141C2×C4×C7#10314yes—
142C2×C4×C7C4×D412yes—
143C2×C4×C7#12112yes—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151Dic14—82no—
152C7⋊C8—82no—
153C22×Q8—71no—
154C23×C4—22no—
155C23×C4C22⋊D711yescharacteristic
156C23×C4C22⋊D711yescharacteristic
157C23×C4C22×D711yescharacteristic
158C22×C8—142no—
159C22⋊Q8—142no—
160C22⋊Q8—142no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—72no—
166C2×C4⋊C4—142no—
167C2×C4⋊C4—141no—
168C2×C22⋊C4—141no—
169C2×C22⋊C4—141no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C4×D712yes—
173C2×C42D2812yes—
174C22⋊C8—142no—
175C22⋊C8—142no—
176#95—24no—
177#95—22no—
178#95—142no—
179#95—142no—
180#95—142no—
181#95—141no—
182#95C22⋊D714yes—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—22no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7#38611yescharacteristic
192C22×C7#41012yes—
193C22×C7C2×#10311yescharacteristic
194C22×C7#41711yescharacteristic
195C22×C7#42112yes—
196C22×C7#45011yescharacteristic
197C22×C7#45111yescharacteristic
198C22×C7#48311yescharacteristic
199C22×C7#48411yescharacteristic
200C4×C7—24no—
201C4×C7—22no—
202C4×C7—22no—
203C4×C7—42no—
204C4×C7—84no—
205Dic7—81no—
206Dic7—81no—
207Dic7—82no—
208C24C2×C22⋊D711yescharacteristic
209C2×Q8—142no—
210C2×Q8—282no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—142no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4(C2×D7)⋊C412yes—
228C22×C4(C2×D7)⋊C412yes—
229C22×C4#114212yes—
230C22×C4#114412yes—
231C22×C4C2×C4×D711yescharacteristic
232C22×C4C2×D2811yescharacteristic
233C22×C4#115712yes—
234C22×C4C2×C22⋊D712yes—
235C2×C8—284no—
236C2×C8—282no—
237C4⋊C4—48no—
238C4⋊C4—142no—
239C4⋊C4—282no—
240C4⋊C4—284no—
241C4⋊C4—282no—
242C22⋊C4—281no—
243C22⋊C4—281no—
244C22⋊C4—281no—
245C22⋊C4—281no—
246C42—24no—
247C42—24no—
248C2×C7—21no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7#151212yes—
253C2×C7#151812yes—
254C2×C7#187211yescharacteristic
255C2×C7#189711yescharacteristic
256C2×C7#189911yescharacteristic
257C23—22no—
258C23—22no—
259C23C2×(C2×D7)⋊C411yescharacteristic
260C23#649211yescharacteristic
261C23#649511yescharacteristic
262C23C2×#114211yescharacteristic
263C23#650511yescharacteristic
264C23C2×#114411yescharacteristic
265C23#651611yescharacteristic
266Q8—562no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—282no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—282no—
293C2×C4—282no—
294C2×C4#638414yes—
295C2×C4C4×C22⋊D712yes—
296C2×C4#649312yes—
297C8—562no—
298C7#915211yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22#8973811yescharacteristic
308C22C2×#638411yescharacteristic
309C22#8974211yescharacteristic
310C22#8974311yescharacteristic
311C22#8977712yes—
312C22#8977812yes—
313C22#8998511yescharacteristic
314C22#8999011yescharacteristic
315C22#9000011yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—84no—
321C4—561no—
322C4—562no—
323C4—561no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#1172729812yes—
329C2#1172730012yes—
330C2#1172931311yescharacteristic
331C2#1172934511yescharacteristic
332C2#1172934911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#382C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenterC2×#382D7
hypocenterC7#9152
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
17. #89701C4semidirect
18. #89700C4semidirect
298. C7#9152semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3822
#117395202
#117292852
C2×#892472
#117254182
#91527