Finite Group Databasebeta

← #13507300

Subgroups — #13507300

Order 1792.

498 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11735987C211yesmaximal, characteristic
2C7×#2108C211yesmaximal, characteristic
3#11729173C211yesmaximal, characteristic
4Dic7×M32C211yesmaximal, characteristic
5#11729155C211yesmaximal, characteristic
6C2×Dic7⋊C16C211yesmaximal, characteristic
7#11727636C211yesmaximal, characteristic
8#11726822C212yesmaximal, complemented
9#11726822C212yesmaximal, complemented
10#11726802C212yesmaximal, complemented
11C16×Dic14C212yesmaximal, complemented
12#90208C411yescharacteristic
13C2×C7×M32C2211yescharacteristic
14C22×C7×C16C2211yescharacteristic
15C7×#405C2211yescharacteristic
16#89891C411yescharacteristic
17#89890C411yescharacteristic
18#89873C2211yescharacteristic
19C2×C8×Dic7C2211yescharacteristic
20C2×#6386C2211yescharacteristic
21C22×C7⋊C16C2211yescharacteristic
22#89771C411yescharacteristic
23#89473C412yes—
24#89473C412yes—
25C8×Dic14C2212yes—
26C8×Dic14C2212yes—
27C7×#346C2212yes—
28C4×C7×C16C2212yes—
29#89297C2212yes—
30#89297C2212yes—
31Dic7⋊C16C2211yescharacteristic
32Dic7⋊C16C2211yescharacteristic
33Dic7⋊C16C2211yescharacteristic
34Dic7⋊C16C2211yescharacteristic
35Dic7⋊C16C2212yes—
36Dic7⋊C16C2212yes—
37C16×Dic7C2212yes—
38C16×Dic7C2212yes—
39C4⋊(C7⋊C16)C2212yes—
40C4⋊(C7⋊C16)C2212yes—
41#33476—71nomaximal, Sylow
42C2×C7×M16C2×C411yescharacteristic
43C22×C7×C8C2311yescharacteristic
44C7×#117C2×C411yescharacteristic
45#6489C2×C411yescharacteristic
46C2×C4×Dic7C2×C411yescharacteristic
47C2×#1136C2×C411yescharacteristic
48C22×C7⋊C8C2×C411yescharacteristic
49#6453C811yescharacteristic
50#6453C811yescharacteristic
51#6451C811yescharacteristic
52#6451C811yescharacteristic
53#6441C812yes—
54#6441C812yes—
55C4×Dic14C2×C412yes—
56C4×Dic14C2×C412yes—
57C7×M32Q814yescomplemented
58C2×C7×C16—22no—
59C2×C7×C16C2311yescharacteristic
60C2×C7×C16C2311yescharacteristic
61C2×C7×C16C2312yes—
62C7×#97C2×C412yes—
63C4×C7×C8C2312yes—
64#6389C2×C412yes—
65Dic7⋊C8C2×C412yes—
66Dic7⋊C8C2×C412yes—
67Dic7⋊C8C2×C412yes—
68Dic7⋊C8C2×C412yes—
69C8×Dic7C2×C411yescharacteristic
70C8×Dic7C2×C411yescharacteristic
71C8×Dic7C2311yescharacteristic
72C8×Dic7C2311yescharacteristic
73C8×Dic7C2312yes—
74#6386—22no—
75C2×C7⋊C16—22no—
76C2×C7⋊C16C2311yescharacteristic
77C2×C7⋊C16C2311yescharacteristic
78C2×C7⋊C16C2312yes—
79C4⋊(C7⋊C8)C2×C412yes—
80C4⋊(C7⋊C8)C2×C412yes—
81#2960—71no—
82#2183—72no—
83#2183—72no—
84#2183—72no—
85Q8×C16—72no—
86#2151—71no—
87#2151—71no—
88C2×C4⋊C16—71no—
89#2108—71no—
90#2108D711yescharacteristic
91C4×M32—71no—
92C22×C4×C7C22×C411yescharacteristic
93C22×Dic7C2×C811yescharacteristic
94C22×Dic7C2×C811yescharacteristic
95C2×Dic14C2×C812yes—
96C2×Dic14C2×C812yes—
97C7×M16—22no—
98C2×C7×C8C22×C412yes—
99C2×C7×C8C22×C411yescharacteristic
100C2×C7×C8C2×Q812yes—
101C2×C7×C8#4112yes—
102C2×C7×C8C2411yescharacteristic
103C7×C4⋊C4C2×C811yescharacteristic
104C7×C4⋊C4C2×C811yescharacteristic
105C7×C22⋊C4C2×C812yes—
106C42×C7C22×C412yes—
107C22⋊Dic7C2×C812yes—
108C4⋊Dic7C2×C812yes—
109C4⋊Dic7C2×C812yes—
110C4⋊Dic7C2×C811yescharacteristic
111C4⋊Dic7C2×C811yescharacteristic
112Dic7⋊C4C2×C812yes—
113Dic7⋊C4C2×C812yes—
114Dic7⋊C4C2×C812yes—
115Dic7⋊C4C2×C812yes—
116C4×Dic7C2×C811yescharacteristic
117C4×Dic7C2×C811yescharacteristic
118C4×Dic7C22×C411yescharacteristic
119C4×Dic7C22×C411yescharacteristic
120C4×Dic7C22×C412yes—
121#1136—22no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8C22×C411yescharacteristic
124C2×C7⋊C8C22×C411yescharacteristic
125C2×C7⋊C8C22×C412yes—
126C7×C16—21no—
127C7×C16—21no—
128C7×C16C2×Q814yes—
129C7⋊C16—21no—
130C7⋊C16—21no—
131C7⋊C16—22no—
132#533—71no—
133C2×M32—71no—
134C2×M32C2×D711yescharacteristic
135C22×C16—71no—
136C22×C16C2×D711yescharacteristic
137#446—72no—
138#446—72no—
139C8×Q8—72no—
140C8×Q8—72no—
141#424—71no—
142#424—71no—
143#423—71no—
144#405—71no—
145#405C2×D711yescharacteristic
146C2×C4×C8—71no—
147C4⋊C16—71no—
148C4⋊C16—71no—
149C4⋊C16—71no—
150C4⋊C16—71no—
151C4⋊C16—72no—
152C4⋊C16—72no—
153C4⋊C16—72no—
154C4⋊C16—72no—
155#346—72no—
156#346—72no—
157#346C2×D712yes—
158C4×C16—72no—
159C4×C16—72no—
160C4×C16C2×D712yes—
161C23×C7C22×C811yescharacteristic
162C2×C4×C7C4×Q812yes—
163C2×C4×C7C22×C812yes—
164C2×C4×C7C22×C812yes—
165C2×C4×C7C22×C811yescharacteristic
166C2×C4×C7#13112yes—
167C2×C4×C7C23×C411yescharacteristic
168C2×Dic7—22no—
169C2×Dic7—41no—
170C2×Dic7C22×C811yescharacteristic
171C2×Dic7C22×C811yescharacteristic
172C2×Dic7C22×C811yescharacteristic
173C2×Dic7C22×C812yes—
174C2×Dic7C22×C812yes—
175C2×Dic7C22×C811yescharacteristic
176Dic14—24no—
177Dic14—42no—
178C7×C8—22no—
179C7×C8C4×Q812yes—
180C7×C8C22×Q811yescharacteristic
181C7×C8C2×#4111yescharacteristic
182C7⋊C8—21no—
183C7⋊C8—21no—
184C7⋊C8—22no—
185C2×M16—71no—
186C2×M16C4×D711yescharacteristic
187C22×C8—71no—
188C22×C8C22×D711yescharacteristic
189C4⋊Q8—71no—
190C4⋊Q8—71no—
191#125—71no—
192#125—71no—
193C22⋊Q8—72no—
194C22⋊Q8—72no—
195C4×Q8—72no—
196C4×Q8—72no—
197#117—71no—
198#117C4×D711yescharacteristic
199C2×C42—71no—
200M32—142no—
201M32Dic1414yescomplemented
202C2×C16—22no—
203C2×C16—71no—
204C2×C16—72no—
205C2×C16—71no—
206C2×C16—142no—
207C2×C16C22×D711yescharacteristic
208C2×C16C22×D711yescharacteristic
209C2×C16C22×D712yes—
210C4⋊C8—72no—
211C4⋊C8—72no—
212C4⋊C8—72no—
213C4⋊C8—72no—
214C4⋊C8—72no—
215C4⋊C8—72no—
216#97—72no—
217#97C4×D712yes—
218C4×C8—71no—
219C4×C8—71no—
220C4×C8—71no—
221C4×C8—72no—
222C4×C8—71no—
223C4×C8C22×D712yes—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7C8×Q812yes—
227C22×C7C23×C811yescharacteristic
228C4×C7—22no—
229C4×C7—22no—
230C4×C7C8×Q812yes—
231C4×C7C2×C4×Q811yescharacteristic
232C4×C7C2×#13111yescharacteristic
233Dic7—22no—
234Dic7—21no—
235Dic7—21no—
236Dic7—22no—
237Dic7—21no—
238Dic7#50412yes—
239C2×Q8—72no—
240C2×Q8—72no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4C2×C4×D711yescharacteristic
244M16—22no—
245M16—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8—72no—
249C2×C8—142no—
250C2×C8C2×Dic1412yes—
251C2×C8C2×C4×D712yes—
252C2×C8C2×C4×D711yescharacteristic
253C2×C8#115712yes—
254C2×C8C23×D711yescharacteristic
255C4⋊C4—72no—
256C4⋊C4—72no—
257C4⋊C4—72no—
258C4⋊C4—72no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—72no—
262C4⋊C4—72no—
263C4⋊C4C8×D711yescharacteristic
264C4⋊C4C8×D711yescharacteristic
265C22⋊C4—72no—
266C22⋊C4C8×D712yes—
267C42—71no—
268C42—71no—
269C42—71no—
270C42—71no—
271C42—72no—
272C42C2×C4×D712yes—
273C16—21no—
274C16—21no—
275C16—141no—
276C16—141no—
277C16—142no—
278C16C2×Dic1414yes—
279C2×C7—22no—
280C2×C7C2×C8×Q811yescharacteristic
281C2×C7C2×#50411yescharacteristic
282C2×C7#340711yescharacteristic
283C23C2×C8×D711yescharacteristic
284Q8—144no—
285Q8—282no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—71no—
290C2×C4—72no—
291C2×C4—72no—
292C2×C4—142no—
293C2×C4—281no—
294C2×C4C4×Dic1412yes—
295C2×C4C2×C8×D712yes—
296C2×C4C2×C8×D712yes—
297C2×C4C2×C8×D711yescharacteristic
298C2×C4#646412yes—
299C2×C4C22×C4×D711yescharacteristic
300C8—22no—
301C8—141no—
302C8—141no—
303C8—142no—
304C8C4×Dic1412yes—
305C8C22×Dic1411yescharacteristic
306C8C2×#115711yescharacteristic
307C7#3347611yescharacteristic, complemented, Sylow
308C22—21no—
309C22—21no—
310C22C8×Dic1412yes—
311C22C22×C8×D711yescharacteristic
312C4—22no—
313C4—22no—
314C4—72no—
315C4—141no—
316C4—142no—
317C4—141no—
318C4—141no—
319C4—142no—
320C4C8×Dic1412yes—
321C4C2×C4×Dic1411yescharacteristic
322C4C2×#646411yescharacteristic
323C2—22no—
324C2C2×C8×Dic1411yescharacteristic
325C2D7×#50411yescharacteristic
326C2#1173945711yescharacteristic
subgroupG/H
center, FrattiniC2×C8C23×D7
commutator, socleC22×C7C23×C8
FittingC7×#2108C2
prime radicalC23×C7C22×C8
cosocleC2×C7×C8C24
hypercenter#2108D7
hypocenterC7#33476
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C8 ◁ #2108

Complements

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

NH ≅ G/Ntype
8. #11726822C2semidirect
9. #11726822C2semidirect
10. #11726802C2semidirect
11. C16×Dic14C2semidirect
57. C7×M32Q8semidirect
201. M32Dic14semidirect
307. C7#33476semidirect

Maximal subgroups

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

subgroupindex
#117359872
C7×#21082
#117291732
Dic7×M322
#117291552
C2×Dic7⋊C162
#117276362
#117268222
#117268022
C16×Dic142
#334767