Subgroups — C₂×#23750199 #34058008

Order 19072.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14558736	C₂	1	1	yes	maximal, characteristic
2	#23750199	C₂	1	16	yes	maximal, direct product factor
3	C₂×#18011909	C₂	1	2	yes	maximal, complemented
4	C₂×#18011900	C₂	1	2	yes	maximal, complemented
5	C₂×#18011894	C₂	1	4	yes	maximal, complemented
6	C₂×#18011887	C₂	1	4	yes	maximal, complemented
7	C₂×#18011870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14558736	—	2	8	no	—
10	C₂×#14558736	C₂²	1	8	yes	—
11	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
12	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
15	#18011909	C₂²	1	16	yes	complemented
16	#18011900	C₂²	1	16	yes	complemented
17	#18011894	C₂²	1	32	yes	complemented
18	#18011887	C₂²	1	32	yes	complemented
19	#18011870	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
21	C₂×#14558715	C₂²	1	8	yes	—
22	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
23	C₂×#14558713	C₂²	1	8	yes	—
24	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₄₉	—	2	8	no	—
26	C₂²×Q₈×C₁₄₉	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₄₉	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₄₉	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
31	C₂³×D₁₄₉	C₂³	1	2	yes	—
32	#14558736	—	2	32	no	—
33	C₂×Dic₂₉₈	—	2	4	no	—
34	C₂×Dic₂₉₈	C₂³	1	4	yes	—
35	C₂×D₅₉₆	—	2	4	no	—
36	C₂×D₅₉₆	C₂³	1	4	yes	—
37	C₂×C₂²⋊D₁₄₉	—	2	8	no	—
38	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
39	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
40	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
41	C₂×C₄×D₁₄₉	—	2	8	no	—
42	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
43	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
44	#14558715	C₂³	1	32	yes	—
45	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
46	#14558713	C₂³	1	32	yes	—
47	C₄×Dic₁₄₉	C₂³	1	8	yes	—
48	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₄₉	—	2	8	no	—
50	C₂×Q₈×C₁₄₉	—	2	8	no	—
51	C₂×Q₈×C₁₄₉	—	2	32	no	—
52	C₂×Q₈×C₁₄₉	C₂³	1	8	yes	—
53	C₂×Q₈×C₁₄₉	C₂³	1	8	yes	—
54	C₂²×C₄×C₁₄₉	—	2	8	no	—
55	C₂²×C₄×C₁₄₉	D₄	1	8	yes	—
56	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₄₉	C₂³	1	8	yes	—
58	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
61	C₂²×D₁₄₉	—	2	4	no	—
62	C₂²×D₁₄₉	—	2	8	no	—
63	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
64	Dic₂₉₈	—	2	8	no	—
65	D₅₉₆	—	2	8	no	—
66	C₂²⋊D₁₄₉	—	2	16	no	—
67	C₂×Dic₁₄₉	—	2	4	no	—
68	C₂×Dic₁₄₉	—	2	8	no	—
69	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
70	C₂×Dic₁₄₉	C₂⁴	1	8	yes	—
71	C₄×D₁₄₉	—	2	16	no	—
72	C₂³×C₁₄₉	—	2	4	no	—
73	C₂³×C₁₄₉	C₂×D₄	1	4	yes	—
74	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
75	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₄₉	—	2	16	no	—
77	Q₈×C₁₄₉	—	2	16	no	—
78	C₂×C₄×C₁₄₉	—	2	16	no	—
79	C₂×C₄×C₁₄₉	—	2	8	no	—
80	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
82	C₂×C₄×C₁₄₉	C₂×D₄	1	16	yes	—
83	C₂×C₄×C₁₄₉	C₂⁴	1	8	yes	—
84	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
85	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
86	C₂²×C₁₄₉	—	2	8	no	—
87	C₂²×C₁₄₉	—	2	4	no	—
88	C₂²×C₁₄₉	C₂²×D₄	1	4	yes	—
89	C₂²×C₁₄₉	C₂²×D₄	1	2	yes	—
90	C₂²×C₁₄₉	2₋¹⁺⁴	1	4	yes	—
91	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₄₉	—	2	4	no	—
93	C₂×D₁₄₉	—	2	8	no	—
94	C₂×D₁₄₉	—	4	8	no	—
95	C₄×C₁₄₉	—	2	8	no	—
96	C₄×C₁₄₉	C₂²×D₄	1	4	yes	—
97	C₄×C₁₄₉	C₂²×D₄	1	4	yes	—
98	Dic₁₄₉	—	2	4	no	—
99	Dic₁₄₉	—	2	8	no	—
100	C₂×C₁₄₉	—	2	4	no	—
101	C₂×C₁₄₉	#536	1	4	yes	complemented
102	C₂×C₁₄₉	C₂³×D₄	1	1	yes	characteristic
103	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	2	yes	—
104	D₁₄₉	—	4	4	no	—
105	C₁₄₉	C₂×#536	1	1	yes	characteristic, complemented, Sylow
106	C₂×#536	—	149	1	no	maximal, Sylow
107	C₂²×#41	—	149	1	no	—
108	C₂³×Q₈	D₁₄₉	1	1	yes	characteristic, complemented
109	#536	—	149	16	no	—
110	C₂×C₄⋊Q₈	—	149	2	no	—
111	C₂×#124	—	149	2	no	—
112	C₂×#123	—	149	4	no	—
113	C₂×C₂²⋊Q₈	—	149	4	no	—
114	C₂×#117	—	149	1	no	—
115	C₂×#41	—	298	8	no	—
116	C₂×#41	—	149	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	—	149	1	no	—
119	C₂²×Q₈	C₂×D₁₄₉	1	2	yes	—
120	C₂²×Q₈	C₂×D₁₄₉	1	2	yes	—
121	C₂²×Q₈	C₂×D₁₄₉	1	8	yes	complemented
122	C₂²×D₄	—	149	2	no	—
123	C₂²×D₄	—	149	1	no	—
124	C₂³×C₄	—	149	2	no	—
125	C₂³×C₄	C₂×D₁₄₉	1	2	yes	—
126	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
127	C₄⋊Q₈	—	149	16	no	—
128	#124	—	149	16	no	—
129	#123	—	149	32	no	—
130	C₂²⋊Q₈	—	149	32	no	—
131	#117	—	149	8	no	—
132	C₂×C₄⋊C₄	—	149	2	no	—
133	C₂×C₄⋊C₄	—	149	8	no	—
134	C₂×C₂²⋊C₄	—	149	2	no	—
135	C₂×C₂²⋊C₄	—	149	8	no	—
136	C₂×C₄²	—	149	2	no	—
137	C₂⁴	—	149	2	no	—
138	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
139	#41	—	298	32	no	—
140	C₂×Q₈	—	2	8	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	2	32	no	—
143	C₂×Q₈	—	298	4	no	—
144	C₂×Q₈	—	149	4	no	—
145	C₂×Q₈	C₂²×D₁₄₉	1	8	yes	—
146	C₂×Q₈	C₂²×D₁₄₉	1	8	yes	—
147	C₂×D₄	—	298	8	no	—
148	C₂×D₄	—	298	4	no	—
149	C₂×D₄	—	149	8	no	—
150	C₂×D₄	—	149	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	298	8	no	—
153	C₂²×C₄	—	149	4	no	—
154	C₂²×C₄	—	149	2	no	—
155	C₂²×C₄	—	149	8	no	—
156	C₂²×C₄	C₂²⋊D₁₄₉	1	8	yes	—
157	C₂²×C₄	C₂²×D₁₄₉	1	8	yes	—
158	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
159	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
160	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
162	C₄⋊C₄	—	149	8	no	—
163	C₄⋊C₄	—	149	32	no	—
164	C₂²⋊C₄	—	149	8	no	—
165	C₂²⋊C₄	—	149	32	no	—
166	C₄²	—	149	8	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	298	4	no	—
169	C₂³	—	298	8	no	—
170	C₂³	—	149	4	no	—
171	C₂³	C₂×C₂²⋊D₁₄₉	1	4	yes	—
172	C₂³	C₂³×D₁₄₉	1	2	yes	—
173	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
174	Q₈	—	2	16	no	—
175	Q₈	—	2	16	no	—
176	Q₈	—	298	8	no	—
177	D₄	—	298	16	no	—
178	D₄	—	298	8	no	—
179	C₂×C₄	—	2	16	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	298	4	no	—
182	C₂×C₄	—	298	8	no	—
183	C₂×C₄	—	298	16	no	—
184	C₂×C₄	—	149	8	no	—
185	C₂×C₄	—	149	4	no	—
186	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	4	yes	—
188	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	16	yes	—
189	C₂×C₄	C₂³×D₁₄₉	1	8	yes	—
190	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
191	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	298	4	no	—
195	C₂²	—	298	8	no	—
196	C₂²	—	596	8	no	—
197	C₂²	C₂²×C₂²⋊D₁₄₉	1	4	yes	—
198	C₂²	C₂²×C₂²⋊D₁₄₉	1	2	yes	—
199	C₂²	#18011959	1	4	yes	—
200	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
201	C₄	—	2	8	no	—
202	C₄	—	298	4	no	—
203	C₄	—	298	8	no	—
204	C₄	C₂²×C₂²⋊D₁₄₉	1	4	yes	—
205	C₄	C₂²×C₂²⋊D₁₄₉	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	596	4	no	—
208	C₂	#23750199	1	4	yes	direct product factor
209	C₂	C₂³×C₂²⋊D₁₄₉	1	1	yes	characteristic
210	C₂	C₂×#18011959	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting	C₂³×Q₈×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂²×#14558736	C₂
hypercenter	C₂³×Q₈	D₁₄₉
hypocenter	C₁₄₉	C₂×#536
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. #23750199	C₂	direct (N × H)
3. C₂×#18011909	C₂	semidirect
4. C₂×#18011900	C₂	semidirect
5. C₂×#18011894	C₂	semidirect
6. C₂×#18011887	C₂	semidirect
7. C₂×#18011870	C₂	semidirect
8. C₂³×Q₈×C₁₄₉	C₂	semidirect
15. #18011909	C₂²	semidirect
16. #18011900	C₂²	semidirect
17. #18011894	C₂²	semidirect
18. #18011887	C₂²	semidirect
19. #18011870	C₂²	semidirect
28. C₂²×Q₈×C₁₄₉	C₂²	semidirect
101. C₂×C₁₄₉	#536	semidirect
105. C₁₄₉	C₂×#536	semidirect
108. C₂³×Q₈	D₁₄₉	semidirect
121. C₂²×Q₈	C₂×D₁₄₉	semidirect
208. C₂	#23750199	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×#14558736	2
#23750199	2
C₂×#18011909	2
C₂×#18011900	2
C₂×#18011894	2
C₂×#18011887	2
C₂×#18011870	2
C₂³×Q₈×C₁₄₉	2
C₂×#536	149

Subgroups — C2×#23750199 #34058008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750199 #34058008