Subgroups — C₂×#23750652 #34061393

Order 19072.

616 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₄₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750652	C₂	1	8	yes	maximal, direct product factor
4	C₂×#18011978	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#18011968	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊#11924753	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#18011871	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₄₉⋊C₈	—	2	4	no	—
10	C₂×D₁₄₉⋊C₈	C₂²	1	4	yes	complemented
11	C₂²×#11924753	—	2	4	no	—
12	C₂²×#11924753	C₂²	1	1	yes	characteristic
13	C₂²×#11924753	C₂²	1	1	yes	characteristic
14	C₂²×#11924753	C₂²	1	1	yes	characteristic
15	C₂²×#11924753	C₂²	1	1	yes	characteristic
16	C₂²×#11924753	C₂²	1	4	yes	complemented
17	#18011978	C₂²	1	4	yes	complemented
18	#18011968	C₂²	1	4	yes	complemented
19	C₂²⋊#11924753	C₂²	1	4	yes	complemented
20	C₄×#11924753	C₂²	1	4	yes	complemented
21	C₂²×C₂²⋊D₁₄₉	C₄	1	1	yes	characteristic
22	C₂³×Dic₁₄₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
24	#18011871	C₄	1	4	yes	—
25	#18011871	C₂²	1	4	yes	—
26	C₂×#14558715	C₄	1	1	yes	characteristic
27	C₂×#14558713	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₄₉	C₄	1	1	yes	characteristic
30	D₁₄₉⋊C₈	—	2	8	no	—
31	C₂×#11924753	—	2	8	no	—
32	C₂×#11924753	—	2	2	no	—
33	C₂×#11924753	—	2	2	no	—
34	C₂×#11924753	—	2	2	no	—
35	C₂×#11924753	D₄	1	4	yes	complemented
36	C₂×#11924753	C₂³	1	2	yes	—
37	C₂×#11924753	C₂³	1	2	yes	—
38	C₂×#11924753	C₂³	1	2	yes	—
39	C₂×#11924753	C₂³	1	2	yes	—
40	C₂³×D₁₄₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₄₉	C₈	1	8	yes	complemented
42	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	4	yes	—
43	C₂²×Dic₁₄₉	—	2	2	no	—
44	C₂²×Dic₁₄₉	—	2	2	no	—
45	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
47	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
49	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₄₉	—	2	2	no	—
51	C₂×C₄×D₁₄₉	—	2	2	no	—
52	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
54	#14558715	C₂×C₄	1	4	yes	—
55	#14558713	C₂×C₄	1	4	yes	—
56	C₄×Dic₁₄₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₄₉	C₂³	1	2	yes	—
58	C₂⁴×C₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	—
62	#11924753	—	2	2	no	—
63	#11924753	—	2	2	no	—
64	#11924753	—	2	2	no	—
65	#11924753	C₂×D₄	1	4	yes	complemented
66	C₂²×D₁₄₉	—	2	2	no	—
67	C₂²×D₁₄₉	—	2	2	no	—
68	C₂²×D₁₄₉	C₂×C₈	1	4	yes	complemented
69	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
70	C₂²⋊D₁₄₉	C₂×C₈	1	16	yes	complemented
71	C₂×Dic₁₄₉	—	2	2	no	—
72	C₂×Dic₁₄₉	—	2	4	no	—
73	C₂×Dic₁₄₉	—	2	2	no	—
74	C₂×Dic₁₄₉	—	2	2	no	—
75	C₂×Dic₁₄₉	C₂×C₈	1	4	yes	complemented
76	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₄₉	C₂×D₄	1	2	yes	—
80	C₂×Dic₁₄₉	#41	1	2	yes	—
81	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
82	C₄×D₁₄₉	—	2	4	no	—
83	C₄×D₁₄₉	—	2	2	no	—
84	C₄×D₁₄₉	—	2	2	no	—
85	C₂³×C₁₄₉	—	2	2	no	—
86	C₂³×C₁₄₉	—	2	2	no	—
87	C₂³×C₁₄₉	C₂×C₈	1	4	yes	complemented
88	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₄₉	—	2	2	no	—
91	C₂×C₄×C₁₄₉	—	2	2	no	—
92	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
94	C₂²×C₁₄₉	—	2	2	no	—
95	C₂²×C₁₄₉	—	2	2	no	—
96	C₂²×C₁₄₉	—	2	8	no	—
97	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
98	C₂²×C₁₄₉	C₂²×C₈	1	2	yes	—
99	C₂²×C₁₄₉	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₁₄₉	#131	1	2	yes	—
101	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₄₉	—	2	2	no	—
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	2	8	no	—
105	C₂×D₁₄₉	C₂²×C₈	1	4	yes	complemented
106	C₄×C₁₄₉	—	2	2	no	—
107	C₄×C₁₄₉	—	2	2	no	—
108	Dic₁₄₉	—	2	2	no	—
109	Dic₁₄₉	C₄×D₄	1	2	yes	—
110	Dic₁₄₉	C₂²×C₈	1	4	yes	complemented
111	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
112	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₁₄₉	—	2	4	no	—
114	C₂×C₁₄₉	C₈×D₄	1	4	yes	complemented
115	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×C₁₄₉	C₂³×C₈	1	1	yes	characteristic
117	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
118	D₁₄₉	—	2	4	no	—
119	C₁₄₉	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₈×D₄	—	149	1	no	maximal, Sylow
121	C₂³×C₈	—	149	1	no	—
122	C₂³×C₈	—	149	1	no	—
123	C₂×C₄×D₄	—	149	1	no	—
124	C₈×D₄	—	149	8	no	—
125	C₂×C₄⋊C₈	—	149	1	no	—
126	C₂×C₂²⋊C₈	—	149	1	no	—
127	C₂×C₂²⋊C₈	—	149	1	no	—
128	C₂×C₄×C₈	—	149	1	no	—
129	C₂²×D₄	—	149	1	no	—
130	C₂³×C₄	—	149	1	no	—
131	C₂³×C₄	—	149	1	no	—
132	C₂²×C₈	—	298	4	no	—
133	C₂²×C₈	—	298	4	no	—
134	C₂²×C₈	—	149	1	no	—
135	C₂²×C₈	—	149	1	no	—
136	C₂²×C₈	—	149	4	no	—
137	C₂²×C₈	—	149	4	no	—
138	C₂²×C₈	—	149	1	no	—
139	C₂²×C₈	—	149	1	no	—
140	C₄×D₄	—	149	4	no	—
141	C₄×D₄	—	149	4	no	—
142	C₂×C₄⋊C₄	—	149	1	no	—
143	C₂×C₂²⋊C₄	—	149	1	no	—
144	C₂×C₂²⋊C₄	#10597010	1	1	yes	characteristic
145	C₂×C₄²	—	149	1	no	—
146	C₄⋊C₈	—	149	4	no	—
147	C₂²⋊C₈	—	149	4	no	—
148	C₂²⋊C₈	—	149	4	no	—
149	C₄×C₈	—	149	4	no	—
150	C₂⁴	—	149	1	no	—
151	C₂⁴	C₂×#10597010	1	1	yes	characteristic
152	C₂×D₄	—	149	8	no	—
153	C₂×D₄	—	149	4	no	—
154	C₂²×C₄	—	298	2	no	—
155	C₂²×C₄	—	298	2	no	—
156	C₂²×C₄	—	298	2	no	—
157	C₂²×C₄	—	298	2	no	—
158	C₂²×C₄	—	149	2	no	—
159	C₂²×C₄	—	149	2	no	—
160	C₂²×C₄	—	149	1	no	—
161	C₂²×C₄	—	149	2	no	—
162	C₂²×C₄	—	149	2	no	—
163	C₂²×C₄	—	149	1	no	—
164	C₂²×C₄	—	149	1	no	—
165	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
166	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
167	C₂×C₈	—	298	2	no	—
168	C₂×C₈	—	298	2	no	—
169	C₂×C₈	—	298	2	no	—
170	C₂×C₈	—	298	8	no	—
171	C₂×C₈	—	298	8	no	—
172	C₂×C₈	—	149	2	no	—
173	C₂×C₈	—	149	2	no	—
174	C₂×C₈	—	149	2	no	—
175	C₂×C₈	—	149	4	no	—
176	C₂×C₈	—	149	2	no	—
177	C₄⋊C₄	—	149	4	no	—
178	C₂²⋊C₄	—	149	4	no	—
179	C₂²⋊C₄	C₂×#10597010	1	4	yes	—
180	C₄²	—	149	2	no	—
181	C₄²	—	149	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	298	2	no	—
185	C₂³	—	298	2	no	—
186	C₂³	—	149	4	no	—
187	C₂³	—	149	2	no	—
188	C₂³	D₁₄₉⋊C₈	1	4	yes	complemented
189	C₂³	C₂²×#10597010	1	2	yes	—
190	C₂³	C₂²×#10597010	1	1	yes	characteristic
191	D₄	—	149	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	298	2	no	—
195	C₂×C₄	—	298	2	no	—
196	C₂×C₄	—	298	4	no	—
197	C₂×C₄	—	298	2	no	—
198	C₂×C₄	—	298	2	no	—
199	C₂×C₄	—	298	2	no	—
200	C₂×C₄	—	298	4	no	—
201	C₂×C₄	—	149	2	no	—
202	C₂×C₄	—	149	2	no	—
203	C₂×C₄	—	149	2	no	—
204	C₂×C₄	—	149	4	no	—
205	C₂×C₄	—	149	2	no	—
206	C₂×C₄	—	149	1	no	—
207	C₂×C₄	—	149	1	no	—
208	C₂×C₄	C₂²×#10597010	1	2	yes	—
209	C₂×C₄	C₂²×#10597010	1	2	yes	—
210	C₈	—	298	2	no	—
211	C₈	—	298	2	no	—
212	C₈	—	298	2	no	—
213	C₈	—	149	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	298	2	no	—
218	C₂²	—	298	2	no	—
219	C₂²	—	298	8	no	—
220	C₂²	—	149	4	no	—
221	C₂²	D₄×#10597010	1	2	yes	—
222	C₂²	C₂×D₁₄₉⋊C₈	1	2	yes	—
223	C₂²	C₂×D₁₄₉⋊C₈	1	4	yes	complemented
224	C₂²	#18011998	1	2	yes	—
225	C₂²	C₂³×#10597010	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	298	2	no	—
229	C₄	—	149	1	no	—
230	C₄	—	149	1	no	—
231	C₄	—	149	2	no	—
232	C₄	—	149	4	no	—
233	C₂	—	2	4	no	—
234	C₂	—	298	4	no	—
235	C₂	#23750652	1	4	yes	direct product factor
236	C₂	C₂×D₄×#10597010	1	1	yes	characteristic
237	C₂	C₂²×D₁₄₉⋊C₈	1	1	yes	characteristic
238	C₂	C₂×#18011998	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#10597010
commutator	C₂×C₁₄₉	C₂³×C₈
Frattini	C₂²	C₂³×#10597010
Fitting	C₂×C₂²⋊C₄×C₁₄₉	C₄
socle	C₂³×C₁₄₉	C₂²×C₄
prime radical	C₂²×C₂²⋊D₁₄₉	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#10597010
hypocenter	C₁₄₉	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₄₉⋊C₈	C₂	semidirect
2. C₂³×#11924753	C₂	semidirect
3. #23750652	C₂	direct (N × H)
4. C₂×#18011978	C₂	semidirect
5. C₂×#18011968	C₂	semidirect
6. C₂×C₂²⋊#11924753	C₂	semidirect
7. C₂×C₄×#11924753	C₂	semidirect
10. C₂×D₁₄₉⋊C₈	C₂²	semidirect
16. C₂²×#11924753	C₂²	semidirect
17. #18011978	C₂²	semidirect
18. #18011968	C₂²	semidirect
19. C₂²⋊#11924753	C₂²	semidirect
20. C₄×#11924753	C₂²	semidirect
35. C₂×#11924753	D₄	semidirect
41. C₂×C₂²⋊D₁₄₉	C₈	semidirect
65. #11924753	C₂×D₄	semidirect
68. C₂²×D₁₄₉	C₂×C₈	semidirect
70. C₂²⋊D₁₄₉	C₂×C₈	semidirect
75. C₂×Dic₁₄₉	C₂×C₈	semidirect
87. C₂³×C₁₄₉	C₂×C₈	semidirect
99. C₂²×C₁₄₉	C₂²×C₈	semidirect
105. C₂×D₁₄₉	C₂²×C₈	semidirect
110. Dic₁₄₉	C₂²×C₈	semidirect
114. C₂×C₁₄₉	C₈×D₄	semidirect
119. C₁₄₉	C₂×C₈×D₄	semidirect
188. C₂³	D₁₄₉⋊C₈	semidirect
223. C₂²	C₂×D₁₄₉⋊C₈	semidirect
235. C₂	#23750652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₄₉⋊C₈	2
C₂³×#11924753	2
#23750652	2
C₂×#18011978	2
C₂×#18011968	2
C₂×C₂²⋊#11924753	2
C₂×C₄×#11924753	2
C₂×#18011871	2
C₂×C₈×D₄	149

Subgroups — C2×#23750652 #34061393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750652 #34061393