Finite Group Databasebeta

← #13249554

Subgroups — #13249554

Order 1792.

630 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 143 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#11739520C211yesmaximal, characteristic
2C2×#89708C211yesmaximal, characteristic
3C7×#1817C211yesmaximal, characteristic
4C2×#89262C212yesmaximal
5#11725481C212yesmaximal
6C22×C42×C7C2211yescharacteristic
7C22×C4⋊Dic7—21no—
8C2×#6487C2211yescharacteristic
9C2×C7×C22⋊C8C2212yes—
10#89984C2211yescharacteristic
11#89983—21no—
12#89982—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89708—24no—
16#89705—22no—
17#89701C412yescomplemented
18#89700C412yescomplemented
19C7×#336—22no—
20#89262—24no—
21#9152—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C23×Dic7—22no—
26C22×Dic14C2×C411yescharacteristic
27C22×C7×C8—22no—
28C2×C42×C7—22no—
29C2×C42×C7—22no—
30C2×C42×C7C2×C412yes—
31C2×C42×C7D412yes—
32C2×C22⋊Dic7—21no—
33C2×C22⋊Dic7—21no—
34C2×C4⋊Dic7—24no—
35C2×C4⋊Dic7—42no—
36C2×C4⋊Dic7—42no—
37C2×C4⋊Dic7C2×C411yescharacteristic
38C2×C4⋊Dic7C2×C412yes—
39#6487—22no—
40#6487—22no—
41C2×Dic7⋊C4—22no—
42C2×Dic7⋊C4—21no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—22no—
45C7⋊#95—24no—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C7⋊#95—22no—
51C7⋊#95D414yes—
52#2855—71no—
53C2×#382—71no—
54#1817D711yescharacteristic
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—42no—
65C22×C4×C7—42no—
66C22×C4×C7C22⋊C412yes—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7SD1612yes—
69C22×C4×C7Q1612yes—
70C22×C4×C7C22×C411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7C2×D412yes—
73C22×C4×C7#4112yes—
74C22×Dic7—22no—
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7—24no—
78C22×Dic7—44no—
79C22×Dic7—44no—
80C22×Dic7—42no—
81C22×Dic7—42no—
82C2×Dic14—22no—
83C2×Dic14—42no—
84C2×C7×C8—44no—
85C2×C7×C8—42no—
86C42×C7—24no—
87C42×C7—24no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92C4⋊Dic7—22no—
93C4⋊Dic7—48no—
94C4⋊Dic7—42no—
95Dic7⋊C4—44no—
96Dic7⋊C4—42no—
97C2×C22⋊Q8—71no—
98C22×C4⋊C4—141no—
99C22×C42C2×D711yescharacteristic
100C2×C22⋊C8C2×D712yes—
101#391—72no—
102#389—72no—
103#387—142no—
104#385—141no—
105#382—144no—
106#381—141no—
107#380—71no—
108C2×#95—72no—
109C2×#95—141no—
110#336—22no—
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#10314yes—
137C2×C4×C7C4×D412yes—
138C2×C4×C7#12112yes—
139C2×Dic7—44no—
140C2×Dic7—44no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—44no—
149C2×Dic7—84no—
150C2×Dic7—84no—
151Dic14—82no—
152C7×C8—82no—
153C22×Q8—71no—
154C23×C4—142no—
155C23×C4D2811yescharacteristic
156C23×C4D2811yescharacteristic
157C23×C4C22×D711yescharacteristic
158C22×C8—22no—
159C22⋊Q8—142no—
160C22⋊Q8—142no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—72no—
163C2×C4⋊C4—144no—
164C2×C4⋊C4—142no—
165C2×C4⋊C4—141no—
166C2×C4⋊C4—282no—
167C2×C4⋊C4—282no—
168C2×C22⋊C4—141no—
169C2×C22⋊C4—141no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C4×D712yes—
173C2×C42C22⋊D712yes—
174C22⋊C8—22no—
175C22⋊C8—22no—
176#95—74no—
177#95—144no—
178#95—142no—
179#95—142no—
180#95—142no—
181#95—142no—
182#95—141no—
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—
204Dic7—84no—
205Dic7—81no—
206Dic7—81no—
207Dic7—82no—
208C24C2×D2811yescharacteristic
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—42no—
218C22×C4—42no—
219C22×C4—144no—
220C22×C4—142no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—284no—
224C22×C4—284no—
225C22×C4—282no—
226C22×C4—282no—
227C22×C4#113212yes—
228C22×C4Dic2812yes—
229C22×C4(C2×D7)⋊C412yes—
230C22×C4(C2×D7)⋊C412yes—
231C22×C4C2×C4×D711yescharacteristic
232C22×C4D4×D712yes—
233C22×C4#115912yes—
234C22×C4C2×C22⋊D711yescharacteristic
235C2×C8—44no—
236C2×C8—42no—
237C4⋊C4—142no—
238C4⋊C4—282no—
239C4⋊C4—284no—
240C4⋊C4—288no—
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—
259C23#644511yescharacteristic
260C23#644911yescharacteristic
261C23C2×#113211yescharacteristic
262C23C2×Dic2811yescharacteristic
263C23#647111yescharacteristic
264C23#647211yescharacteristic
265C23C2×(C2×D7)⋊C411yescharacteristic
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—284no—
283C2×C4—284no—
284C2×C4—282no—
285C2×C4—282no—
286C2×C4—282no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—284no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—564no—
293C2×C4—564no—
294C2×C4#639014yes—
295C2×C4(C22⋊D7)⋊C412yes—
296C2×C4#650212yes—
297C8—82no—
298C7#915211yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22#8949311yescharacteristic
308C22#8949411yescharacteristic
309C22#8950311yescharacteristic
310C22#8951111yescharacteristic
311C22#8973011yescharacteristic
312C22#8977812yes—
313C22C2×#639011yescharacteristic
314C22#8990511yescharacteristic
315C22#8991112yes—
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—564no—
321C4—561no—
322C4—561no—
323C4—562no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#1172728911yescharacteristic
329C2#1172768411yescharacteristic
330C2#1172772011yescharacteristic
331C2#1172854712yes—
332C2#1172858812yes—
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1817C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1817D7
hypocenterC7#9152
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

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
#117395202
C2×#897082
C7×#18172
C2×#892622
#117254812
#91527