Subgroups — C₄×#18011898 #34047233

Order 19072.

652 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011898	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23749530	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749528	C₂	1	1	yes	maximal, characteristic
5	#23749519	C₂	1	4	yes	maximal, complemented
6	#23749490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊Dic₁₄₉	C₂	1	4	yes	maximal, complemented
8	C₄²×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
11	#18011898	C₄	1	8	yes	direct product factor
12	C₄×Dic₂₉₈	—	2	4	no	—
13	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×#14558715	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
22	#18011818	C₂²	1	4	yes	—
23	#18011818	C₂²	1	4	yes	—
24	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	—
26	C₄×D₄×C₁₄₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
30	C₂×Dic₂₉₈	—	2	2	no	—
31	C₂×Dic₂₉₈	—	2	2	no	—
32	C₂×Dic₂₉₈	C₂×C₄	1	4	yes	complemented
33	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
34	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
35	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
36	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
37	C₄⋊Dic₁₄₉	—	2	2	no	—
38	#14558715	—	2	4	no	—
39	C₂²⋊Dic₁₄₉	—	2	8	no	—
40	C₂²⋊Dic₁₄₉	C₂×C₄	1	16	yes	complemented
41	C₄×Dic₁₄₉	—	2	4	no	—
42	C₄×Dic₁₄₉	—	2	4	no	—
43	C₄×Dic₁₄₉	—	2	4	no	—
44	C₄×Dic₁₄₉	—	2	2	no	—
45	C₄×Dic₁₄₉	C₂×C₄	1	4	yes	complemented
46	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
47	C₂×D₄×C₁₄₉	—	2	2	no	—
48	C₂×D₄×C₁₄₉	—	2	2	no	—
49	C₂×D₄×C₁₄₉	C₂×C₄	1	4	yes	—
50	C₂²×C₄×C₁₄₉	—	2	4	no	—
51	C₂²×C₄×C₁₄₉	—	2	8	no	—
52	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₄₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₄₉	—	2	4	no	—
58	C₄²×C₁₄₉	D₄	1	4	yes	—
59	Dic₂₉₈	—	2	8	no	—
60	C₂×Dic₁₄₉	—	2	2	no	—
61	C₂×Dic₁₄₉	—	2	2	no	—
62	C₂×Dic₁₄₉	—	2	2	no	—
63	C₂×Dic₁₄₉	—	2	2	no	—
64	C₂×Dic₁₄₉	—	2	2	no	—
65	C₂×Dic₁₄₉	—	2	2	no	—
66	C₂×Dic₁₄₉	—	2	2	no	—
67	C₂×Dic₁₄₉	—	2	4	no	—
68	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
69	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
70	C₂×Dic₁₄₉	#41	1	4	yes	—
71	C₂³×C₁₄₉	—	2	2	no	—
72	C₂³×C₁₄₉	—	2	4	no	—
73	C₂³×C₁₄₉	—	2	2	no	—
74	C₂³×C₁₄₉	—	2	4	no	—
75	C₂³×C₁₄₉	C₂²×C₄	1	4	yes	—
76	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
77	D₄×C₁₄₉	—	2	8	no	—
78	C₂×C₄×C₁₄₉	—	2	2	no	—
79	C₂×C₄×C₁₄₉	—	2	2	no	—
80	C₂×C₄×C₁₄₉	—	2	4	no	—
81	C₂×C₄×C₁₄₉	—	4	8	no	—
82	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₄₉	#41	1	4	yes	—
87	C₂×C₄×C₁₄₉	#41	1	2	yes	—
88	C₂²×C₁₄₉	—	2	4	no	—
89	C₂²×C₁₄₉	—	2	8	no	—
90	C₂²×C₁₄₉	—	4	2	no	—
91	C₂²×C₁₄₉	—	4	4	no	—
92	C₂²×C₁₄₉	—	4	2	no	—
93	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
96	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
97	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
98	C₄×C₁₄₉	—	2	2	no	—
99	C₄×C₁₄₉	—	4	2	no	—
100	C₄×C₁₄₉	C₄×D₄	1	4	yes	—
101	C₄×C₁₄₉	#124	1	4	yes	complemented
102	Dic₁₄₉	—	2	4	no	—
103	Dic₁₄₉	—	2	4	no	—
104	Dic₁₄₉	—	2	4	no	—
105	Dic₁₄₉	—	4	2	no	—
106	C₂×C₁₄₉	—	4	4	no	—
107	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₄₉	C₄×#41	1	2	yes	—
109	C₂×C₁₄₉	C₂×#124	1	1	yes	characteristic
110	C₂×C₁₄₉	#529	1	2	yes	—
111	C₂×C₁₄₉	#532	1	1	yes	characteristic
112	C₁₄₉	C₄×#124	1	1	yes	characteristic, complemented, Sylow
113	C₄×#124	—	149	1	no	maximal, Sylow
114	C₂×#124	—	149	1	no	—
115	C₂×C₄×Q₈	—	149	1	no	—
116	C₂×C₄×D₄	D₁₄₉	1	1	yes	characteristic
117	#391	—	149	1	no	—
118	#390	—	149	1	no	—
119	#388	—	149	4	no	—
120	#382	—	149	1	no	—
121	C₄×C₂²⋊C₄	—	149	4	no	—
122	C₄³	—	149	1	no	—
123	C₂²×Q₈	—	149	1	no	—
124	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₁₄₉	1	2	yes	—
126	#124	—	149	8	no	—
127	C₄×Q₈	—	298	4	no	—
128	C₄×D₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	149	1	no	—
130	C₂×C₄⋊C₄	—	149	2	no	—
131	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	—	149	4	no	—
133	C₂×C₂²⋊C₄	—	149	4	no	—
134	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
135	C₂×C₄²	—	149	2	no	—
136	C₂×C₄²	—	149	1	no	—
137	C₂×C₄²	—	149	2	no	—
138	C₂×C₄²	—	149	2	no	—
139	C₂×C₄²	—	149	1	no	—
140	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
141	#95	—	149	4	no	—
142	#95	—	149	4	no	—
143	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
144	C₂×Q₈	—	298	2	no	—
145	C₂×Q₈	—	298	2	no	—
146	C₂×Q₈	—	149	4	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₄×D₁₄₉	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	149	2	no	—
153	C₂²×C₄	—	149	2	no	—
154	C₂²×C₄	—	149	2	no	—
155	C₂²×C₄	—	149	2	no	—
156	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
157	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	298	4	no	—
162	C₄⋊C₄	—	298	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	298	8	no	—
165	C₂²⋊C₄	—	149	16	no	—
166	C₄²	—	298	4	no	—
167	C₄²	—	298	4	no	—
168	C₄²	—	298	4	no	—
169	C₄²	—	298	2	no	—
170	C₄²	—	149	4	no	—
171	C₄²	C₂²⋊D₁₄₉	1	4	yes	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	2	no	—
176	C₂³	C₂×C₄×D₁₄₉	1	4	yes	—
177	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
178	Q₈	—	298	8	no	—
179	D₄	—	2	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	298	2	no	—
185	C₂×C₄	—	298	2	no	—
186	C₂×C₄	—	298	2	no	—
187	C₂×C₄	—	298	2	no	—
188	C₂×C₄	—	298	2	no	—
189	C₂×C₄	—	298	2	no	—
190	C₂×C₄	—	298	2	no	—
191	C₂×C₄	—	298	4	no	—
192	C₂×C₄	—	149	4	no	—
193	C₂×C₄	—	149	4	no	—
194	C₂×C₄	—	149	4	no	—
195	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
196	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
199	C₂×C₄	#14558735	1	4	yes	—
200	C₂×C₄	#14558736	1	2	yes	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
207	C₂²	C₂²×C₂²⋊D₁₄₉	1	1	yes	characteristic
208	C₂²	#41×D₁₄₉	1	2	yes	—
209	C₂²	C₂×#14558735	1	2	yes	—
210	C₂²	C₂×#14558736	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	298	4	no	—
214	C₄	—	298	4	no	—
215	C₄	—	298	4	no	—
216	C₄	—	596	2	no	—
217	C₄	C₄×C₂²⋊D₁₄₉	1	4	yes	—
218	C₄	#18011898	1	4	yes	direct product factor
219	C₂	—	4	4	no	—
220	C₂	C₂×C₄×C₂²⋊D₁₄₉	1	1	yes	characteristic
221	C₂	C₄×#14558735	1	2	yes	—
222	C₂	C₂×#18011898	1	1	yes	characteristic
223	C₂	#23750157	1	2	yes	—
224	C₂	#23750171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Frattini	C₂³	C₂³×D₁₄₉
Fitting	C₂×C₄×D₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂²×D₄×C₁₄₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₄₉
hypocenter	C₁₄₉	C₄×#124
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₉₈	C₂	semidirect
3. #23749530	C₂	semidirect
5. #23749519	C₂	semidirect
6. #23749490	C₂	semidirect
7. C₄×C₂²⋊Dic₁₄₉	C₂	semidirect
8. C₄²×Dic₁₄₉	C₂	semidirect
11. #18011898	C₄	direct (N × H)
32. C₂×Dic₂₉₈	C₂×C₄	semidirect
40. C₂²⋊Dic₁₄₉	C₂×C₄	semidirect
45. C₄×Dic₁₄₉	C₂×C₄	semidirect
101. C₄×C₁₄₉	#124	semidirect
112. C₁₄₉	C₄×#124	semidirect
218. C₄	#18011898	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#18011898	2
C₂×C₄×Dic₂₉₈	2
#23749530	2
#23749528	2
#23749519	2
#23749490	2
C₄×C₂²⋊Dic₁₄₉	2
C₄²×Dic₁₄₉	2
C₂×C₄×D₄×C₁₄₉	2
C₄×#124	149

Subgroups — C4×#18011898 #34047233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#18011898 #34047233