Finite Group Databasebeta

← #13248983

Subgroups — #13248983

Order 1792.

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

normal subgroups only

This group has 424 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#11736755C211yesmaximal, characteristic
2#11736490C211yesmaximal, characteristic
3#11728254C211yesmaximal, characteristic
4#11728227C211yesmaximal, characteristic
5#11728191C211yesmaximal, characteristic
6C7×#1301C211yesmaximal, characteristic
7#11725547C211yesmaximal, characteristic
8#90161C2211yescharacteristic
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13#89894—21no—
14#89891—21no—
15#89697C2211yescharacteristic
16C2×#6377C2211yescharacteristic
17#89622—21no—
18#89621—21no—
19#89600—22no—
20#89600—22no—
21#89511C412yes—
22#89494C412yes—
23C4×Dic28—22no—
24#89345—21no—
25#89345—21no—
26#89281—21no—
27#89279—22no—
28#89276—21no—
29#9048—71nomaximal, Sylow
30C22×Dic14C2×C411yescharacteristic
31C2×C7×M16—21no—
32C22×C7×C8—21no—
33C2×C42×C7C2311yescharacteristic
34#6489—21no—
35C2×C4⋊Dic7C2×C411yescharacteristic
36#6487—21no—
37#6487C2×C412yes—
38C2×Dic7⋊C4—21no—
39C2×C4×Dic7—21no—
40C22×C7⋊C8D411yescharacteristic
41C22×C7⋊C8D411yescharacteristic
42C2×Dic28—22no—
43#6433—21no—
44#6432D411yescharacteristic
45#6432D411yescharacteristic
46C4×Dic14—21no—
47C4×Dic14—21no—
48C4×Dic14—21no—
49C4×Dic14—21no—
50C4×Dic14—21no—
51C4×Dic14D412yes—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8D412yes—
57C7×C22⋊C8C2×C412yes—
58C4×C7×C8—21no—
59C22⋊(C7⋊C8)—21no—
60#6392—22no—
61#6390—22no—
62#6390—22no—
63#6389—22no—
64Dic7⋊C8—24no—
65C8×Dic7—22no—
66C4⋊(C7⋊C8)—21no—
67#6377—21no—
68#6377—21no—
69C4×C7⋊C8—21no—
70#3104—71no—
71#2961—71no—
72#1985—71no—
73#1542—71no—
74#1487—71no—
75#1301D711yescharacteristic
76#1281—71no—
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7#4111yescharacteristic
80C22×Dic7—21no—
81C22×Dic7—21no—
82C2×Dic14—21no—
83C2×Dic14—21no—
84C2×Dic14—21no—
85C2×Dic14—42no—
86C2×Dic14C22⋊C412yes—
87C7×M16—41no—
88C7×M16—41no—
89C2×C7×C8—22no—
90C2×C7×C8—22no—
91C2×C7×C8—41no—
92C2×C7×C8—41no—
93C2×C7×C8—41no—
94C42×C7C2×D411yescharacteristic
95C42×C7C2×D411yescharacteristic
96C42×C7C2×D411yescharacteristic
97C42×C7#4111yescharacteristic
98C22⋊Dic7—22no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7C22⋊C412yes—
105Dic7⋊C4—22no—
106Dic7⋊C4—22no—
107Dic7⋊C4—41no—
108Dic7⋊C4—41no—
109C4×Dic7—22no—
110C4×Dic7—41no—
111C4×Dic7—41no—
112C2×C7⋊C8—21no—
113C2×C7⋊C8—21no—
114C2×C7⋊C8—21no—
115C2×C7⋊C8—21no—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8Q1612yescomplemented
119Dic28—44no—
120#533—71no—
121C2×C4×Q8—71no—
122#484—72no—
123#477—141no—
124#462—141no—
125#451—72no—
126#446—142no—
127#446—142no—
128C4×Q16—142no—
129#431—71no—
130#431C2×D711yescharacteristic
131#424—141no—
132#423C2×D711yescharacteristic
133C2×C4⋊C8C2×D711yescharacteristic
134#405—141no—
135C2×#97—71no—
136#336—141no—
137#336—141no—
138#333—141no—
139#332—141no—
140Q8⋊C8—142no—
141C23×C7C4×D411yescharacteristic
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7C2×C22⋊C411yescharacteristic
145C2×C4×C7C4×D411yescharacteristic
146C2×C4×C7#12011yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12311yescharacteristic
150C2×C4×C7C2×Q1611yescharacteristic
151C2×C4×C7#13511yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×C4×C7#13711yescharacteristic
154C2×Dic7—22no—
155C2×Dic7—22no—
156C2×Dic7—22no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—42no—
160C2×Dic7—41no—
161Dic14—22no—
162Dic14—22no—
163Dic14—41no—
164Dic14—41no—
165Dic14—42no—
166Dic14—81no—
167C7×C8—42no—
168C7×C8—42no—
169C7×C8—81no—
170C7⋊C8—22no—
171C7⋊C8—41no—
172C7⋊C8—81no—
173C22×Q8—71no—
174C2×Q16—142no—
175C2×M16—21no—
176C22×C8—21no—
177C22×C8—71no—
178C22×C8—71no—
179C4⋊Q8—71no—
180C4⋊Q8—71no—
181#125—141no—
182C22⋊Q8—72no—
183C22⋊Q8—141no—
184C4×Q8—72no—
185C4×Q8—141no—
186C4×Q8—141no—
187C4×Q8—141no—
188C4×Q8—141no—
189C4×Q8—141no—
190#117—141no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—141no—
193C2×C42—141no—
194C2×C42C22×D711yescharacteristic
195#107—142no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—144no—
201C4⋊C8—141no—
202C4⋊C8D2812yes—
203#103—142no—
204#103—142no—
205C22⋊C8—141no—
206C22⋊C8C4×D712yes—
207#97—142no—
208#97—141no—
209#97—141no—
210C4×C8—21no—
211C4×C8—142no—
212C4×C8—141no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7#38611yescharacteristic
216C22×C7C4×Q1611yescharacteristic
217C22×C7#44111yescharacteristic
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7#45111yescharacteristic
223C4×C7#45211yescharacteristic
224C4×C7#46211yescharacteristic
225C4×C7#46411yescharacteristic
226Dic7—42no—
227Dic7—42no—
228Dic7—42no—
229Dic7—81no—
230C2×Q8—72no—
231C2×Q8—141no—
232C2×Q8—141no—
233C2×Q8—141no—
234C2×Q8—282no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4C2×C4×D711yescharacteristic
238C22×C4D4×D711yescharacteristic
239C22×C4#116111yescharacteristic
240Q16—284no—
241M16—41no—
242M16—41no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—72no—
249C2×C8—141no—
250C2×C8—141no—
251C2×C8—141no—
252C2×C8—141no—
253C2×C8—141no—
254C2×C8—281no—
255C4⋊C4—72no—
256C4⋊C4—142no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—142no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—281no—
264C4⋊C4—281no—
265C22⋊C4—142no—
266C42—142no—
267C42—281no—
268C42—281no—
269C42C2×D2811yescharacteristic
270C42#115711yescharacteristic
271C42D4×D711yescharacteristic
272C42D4×D711yescharacteristic
273C2×C7—21no—
274C2×C7—21no—
275C2×C7#187811yescharacteristic
276C2×C7#188111yescharacteristic
277C2×C7#189111yescharacteristic
278C23#645611yescharacteristic
279Q8—142no—
280Q8—142no—
281Q8—282no—
282Q8—281no—
283Q8—281no—
284Q8—561no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—142no—
288C2×C4—142no—
289C2×C4—142no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—282no—
293C2×C4—281no—
294C2×C4C4×D2811yescharacteristic
295C2×C4D7×C22⋊C411yescharacteristic
296C2×C4#644511yescharacteristic
297C2×C4#644711yescharacteristic
298C2×C4#645711yescharacteristic
299C2×C4#645811yescharacteristic
300C2×C4#647511yescharacteristic
301C2×C4#647811yescharacteristic
302C2×C4#647811yescharacteristic
303C2×C4D7×Q1611yescharacteristic
304C8—42no—
305C8—42no—
306C8—81no—
307C8—142no—
308C8—281no—
309C8—561no—
310C7#904811yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22#8943711yescharacteristic
314C22#8962511yescharacteristic
315C22#8964511yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—282no—
321C4—282no—
322C4—282no—
323C4—561no—
324C4#8953911yescharacteristic
325C4#8955511yescharacteristic
326C4#8956511yescharacteristic
327C4#8958011yescharacteristic
328C2—21no—
329C2—21no—
330C2#1172666211yescharacteristic
331C2#1172670111yescharacteristic
332C2#1172679011yescharacteristic
subgroupG/H
centerC2×C4#6445
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1301C2
socleC22×C7#386
prime radicalC23×C7C4×D4
cosocleC2×C42×C7C23
hypercenter#1301D7
hypocenterC7#9048
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
118. C2×C7⋊C8Q16semidirect
310. C7#9048semidirect

Maximal subgroups

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

subgroupindex
#117367552
#117364902
#117282542
#117282272
#117281912
C7×#13012
#117255472
#90487