Finite Group Databasebeta

← #13263014

Subgroups — #13263014

Order 1792.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#11736515C211yesmaximal, characteristic
2#11726847C211yesmaximal, characteristic, complemented
3#11726846C211yesmaximal, characteristic, complemented
4C7×#1323C211yesmaximal, characteristic, complemented
5#11725524C211yesmaximal, characteristic, complemented
6#11725329C211yesmaximal, characteristic
7#11725329C211yesmaximal, characteristic
8#90213C411yescharacteristic
9C7×#424C2211yescharacteristic
10#89901C411yescharacteristic
11M16×Dic7C411yescharacteristic
12#89882C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14#89771C411yescharacteristic
15#89612—21no—
16#89612—21no—
17#89605—21no—
18#89605—21no—
19C7×C22⋊C16C2211yescharacteristic
20C7×C22⋊C16C2211yescharacteristic
21C22⋊(C7⋊C16)C2211yescharacteristic
22C22⋊(C7⋊C16)C2211yescharacteristic
23#89297—21no—
24Dic7⋊C16—21no—
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11464—71nomaximal, Sylow
28C2×#1157C2×C411yescharacteristic
29C2×C7×M16C2×C411yescharacteristic
30C22×C7×C8C2311yescharacteristic
31C7×#117C2×C411yescharacteristic
32C2×C4×Dic7C2×C411yescharacteristic
33C2×#1136C2×C411yescharacteristic
34C22×C7⋊C8C2×C411yescharacteristic
35#6456C811yescharacteristic
36#6456C811yescharacteristic
37#6450C811yescharacteristic
38#6450C811yescharacteristic
39(C22⋊D7)⋊C4—21no—
40(C22⋊D7)⋊C4—21no—
41#6435—21no—
42C42×D7—21no—
43C2×C7×C16—21no—
44C2×C7×C16—21no—
45C7×C4⋊C8—21no—
46C7×C4⋊C8—21no—
47(C2×D7)⋊C8—21no—
48(C2×D7)⋊C8—21no—
49(C2×D7)⋊C8—21no—
50(C2×D7)⋊C8—21no—
51#6389—21no—
52C8×Dic7—21no—
53C8×Dic7D411yescharacteristic
54C8×Dic7D411yescharacteristic
55C8×Dic7D411yescharacteristic
56C8×Dic7D411yescharacteristic
57C2×C7⋊C16—21no—
58C2×C7⋊C16—21no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61#2922—71no—
62#2163—71no—
63#2162—71no—
64#1323—71no—
65#1323D711yescharacteristic, complemented
66#1315—71no—
67#1315—71no—
68C22×C4×C7C22×C411yescharacteristic
69C2×C22⋊D7—21no—
70C22×Dic7M1611yescharacteristic
71C22×Dic7M1611yescharacteristic
72#1157—41no—
73#1157—41no—
74C2×D28C2×C811yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×Dic14C2×C811yescharacteristic
79C7×M16—41no—
80C2×C7×C8—21no—
81C2×C7×C8—41no—
82C2×C7×C8C2×D411yescharacteristic
83C2×C7×C8C2×D411yescharacteristic
84C7×C4⋊C4C2×C811yescharacteristic
85C7×C4⋊C4C2×C811yescharacteristic
86C7×C22⋊C4—21no—
87C42×C7—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92C4×Dic7—21no—
93C4×Dic7—21no—
94C4×Dic7C22⋊C411yescharacteristic
95C4×Dic7C22⋊C411yescharacteristic
96C4×Dic7C2×C811yescharacteristic
97C4×Dic7C2×C811yescharacteristic
98#1136—41no—
99C2×C7⋊C8—21no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8C22⋊C411yescharacteristic
102C2×C7⋊C8C22⋊C411yescharacteristic
103C7×C16—41no—
104C7×C16—41no—
105C7⋊C16—41no—
106C7⋊C16—41no—
107C4×#41—71no—
108#433—141no—
109#433—141no—
110#431—141no—
111#431—141no—
112#424—71no—
113#424C2×D711yescharacteristic
114#408—71no—
115#408—71no—
116C4×M16—71no—
117C2×C4×C8—71no—
118C4⋊C16—141no—
119C4⋊C16—141no—
120C22⋊C16—71no—
121C22⋊C16—71no—
122C22⋊C16C2×D711yescharacteristic
123C22⋊C16C2×D711yescharacteristic
124#346—141no—
125C4×C16—141no—
126C23×C7C2×M1611yescharacteristic
127C22×D7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C2×C22⋊C411yescharacteristic
133C2×C4×C7C22×C811yescharacteristic
134C22⋊D7—42no—
135C2×Dic7—22no—
136C2×Dic7—21no—
137C2×Dic7—41no—
138C2×Dic7C22⋊C811yescharacteristic
139C2×Dic7C22⋊C811yescharacteristic
140C2×Dic7C22⋊C811yescharacteristic
141C2×Dic7C22⋊C811yescharacteristic
142D28—41no—
143C4×D7—42no—
144C4×D7—41no—
145C4×D7—41no—
146C4×D7—42no—
147Dic14—41no—
148C7×C8—21no—
149C7×C8—21no—
150C7×C8—41no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8—41no—
154C2×#41—71no—
155C2×M16—71no—
156C2×M16C4×D711yescharacteristic
157C22×C8—71no—
158C22×C8C22×D711yescharacteristic
159C4×Q8—71no—
160C4×Q8—71no—
161C4×D4—71no—
162C4×D4—71no—
163C4×D4—141no—
164C4×D4—141no—
165#117—141no—
166#117C4×D711yescharacteristic
167C2×C42—71no—
168C2×C42—141no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—141no—
172C2×C16—141no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—141no—
176C4⋊C8—141no—
177C22⋊C8—141no—
178C22⋊C8—141no—
179C22⋊C8—141no—
180C22⋊C8—141no—
181#97—141no—
182C4×C8—71no—
183C4×C8—71no—
184C4×C8—71no—
185C4×C8—71no—
186C4×C8—141no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7C2×C22⋊C811yescharacteristic
191C2×D7—41no—
192C2×D7—41no—
193C2×D7—41no—
194C4×C7—21no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7#41011yescharacteristic
198C4×C7#41311yescharacteristic
199Dic7—21no—
200Dic7—21no—
201Dic7—21no—
202Dic7—41no—
203Dic7#35012yes—
204#41—281no—
205#41—281no—
206C2×Q8—71no—
207C2×D4—71no—
208C2×D4—141no—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—141no—
212C22×C4—141no—
213C22×C4—141no—
214C22×C4C2×C4×D711yescharacteristic
215M16—41no—
216M16—281no—
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—71no—
220C2×C8—71no—
221C2×C8—141no—
222C2×C8—281no—
223C2×C8D4×D711yescharacteristic
224C2×C8D4×D711yescharacteristic
225C4⋊C4—141no—
226C4⋊C4—141no—
227C4⋊C4C8×D711yescharacteristic
228C4⋊C4C8×D711yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—141no—
231C22⋊C4—141no—
232C42—21no—
233C42—71no—
234C42—71no—
235C42—71no—
236C42—71no—
237C42—141no—
238C42—141no—
239C16—41no—
240C16—41no—
241C16—281no—
242C16—281no—
243C2×C7—41no—
244C2×C7#145811yescharacteristic
245C2×C7C2×#35011yescharacteristic
246C2×C7#211711yescharacteristic
247D7—81no—
248C23—141no—
249C23D7×M1611yescharacteristic
250Q8—281no—
251D4—281no—
252D4—282no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—71no—
258C2×C4—71no—
259C2×C4—71no—
260C2×C4—71no—
261C2×C4—142no—
262C2×C4—141no—
263C2×C4—282no—
264C2×C4—282no—
265C2×C4—281no—
266C2×C4—281no—
267C2×C4—281no—
268C2×C4D7×C22⋊C411yescharacteristic
269C2×C4C2×C8×D711yescharacteristic
270C8—21no—
271C8—21no—
272C8—41no—
273C8—141no—
274C8—141no—
275C8—281no—
276C7#1146411yescharacteristic, complemented, Sylow
277C22—21no—
278C22—21no—
279C22—21no—
280C22—281no—
281C22—281no—
282C22—281no—
283C22D7×C22⋊C811yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—72no—
288C4—141no—
289C4—141no—
290C4—141no—
291C4—281no—
292C4#8951211yescharacteristic
293C4#8952311yescharacteristic
294C2—41no—
295C2—561no—
296C2#1172587811yescharacteristic
297C2D7×#35011yescharacteristic
298C2#1172687911yescharacteristic
subgroupG/H
centerC2×C4D7×C22⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC22×C8C22×D7
FittingC7×#1323C2
socleC22×C7C2×C22⋊C8
prime radicalC2×#1157C2×C4
cosocleC22×C7×C8C23
hypercenter#1323D7
hypocenterC7#11464
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #11726847C2semidirect
3. #11726846C2semidirect
4. C7×#1323C2semidirect
5. #11725524C2semidirect
65. #1323D7semidirect
276. C7#11464semidirect

Maximal subgroups

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

subgroupindex
#117365152
#117268472
#117268462
C7×#13232
#117255242
#117253292
#114647