Finite Group Databasebeta

← #34057722

Subgroups — #34057722

Order 19072.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#23750354C211yesmaximal, characteristic, complemented
2#23750020C212yesmaximal
3#23750004C211yesmaximal, characteristic, complemented
4#23749844C212yesmaximal
5#23749835C212yesmaximal, complemented
6#23749832C211yesmaximal, characteristic, complemented
7#23749830C211yesmaximal, characteristic
8#23749756C211yesmaximal, characteristic, complemented
9#23749730C212yesmaximal, complemented
10C4×#14558724C211yesmaximal, characteristic
11#556×C149C211yesmaximal, characteristic
12C2×#14558726C2212yes—
13C2×#14558726C2212yes—
14C2×#14558726C2212yes—
15C2×#14558724C2211yescharacteristic
16C2×#14558724C2212yes—
17#18011911C2211yescharacteristic
18#18011911C2212yes—
19#18011902—21no—
20#18011902—22no—
21Q8×Dic149—21no—
22C4×Dic298C2211yescharacteristic
23C4×Dic298C2212yes—
24#18011842C2211yescharacteristic
25C4⋊#11924746C2211yescharacteristic
26C4⋊#11924746C2212yes—
27#18011835C2212yes—
28#18011834C2211yescharacteristic
29#18011834C2212yes—
30#18011832C2211yescharacteristic
31#18011822C2212yes—
32C4×#11924746C2211yescharacteristic
33C4⋊Q8×C149C2211yescharacteristic
34#126×C149—22no—
35#124×C149C2212yes—
36C22⋊Q8×C149—21no—
37C4×Q8×C149C2212yes—
38C4×Q8×C149C2211yescharacteristic
39C4×D4×C149C2211yescharacteristic
40#117×C149—21no—
41C2×Dic298C2311yescharacteristic
42C2×Dic298C2312yes—
43#14558726—24no—
44#14558726—42no—
45#14558726—42no—
46#14558724—22no—
47#14558724—42no—
48C2×#11924746C2312yes—
49C2×#11924746C2311yescharacteristic
50C2×#11924746C2311yescharacteristic
51C4⋊Dic149—22no—
52C4⋊Dic149—21no—
53C4⋊Dic149—22no—
54C4⋊Dic149—21no—
55C4⋊Dic149C2311yescharacteristic
56#14558715—22no—
57#14558715—21no—
58C4×Dic149—22no—
59C4×Dic149—21no—
60C2×Q8×C149D411yescharacteristic
61C2×Q8×C149C2312yes—
62C2×D4×C149C2311yescharacteristic
63C22×C4×C149—21no—
64C4⋊C4×C149—21no—
65C4⋊C4×C149—22no—
66C4⋊C4×C149—21no—
67C4⋊C4×C149D411yescharacteristic
68C4⋊C4×C149D412yes—
69C4⋊C4×C149C2311yescharacteristic
70C22⋊C4×C149—22no—
71C22⋊C4×C149—21no—
72C22⋊C4×C149—22no—
73C42×C149—22no—
74C42×C149C2311yescharacteristic
75C42×C149C2312yes—
76Dic298—21no—
77Dic298—22no—
78Dic298—22no—
79Dic298—21no—
80C2×Dic149—21no—
81C2×Dic149—22no—
82C2×Dic149—21no—
83#11924746—22no—
84#11924746—21no—
85#11924746#4112yes—
86C23×C149—21no—
87Q8×C149—22no—
88Q8×C149—22no—
89Q8×C149—22no—
90D4×C149—21no—
91D4×C149—21no—
92C2×C4×C149—21no—
93C2×C4×C149—22no—
94C2×C4×C149—42no—
95C2×C4×C149C2×D412yes—
96C2×C4×C149C2×D411yescharacteristic
97C2×C4×C149C2×D412yes—
98C2×C4×C149C2×D411yescharacteristic
99C2×C4×C149C2411yescharacteristic
100C22×C149—41no—
101C22×C149—41no—
102C22×C149—41no—
103C22×C149C22×D411yescharacteristic
104C4×C149—22no—
105C4×C149—22no—
106C4×C149—21no—
107C4×C149—42no—
108C4×C149—41no—
109C4×C149#13712yes—
110C4×C149C2×#4111yescharacteristic
111C4×C149H(5,𝔽2)11yescharacteristic
112Dic149—41no—
113Dic149—42no—
114Dic149—41no—
115C2×C149—81no—
116C2×C149#55011yescharacteristic
117C2×C149C2×#13711yescharacteristic
118C2×C149#57811yescharacteristic
119C149#339811yescharacteristic, complemented, Sylow
120#3398—1491nomaximal, Sylow
121#557—1491no—
122#556D14911yescharacteristic
123#497—1492no—
124#494—1491no—
125#464—1492no—
126#462—1491no—
127#462—1492no—
128#461—1491no—
129#446—1491no—
130#441—1492no—
131C4×SD16—1491no—
132C2×Q16—1492no—
133C2×Q16—1492no—
134C2×Q16—1492no—
135C2×SD16—1491no—
136C2×SD16—1492no—
137C4⋊Q8—1491no—
138C4⋊Q8—1492no—
139C4⋊Q8C2×D14911yescharacteristic
140#126—22no—
141#125—2981no—
142#125—2982no—
143#124C2×D14912yes—
144C22⋊Q8—21no—
145C4×Q8—2981no—
146C4×Q8—1491no—
147C4×Q8—1492no—
148C4×Q8C2×D14912yes—
149C4×Q8C2×D14911yescharacteristic
150C4×D4C2×D14911yescharacteristic
151#117—21no—
152#106—1491no—
153C4⋊C8—1491no—
154C4⋊C8—1492no—
155#103—1492no—
156#103—1491no—
157#103—1492no—
158#102—1491no—
159#97—1492no—
160C4×C8—1491no—
161C2×Q8—1491no—
162C2×Q8—1492no—
163C2×Q8C22⋊D14911yescharacteristic
164C2×Q8C22×D14912yes—
165C2×D4C22×D14911yescharacteristic
166C22×C4—21no—
167Q16—2984no—
168Q16—5962no—
169Q16—5962no—
170SD16—2982no—
171SD16—5962no—
172C2×C8—1491no—
173C2×C8—1492no—
174C2×C8—1491no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—2982no—
179C4⋊C4—2981no—
180C4⋊C4—2982no—
181C4⋊C4—2981no—
182C4⋊C4—2982no—
183C4⋊C4—2981no—
184C4⋊C4—1491no—
185C4⋊C4C22⋊D14911yescharacteristic
186C4⋊C4C22⋊D14912yes—
187C4⋊C4C22×D14911yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—22no—
192C42—2982no—
193C42—2981no—
194C42C22×D14911yescharacteristic
195C42C22×D14912yes—
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—2981no—
201Q8—2982no—
202Q8—2982no—
203Q8—2981no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2981no—
210C2×C4—2981no—
211C2×C4—2982no—
212C2×C4C2×C22⋊D14912yes—
213C2×C4C2×C22⋊D14912yes—
214C2×C4C2×C22⋊D14911yescharacteristic
215C2×C4C2×C22⋊D14911yescharacteristic
216C2×C4C23×D14911yescharacteristic
217C8—2982no—
218C8—2981no—
219C8—1492no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C22×C22⋊D14911yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—5961no—
230C4—5961no—
231C4—5962no—
232C4#1801194112yes—
233C4#41×D14911yescharacteristic
234C4#1801195711yescharacteristic
235C2—81no—
236C2#2375029411yescharacteristic
237C2C2×#1801194111yescharacteristic
238C2#2375044011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#556×C149C2
socleC22×C149C22×D4
prime radicalC2×D4×C149C23
hypercenter#556D149
hypocenterC149#3398
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23750354C2semidirect
3. #23750004C2semidirect
5. #23749835C2semidirect
6. #23749832C2semidirect
8. #23749756C2semidirect
9. #23749730C2semidirect
119. C149#3398semidirect

Maximal subgroups

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

subgroupindex
#237503542
#237500202
#237500042
#237498442
#237498352
#237498322
#237498302
#237497562
#237497302
C4×#145587242
#556×C1492
#3398149