Subgroups — #34053718

Order 19072.

456 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₁₄₉	C₂	1	1	yes	maximal, characteristic
2	#23749699	C₂	1	4	yes	maximal, complemented
3	#23749694	C₂	1	4	yes	maximal, complemented
4	#23749679	C₂	1	2	yes	maximal, complemented
5	#23749589	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₈×D₁₄₉	C₂	1	2	yes	maximal, complemented
7	#407×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₁₄₉	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
10	C₂×#14558720	C₂²	1	2	yes	complemented
11	C₂×C₈×D₁₄₉	C₂²	1	4	yes	complemented
12	C₄×C₂²⋊D₁₄₉	C₄	1	2	yes	—
13	C₄×C₂²⋊D₁₄₉	C₄	1	2	yes	—
14	C₄×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
16	C₂×#14558715	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₁₄₉	C₄	1	1	yes	characteristic
18	C₂×#14558713	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
20	#18011840	C₂²	1	4	yes	complemented
21	C₂²⋊#11924746	C₂²	1	2	yes	—
22	C₂²⋊#11924746	C₂²	1	2	yes	complemented
23	#18011823	C₂²	1	4	yes	complemented
24	#18011821	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	complemented
27	C₂²⋊C₈×C₁₄₉	C₂²	1	2	yes	—
28	C₂²⋊C₈×C₁₄₉	C₂²	1	2	yes	complemented
29	C₂³×D₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊D₁₄₉	—	2	4	no	—
31	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
33	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₄₉	—	4	2	no	—
37	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
39	#14558720	—	4	2	no	—
40	C₂×#11924746	C₂³	1	4	yes	—
41	C₈×D₁₄₉	—	4	4	no	—
42	#14558715	C₂×C₄	1	2	yes	—
43	#14558715	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
46	#14558713	C₂×C₄	1	2	yes	—
47	#14558713	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₄₉	C₂³	1	2	yes	—
50	C₂⁴×C₁₄₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₄₉	—	4	2	no	—
52	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₄₉	—	4	2	no	—
59	C₂×C₈×C₁₄₉	C₂³	1	4	yes	—
60	C₂²×D₁₄₉	—	4	1	no	—
61	C₂²×D₁₄₉	—	4	1	no	—
62	C₂²×D₁₄₉	M₁₆	1	4	yes	complemented
63	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
64	C₂²⋊D₁₄₉	—	2	4	no	—
65	C₂²⋊D₁₄₉	—	2	4	no	—
66	C₂×Dic₁₄₉	—	2	2	no	—
67	C₂×Dic₁₄₉	—	4	1	no	—
68	C₂×Dic₁₄₉	—	4	1	no	—
69	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
70	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
72	C₄×D₁₄₉	—	2	4	no	—
73	C₄×D₁₄₉	—	4	2	no	—
74	#11924746	—	2	4	no	—
75	C₂³×C₁₄₉	—	2	2	no	—
76	C₂³×C₁₄₉	—	4	1	no	—
77	C₂³×C₁₄₉	—	4	1	no	—
78	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
80	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₄₉	—	2	2	no	—
82	C₂×C₄×C₁₄₉	—	2	2	no	—
83	C₂×C₄×C₁₄₉	—	2	2	no	—
84	C₂×C₄×C₁₄₉	—	4	1	no	—
85	C₂×C₄×C₁₄₉	—	4	1	no	—
86	C₂×C₄×C₁₄₉	—	4	1	no	—
87	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₄₉	—	2	4	no	—
92	C₂²×C₁₄₉	—	2	1	no	—
93	C₂²×C₁₄₉	—	2	1	no	—
94	C₂²×C₁₄₉	—	2	1	no	—
95	C₂²×C₁₄₉	—	2	1	no	—
96	C₂²×C₁₄₉	—	2	1	no	—
97	C₂²×C₁₄₉	—	2	1	no	—
98	C₂²×C₁₄₉	—	2	1	no	—
99	C₂²×C₁₄₉	—	2	1	no	—
100	C₂²×C₁₄₉	—	4	4	no	—
101	C₂²×C₁₄₉	C₂×M₁₆	1	2	yes	—
102	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	2	2	no	—
105	C₂×D₁₄₉	—	4	4	no	—
106	C₂×D₁₄₉	C₂×M₁₆	1	4	yes	—
107	C₄×C₁₄₉	—	2	1	no	—
108	C₄×C₁₄₉	—	2	1	no	—
109	C₄×C₁₄₉	—	2	1	no	—
110	C₄×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
111	Dic₁₄₉	—	2	2	no	—
112	Dic₁₄₉	—	2	2	no	—
113	Dic₁₄₉	—	2	2	no	—
114	C₂×C₁₄₉	—	2	2	no	—
115	C₂×C₁₄₉	—	4	1	no	—
116	C₂×C₁₄₉	—	4	1	no	—
117	C₂×C₁₄₉	#518	1	1	yes	characteristic
118	C₂×C₁₄₉	C₂²×M₁₆	1	1	yes	characteristic
119	C₂×C₁₄₉	#568	1	1	yes	characteristic
120	D₁₄₉	—	2	4	no	—
121	C₁₄₉	#2973	1	1	yes	characteristic, complemented, Sylow
122	#2973	—	149	1	no	maximal, Sylow
123	C₂×C₄×D₄	—	149	1	no	—
124	#435	—	149	4	no	—
125	#435	—	149	4	no	—
126	#432	—	149	2	no	—
127	#407	—	149	1	no	—
128	#407	D₁₄₉	1	1	yes	characteristic, complemented
129	C₂×C₂²⋊C₈	—	149	2	no	—
130	C₂²×D₄	—	149	1	no	—
131	C₂³×C₄	—	149	1	no	—
132	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
133	C₂×M₁₆	—	149	2	no	—
134	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	complemented
135	C₂²×C₈	—	149	4	no	—
136	C₄×D₄	—	149	2	no	—
137	C₄×D₄	—	149	2	no	—
138	C₄×D₄	—	149	2	no	—
139	C₄×D₄	—	149	2	no	—
140	C₂×C₄⋊C₄	—	149	1	no	—
141	C₂×C₂²⋊C₄	—	149	1	no	—
142	C₂×C₂²⋊C₄	—	149	1	no	—
143	C₂×C₄²	—	149	1	no	—
144	C₄⋊C₈	—	149	4	no	—
145	C₂²⋊C₈	—	149	2	no	—
146	C₂²⋊C₈	—	149	2	no	—
147	C₂²⋊C₈	—	149	4	no	—
148	C₂²⋊C₈	C₂×D₁₄₉	1	2	yes	—
149	C₂²⋊C₈	C₂×D₁₄₉	1	2	yes	complemented
150	#97	—	149	4	no	—
151	C₂⁴	—	149	1	no	—
152	C₂⁴	C₄×D₁₄₉	1	1	yes	characteristic
153	C₂×D₄	—	298	4	no	—
154	C₂×D₄	—	149	2	no	—
155	C₂×D₄	—	149	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	596	2	no	—
158	C₂²×C₄	—	149	1	no	—
159	C₂²×C₄	—	149	2	no	—
160	C₂²×C₄	—	149	1	no	—
161	C₂²×C₄	—	149	1	no	—
162	C₂²×C₄	—	149	2	no	—
163	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
165	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
169	M₁₆	—	4	2	no	—
170	M₁₆	—	596	2	no	—
171	C₂×C₈	—	596	4	no	—
172	C₂×C₈	—	149	4	no	—
173	C₂×C₈	C₂²×D₁₄₉	1	4	yes	—
174	C₄⋊C₄	—	149	2	no	—
175	C₄⋊C₄	—	149	2	no	—
176	C₂²⋊C₄	—	149	2	no	—
177	C₂²⋊C₄	—	149	2	no	—
178	C₂²⋊C₄	—	149	2	no	—
179	C₂²⋊C₄	—	149	2	no	—
180	C₄²	—	149	2	no	—
181	C₄²	—	149	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	—	4	1	no	—
185	C₂³	—	596	1	no	—
186	C₂³	—	596	1	no	—
187	C₂³	—	149	2	no	—
188	C₂³	—	149	4	no	—
189	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
190	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
191	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
192	D₄	—	298	4	no	—
193	D₄	—	298	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	298	2	no	—
201	C₂×C₄	—	298	4	no	—
202	C₂×C₄	—	596	2	no	—
203	C₂×C₄	—	596	1	no	—
204	C₂×C₄	—	596	1	no	—
205	C₂×C₄	—	149	2	no	—
206	C₂×C₄	—	149	2	no	—
207	C₂×C₄	—	149	2	no	—
208	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	—	298	4	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	4	4	no	—
223	C₂²	—	298	2	no	—
224	C₂²	—	298	2	no	—
225	C₂²	—	596	4	no	—
226	C₂²	—	149	4	no	—
227	C₂²	M₁₆×D₁₄₉	1	2	yes	—
228	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	298	2	no	—
233	C₄	—	298	2	no	—
234	C₄	—	298	2	no	—
235	C₄	#18011956	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	1	no	—
238	C₂	—	4	1	no	—
239	C₂	—	298	4	no	—
240	C₂	#23750098	1	1	yes	characteristic
241	C₂	C₂×M₁₆×D₁₄₉	1	1	yes	characteristic
242	C₂	#23750391	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#407×C₁₄₉	C₂
prime radical	C₂²×C₂²⋊D₁₄₉	C₄
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#407	D₁₄₉
hypocenter	C₁₄₉	#2973
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #23749699	C₂	semidirect
3. #23749694	C₂	semidirect
4. #23749679	C₂	semidirect
5. #23749589	C₂	semidirect
6. C₂²⋊C₈×D₁₄₉	C₂	semidirect
7. #407×C₁₄₉	C₂	semidirect
10. C₂×#14558720	C₂²	semidirect
11. C₂×C₈×D₁₄₉	C₂²	semidirect
20. #18011840	C₂²	semidirect
22. C₂²⋊#11924746	C₂²	semidirect
23. #18011823	C₂²	semidirect
24. #18011821	C₂²	semidirect
26. C₂×M₁₆×C₁₄₉	C₂²	semidirect
28. C₂²⋊C₈×C₁₄₉	C₂²	semidirect
62. C₂²×D₁₄₉	M₁₆	semidirect
121. C₁₄₉	#2973	semidirect
128. #407	D₁₄₉	semidirect
134. C₂×M₁₆	C₂×D₁₄₉	semidirect
149. C₂²⋊C₈	C₂×D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₁₄₉	2
#23749699	2
#23749694	2
#23749679	2
#23749589	2
C₂²⋊C₈×D₁₄₉	2
#407×C₁₄₉	2
#2973	149