Finite Group Databasebeta

← #13251639

Subgroups — #13251639

Order 1792.

474 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 113 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#11736708C211yesmaximal, characteristic
2#11731636C211yesmaximal, characteristic
3#11728524C211yesmaximal, characteristic
4#11728282C211yesmaximal, characteristic
5#11728254C211yesmaximal, characteristic
6C7×#1286C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C2×C4×Dic14C2211yescharacteristic
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C7×#382C2211yescharacteristic
13C2×#6398—21no—
14#89757—21no—
15#89746—21no—
16#89697C2211yescharacteristic
17#89696C2211yescharacteristic
18#89602—22no—
19#89490—22no—
20C8×Dic14—22no—
21#89431C412yes—
22#89428C412yes—
23#89425—22no—
24#89349—21no—
25#89279—22no—
26#89251—22no—
27#9484—71nomaximal, Sylow
28C22×Dic14C2×C411yescharacteristic
29C22×Dic14D411yescharacteristic
30C2×C7×M16—21no—
31C2×C7×C4⋊C4—21no—
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7—22no—
34C2×C4⋊Dic7C2×C411yescharacteristic
35C2×C4⋊Dic7D411yescharacteristic
36C2×Dic7⋊C4—21no—
37C2×Dic7⋊C4C2×C412yes—
38C2×C4×Dic7—21no—
39C2×C4×Dic7D412yes—
40C2×#1136—21no—
41C4×Dic14—21no—
42C4×Dic14—21no—
43C4×Dic14—21no—
44C4×Dic14—21no—
45C4×Dic14—21no—
46C4×Dic14—21no—
47C4×Dic14—21no—
48C4×Dic14—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8—21no—
52C7×C22⋊C8—21no—
53C4×C7×C8D412yes—
54C7×#95—21no—
55C7×#95C2×C412yes—
56C7⋊#95—22no—
57C7⋊#95—22no—
58C7⋊#95—22no—
59C22⋊(C7⋊C8)—21no—
60#6398—42no—
61Dic7⋊C8—24no—
62C8×Dic7—24no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4×C7⋊C8D412yes—
67#2982—71no—
68#2581—71no—
69#1801—71no—
70#1487—71no—
71#1487—71no—
72#1286—71no—
73#1286D711yescharacteristic
74C22×C4×C7—22no—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×Dic7—21no—
79C22×Dic7—21no—
80C22×Dic7C22⋊C412yes—
81C22×Dic7C22⋊C412yes—
82C2×Dic14—21no—
83C2×Dic14—21no—
84C2×Dic14—21no—
85C2×Dic14—21no—
86C2×Dic14—21no—
87C2×Dic14—21no—
88C2×Dic14—42no—
89C2×Dic14SD1612yes—
90C2×Dic14Q1612yes—
91C7×M16—41no—
92C7×M16—41no—
93C2×C7×C8—22no—
94C2×C7×C8—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C42×C7C2×D411yescharacteristic
98C42×C7C2×D411yescharacteristic
99C42×C7#4111yescharacteristic
100C42×C7#4111yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7—21no—
105C4⋊Dic7—44no—
106Dic7⋊C4—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—41no—
109Dic7⋊C4—41no—
110C4×Dic7—22no—
111C4×Dic7—22no—
112C4×Dic7—41no—
113C4×Dic7—41no—
114#1136—41no—
115#1136—41no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—41no—
118C2×C4×Q8—71no—
119C2×C4×Q8—71no—
120C8×Q8—142no—
121C8×Q8—142no—
122#431—71no—
123#431—142no—
124#431C2×D711yescharacteristic
125#423—71no—
126#423C2×D711yescharacteristic
127C2×#101—141no—
128#400—72no—
129#395—72no—
130#391—141no—
131#389—141no—
132#387—142no—
133#382C2×D711yescharacteristic
134#343—141no—
135Q8⋊C8—142no—
136Q8⋊C8—142no—
137C23×C7C2×C22⋊C411yescharacteristic
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7C4×D411yescharacteristic
144C2×C4×C7C4×D411yescharacteristic
145C2×C4×C7#12011yescharacteristic
146C2×C4×C7#12111yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12311yescharacteristic
149C2×C4×C7C2×SD1611yescharacteristic
150C2×C4×C7C2×Q1611yescharacteristic
151C2×C4×C7#13511yescharacteristic
152C2×C4×C7#13511yescharacteristic
153C2×Dic7—22no—
154C2×Dic7—22no—
155C2×Dic7—22no—
156C2×Dic7—22no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161C2×Dic7—41no—
162C2×Dic7#10314yes—
163Dic14—22no—
164Dic14—22no—
165Dic14—22no—
166Dic14—22no—
167Dic14—22no—
168Dic14—22no—
169Dic14—41no—
170Dic14—41no—
171C7×C8—42no—
172C7×C8—81no—
173C7⋊C8—42no—
174C7⋊C8—81no—
175C22×Q8—71no—
176C22×Q8—71no—
177C2×M16—21no—
178C2×M16—141no—
179C4×Q8—141no—
180C4×Q8—141no—
181C4×Q8—141no—
182C4×Q8—141no—
183C4×Q8—141no—
184C4×Q8—141no—
185C4×Q8—141no—
186C4×Q8—141no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—72no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—142no—
192C2×C4⋊C4—141no—
193C2×C42—72no—
194C2×C42—141no—
195C2×C42C22×D711yescharacteristic
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—144no—
200C4⋊C8—141no—
201C4⋊C8—141no—
202C4⋊C8—141no—
203#101—282no—
204C22⋊C8—21no—
205C22⋊C8—141no—
206C4×C8—72no—
207C4×C8—144no—
208C4×C8D2812yes—
209#95—21no—
210#95—142no—
211#95—142no—
212#95—142no—
213#95C4×D712yes—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7#38611yescharacteristic
217C22×C7C2×#10311yescharacteristic
218C22×C7#41611yescharacteristic
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—82no—
224C4×C7#46511yescharacteristic
225C4×C7#46511yescharacteristic
226C4×C7#46711yescharacteristic
227C4×C7#46711yescharacteristic
228Dic7—24no—
229Dic7—42no—
230Dic7—42no—
231Dic7—81no—
232C2×Q8—72no—
233C2×Q8—72no—
234C2×Q8—141no—
235C2×Q8—141no—
236C2×Q8—141no—
237C2×Q8—141no—
238C2×Q8—141no—
239C2×Q8—141no—
240C2×Q8—282no—
241C22×C4—22no—
242C22×C4—72no—
243C22×C4—72no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4C2×C4×D711yescharacteristic
247C22×C4D4×D711yescharacteristic
248C22×C4D4×D711yescharacteristic
249M16—41no—
250M16—41no—
251M16—281no—
252M16—281no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—142no—
256C2×C8—281no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—142no—
260C4⋊C4—141no—
261C4⋊C4—142no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—284no—
266C4⋊C4—281no—
267C4⋊C4—281no—
268C42—142no—
269C42—142no—
270C42—281no—
271C42—281no—
272C42C2×D2811yescharacteristic
273C42#115711yescharacteristic
274C42D4×D711yescharacteristic
275C42#116111yescharacteristic
276C2×C7—21no—
277C2×C7—21no—
278C2×C7#189111yescharacteristic
279C2×C7#189211yescharacteristic
280C2×C7#191211yescharacteristic
281C23D7×C22⋊C411yescharacteristic
282Q8—142no—
283Q8—142no—
284Q8—142no—
285Q8—142no—
286Q8—142no—
287Q8—142no—
288Q8—281no—
289Q8—281no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—74no—
296C2×C4—142no—
297C2×C4—142no—
298C2×C4—142no—
299C2×C4—142no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4—281no—
305C2×C4C4×D2811yescharacteristic
306C2×C4#644511yescharacteristic
307C2×C4#644711yescharacteristic
308C2×C4#645611yescharacteristic
309C2×C4#645711yescharacteristic
310C2×C4#645811yescharacteristic
311C2×C4#647511yescharacteristic
312C2×C4D7×SD1611yescharacteristic
313C2×C4#647911yescharacteristic
314C2×C4D7×Q1611yescharacteristic
315C8—42no—
316C8—81no—
317C8—282no—
318C8—561no—
319C7#948411yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22#8943711yescharacteristic
323C22#8954411yescharacteristic
324C22D7×#10311yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—144no—
331C4—282no—
332C4—282no—
333C4—561no—
334C4#8955011yescharacteristic
335C4#8958511yescharacteristic
336C4#8963611yescharacteristic
337C4#8965911yescharacteristic
338C2—21no—
339C2—21no—
340C2#1172646211yescharacteristic
341C2#1172663611yescharacteristic
342C2#1172679011yescharacteristic
subgroupG/H
centerC22#89437
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1286C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1286D7
hypocenterC7#9484
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
319. C7#9484semidirect

Maximal subgroups

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

subgroupindex
#117367082
#117316362
#117285242
#117282822
#117282542
C7×#12862
#117254372
#94847