Finite Group Databasebeta

← #13256068

Subgroups — #13256068

Order 1792.

462 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C7×#3230C211yesmaximal, characteristic
2#11740785C211yesmaximal, characteristic
3#11740667C211yesmaximal, characteristic
4#11729592C211yesmaximal, characteristic
5#11729542C211yesmaximal, characteristic
6#11728547C211yesmaximal, characteristic
7#11725547C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10#90161C2211yescharacteristic
11C7×#500—21no—
12C7×#487—21no—
13C7×#484—22no—
14C7×#477—22no—
15C7×#431C2211yescharacteristic
16#90000—22no—
17#89864—21no—
18#89862—22no—
19#89853—21no—
20#89852—21no—
21#89852—21no—
22#89837—21no—
23#89837—21no—
24#89770—22no—
25#89708C2211yescharacteristic
26#89697C2211yescharacteristic
27C2×#6377C2211yescharacteristic
28#89333—22no—
29#89324—22no—
30#89262—22no—
31#10224—71nomaximal, Sylow
32C22×C7×Q8—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8D412yes—
37C7×C4⋊Q8D411yescharacteristic
38C7×C4⋊Q8D411yescharacteristic
39C7×#125—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C7×#117—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C42×C7C2311yescharacteristic
48C2×#1144—24no—
49#6489—21no—
50C2×C4⋊Dic7—21no—
51#6487—21no—
52#6487—21no—
53C22×C7⋊C8D412yes—
54#6433—21no—
55#6432D411yescharacteristic
56#6432D411yescharacteristic
57C4×Dic14—21no—
58C4×Dic14—21no—
59C7×#107—24no—
60C7×C4⋊C8—21no—
61C7×#103—24no—
62C7×C22⋊C8D412yes—
63C4×C7×C8—21no—
64#6409—24no—
65C7⋊#95—21no—
66C7⋊#95—21no—
67C22⋊(C7⋊C8)D412yes—
68#6382—22no—
69#6382—22no—
70#6381—22no—
71#6381—22no—
72C4⋊(C7⋊C8)—21no—
73#6377—21no—
74#6377—21no—
75C4×C7⋊C8—21no—
76#3230D711yescharacteristic
77#3209—71no—
78#3162—71no—
79#1993—71no—
80#1518—71no—
81#1518—71no—
82#1281—71no—
83C2×C7×Q8—22no—
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—41no—
87C2×C7×Q8—82no—
88C22×C4×C7—41no—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×Dic7—41no—
93C2×Dic14—21no—
94C2×Dic14—21no—
95C2×C7×C8—22no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C22⋊C4—41no—
107C42×C7—41no—
108C42×C7C2×D411yescharacteristic
109C42×C7C2×D411yescharacteristic
110C42×C7C2×D411yescharacteristic
111C42×C7#4111yescharacteristic
112C22⋊Dic7—41no—
113#1144—84no—
114C4⋊Dic7—21no—
115C4⋊Dic7—21no—
116C4⋊Dic7—21no—
117C4⋊Dic7—21no—
118C4⋊Dic7—42no—
119Dic7⋊C4—41no—
120Dic7⋊C4—41no—
121C4×Dic7—41no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—22no—
125C2×C7⋊C8Q1614yescomplemented
126#533—71no—
127#533C2×D711yescharacteristic
128C2×C4⋊Q8C2×D711yescharacteristic
129#501—141no—
130#501—141no—
131#501—141no—
132#501—141no—
133#500—21no—
134#494—141no—
135#487—21no—
136#487—141no—
137#484—22no—
138#477—22no—
139#462—142no—
140#451—142no—
141#431—71no—
142#431C2×D711yescharacteristic
143#427—142no—
144C2×#97—71no—
145#382—71no—
146#340—142no—
147#328—142no—
148#328—142no—
149C23×C7#12711yescharacteristic
150C7×Q8—44no—
151C7×Q8—42no—
152C7×Q8—81no—
153C7×Q8—82no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—82no—
160C2×C4×C7#12011yescharacteristic
161C2×C4×C7#12011yescharacteristic
162C2×C4×C7#12011yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7C2×Q1612yes—
167C2×C4×C7#13712yes—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—81no—
171C2×Dic7—81no—
172C2×Dic7—81no—
173Dic14—42no—
174Dic14—81no—
175C7×C8—42no—
176C7⋊C8—24no—
177C7⋊C8—42no—
178C22×Q8—21no—
179C2×Q16—144no—
180C22×C8—72no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—71no—
185C4⋊Q8—71no—
186C4⋊Q8C22⋊D712yes—
187C4⋊Q8C22⋊D711yescharacteristic
188C4⋊Q8C22⋊D711yescharacteristic
189#125—21no—
190#125—141no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—141no—
194C22⋊Q8—141no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—141no—
198C4×Q8—141no—
199#117—21no—
200#117—141no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—141no—
204C2×C42C22×D711yescharacteristic
205#107—24no—
206#107—142no—
207#107—142no—
208#106—142no—
209#106—142no—
210C4⋊C8—21no—
211C4⋊C8—141no—
212#103—24no—
213#103—144no—
214C22⋊C8—72no—
215C22⋊C8C22⋊D712yes—
216#97—141no—
217#97—141no—
218C4×C8—21no—
219C4×C8—141no—
220#95—141no—
221#95—141no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7#39211yescharacteristic
225C22×C7#49411yescharacteristic
226C22×C7#49711yescharacteristic
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—82no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7#45112yes—
235C4×C7#46212yes—
236Dic7—81no—
237Dic7—81no—
238Dic7—161no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×Q8—82no—
244C2×Q8—141no—
245C2×Q8—141no—
246C22×C4—41no—
247C22×C4—281no—
248C22×C4D4×D711yescharacteristic
249C22×C4D4×D711yescharacteristic
250C22×C4C2×C22⋊D711yescharacteristic
251Q16—564no—
252C2×C8—22no—
253C2×C8—74no—
254C2×C8—142no—
255C2×C8—142no—
256C2×C8—142no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—281no—
272C4⋊C4—281no—
273C4⋊C4—282no—
274C22⋊C4—41no—
275C22⋊C4—281no—
276C42—41no—
277C42—281no—
278C42D4×D711yescharacteristic
279C42#115911yescharacteristic
280C42C2×C22⋊D711yescharacteristic
281C42C2×C22⋊D711yescharacteristic
282C2×C7—21no—
283C2×C7—21no—
284C2×C7#200811yescharacteristic
285C2×C7#200911yescharacteristic
286C2×C7#201611yescharacteristic
287C23#650311yescharacteristic
288Q8—44no—
289Q8—42no—
290Q8—81no—
291Q8—82no—
292Q8—282no—
293Q8—561no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—561no—
303C2×C4—561no—
304C2×C4—561no—
305C2×C4D7×Q1612yes—
306C2×C4#648112yes—
307C2×C4#650011yescharacteristic
308C2×C4#650011yescharacteristic
309C2×C4#650111yescharacteristic
310C2×C4#650211yescharacteristic
311C2×C4#650211yescharacteristic
312C2×C4#651611yescharacteristic
313C8—42no—
314C8—144no—
315C8—282no—
316C7#1022411yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22#8995311yescharacteristic
320C22#8996211yescharacteristic
321C22#8999511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—561no—
330C4—561no—
331C4—1121no—
332C4#8995412yes—
333C4#8995812yes—
334C2—21no—
335C2—21no—
336C2#1172962311yescharacteristic
337C2#1172982711yescharacteristic
338C2#1172985311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3230C2
socleC22×C7#392
prime radicalC23×C7#127
hypercenter#3230D7
hypocenterC7#10224
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
125. C2×C7⋊C8Q16semidirect
316. C7#10224semidirect

Maximal subgroups

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

subgroupindex
C7×#32302
#117407852
#117406672
#117295922
#117295422
#117285472
#117255472
#102247