Subgroups — C₂×D₄×Dic₂₉₈ #34058165

Order 19072.

1568 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₉₈	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₂₉₈	C₂	1	16	yes	maximal, direct product factor
3	C₂×#18011911	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#18011910	C₂	1	2	yes	maximal, complemented
5	C₂×#18011890	C₂	1	2	yes	maximal, complemented
6	C₂×#18011888	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×Dic₁₄₉	C₂	1	2	yes	maximal
9	C₂×C₄×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₉₈	—	2	16	no	—
11	C₂²×Dic₂₉₈	C₂²	1	4	yes	—
12	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
13	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₉₈	C₂²	1	16	yes	complemented
15	C₂³×Dic₁₄₉	C₂²	1	4	yes	—
16	#18011911	C₂²	1	8	yes	complemented
17	#18011910	C₂²	1	16	yes	complemented
18	#18011890	C₂²	1	16	yes	complemented
19	#18011888	C₂²	1	32	yes	complemented
20	C₄×Dic₂₉₈	C₂²	1	8	yes	complemented
21	D₄×Dic₁₄₉	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
25	C₂×#14558715	C₂²	1	4	yes	—
26	C₂×#14558715	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	—
31	C₄×D₄×C₁₄₉	C₂²	1	8	yes	—
32	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₉₈	—	2	64	no	—
36	C₂×Dic₂₉₈	—	2	16	no	—
37	C₂×Dic₂₉₈	—	2	8	no	—
38	C₂×Dic₂₉₈	D₄	1	8	yes	direct product factor
39	C₂×Dic₂₉₈	C₂³	1	16	yes	—
40	C₂×Dic₂₉₈	C₂³	1	8	yes	—
41	C₂×Dic₂₉₈	C₂³	1	4	yes	—
42	C₂²×Dic₁₄₉	—	2	16	no	—
43	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
44	C₂²×Dic₁₄₉	C₂³	1	16	yes	—
45	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
46	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
47	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
48	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
49	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
50	#14558715	C₂³	1	16	yes	—
51	#14558715	C₂³	1	8	yes	—
52	C₂²⋊Dic₁₄₉	C₂³	1	16	yes	—
53	C₄×Dic₁₄₉	C₂³	1	8	yes	—
54	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
55	C₂×D₄×C₁₄₉	Q₈	1	8	yes	complemented
56	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₄₉	—	2	8	no	—
58	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₄₉	C₂³	1	8	yes	—
60	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
64	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
65	C₄²×C₁₄₉	C₂³	1	4	yes	—
66	Dic₂₉₈	—	2	16	no	—
67	Dic₂₉₈	—	2	32	no	—
68	Dic₂₉₈	C₂×D₄	1	16	yes	direct product factor
69	C₂×Dic₁₄₉	—	2	8	no	—
70	C₂×Dic₁₄₉	—	2	4	no	—
71	C₂×Dic₁₄₉	—	2	32	no	—
72	C₂×Dic₁₄₉	C₂×D₄	1	8	yes	—
73	C₂×Dic₁₄₉	C₂⁴	1	8	yes	—
74	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
75	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
76	C₂³×C₁₄₉	—	2	4	no	—
77	C₂³×C₁₄₉	—	2	4	no	—
78	C₂³×C₁₄₉	C₂×Q₈	1	8	yes	complemented
79	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
80	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
81	D₄×C₁₄₉	C₂×Q₈	1	16	yes	complemented
82	C₂×C₄×C₁₄₉	—	2	4	no	—
83	C₂×C₄×C₁₄₉	—	2	2	no	—
84	C₂×C₄×C₁₄₉	—	2	16	no	—
85	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₄₉	C₂×Q₈	1	4	yes	complemented
87	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
88	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
89	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
91	C₂²×C₁₄₉	—	2	16	no	—
92	C₂²×C₁₄₉	—	2	4	no	—
93	C₂²×C₁₄₉	—	2	4	no	—
94	C₂²×C₁₄₉	C₂²×D₄	1	2	yes	—
95	C₂²×C₁₄₉	C₂²×Q₈	1	2	yes	—
96	C₂²×C₁₄₉	C₂²×Q₈	1	8	yes	complemented
97	C₂²×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
98	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
99	C₄×C₁₄₉	—	2	4	no	—
100	C₄×C₁₄₉	—	2	2	no	—
101	C₄×C₁₄₉	C₂²×D₄	1	4	yes	—
102	C₄×C₁₄₉	C₂²×Q₈	1	4	yes	complemented
103	Dic₁₄₉	—	2	8	no	—
104	Dic₁₄₉	—	2	4	no	—
105	Dic₁₄₉	C₂²×D₄	1	8	yes	—
106	C₂×C₁₄₉	—	2	8	no	—
107	C₂×C₁₄₉	D₄×Q₈	1	4	yes	complemented
108	C₂×C₁₄₉	C₂³×D₄	1	1	yes	characteristic
109	C₂×C₁₄₉	C₂³×Q₈	1	1	yes	characteristic
110	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
111	C₁₄₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
112	C₂×D₄×Q₈	—	149	1	no	maximal, Sylow
113	C₂³×Q₈	—	149	2	no	—
114	D₄×Q₈	—	149	16	no	—
115	C₂×C₄⋊Q₈	—	149	2	no	—
116	C₂×C₄⋊Q₈	—	149	1	no	—
117	C₂×C₂²⋊Q₈	—	149	2	no	—
118	C₂×C₂²⋊Q₈	—	149	4	no	—
119	C₂×C₄×Q₈	—	149	1	no	—
120	C₂×C₄×D₄	—	149	2	no	—
121	C₂×C₄×D₄	D₁₄₉	1	1	yes	characteristic
122	C₂²×Q₈	—	298	16	no	—
123	C₂²×Q₈	—	149	2	no	—
124	C₂²×Q₈	—	149	4	no	—
125	C₂²×Q₈	—	149	16	no	—
126	C₂²×Q₈	—	149	1	no	—
127	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
128	C₂³×C₄	—	149	4	no	—
129	C₂³×C₄	C₂×D₁₄₉	1	2	yes	—
130	C₄⋊Q₈	—	149	16	no	—
131	C₄⋊Q₈	—	149	8	no	—
132	C₂²⋊Q₈	—	149	16	no	—
133	C₂²⋊Q₈	—	149	32	no	—
134	C₄×Q₈	—	149	8	no	—
135	C₄×D₄	—	149	16	no	—
136	C₄×D₄	C₂×D₁₄₉	1	8	yes	—
137	C₂×C₄⋊C₄	—	149	2	no	—
138	C₂×C₄⋊C₄	—	149	2	no	—
139	C₂×C₄⋊C₄	—	149	1	no	—
140	C₂×C₄⋊C₄	—	149	4	no	—
141	C₂×C₄⋊C₄	—	149	2	no	—
142	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	149	4	no	—
144	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
145	C₂×C₄²	—	149	2	no	—
146	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
147	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
148	C₂×Q₈	—	298	64	no	—
149	C₂×Q₈	—	298	16	no	—
150	C₂×Q₈	—	298	8	no	—
151	C₂×Q₈	—	149	8	no	—
152	C₂×Q₈	—	149	8	no	—
153	C₂×Q₈	—	149	16	no	—
154	C₂×Q₈	—	149	4	no	—
155	C₂×D₄	Dic₂₉₈	1	8	yes	direct product factor
156	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	298	16	no	—
159	C₂²×C₄	—	149	16	no	—
160	C₂²×C₄	—	149	2	no	—
161	C₂²×C₄	—	149	4	no	—
162	C₂²×C₄	—	149	2	no	—
163	C₂²×C₄	C₂²×D₁₄₉	1	8	yes	—
164	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
168	C₄⋊C₄	—	149	8	no	—
169	C₄⋊C₄	—	149	8	no	—
170	C₄⋊C₄	—	149	4	no	—
171	C₄⋊C₄	—	149	16	no	—
172	C₄⋊C₄	—	149	8	no	—
173	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
174	C₂²⋊C₄	—	149	16	no	—
175	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
176	C₄²	—	149	8	no	—
177	C₄²	C₂²×D₁₄₉	1	4	yes	—
178	C₂³	—	2	4	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×Dic₂₉₈	1	8	yes	complemented
181	C₂³	C₂³×D₁₄₉	1	4	yes	—
182	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
183	Q₈	—	298	32	no	—
184	Q₈	—	298	16	no	—
185	Q₈	—	149	16	no	—
186	D₄	C₂×Dic₂₉₈	1	16	yes	direct product factor
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	298	8	no	—
191	C₂×C₄	—	298	4	no	—
192	C₂×C₄	—	298	32	no	—
193	C₂×C₄	—	149	4	no	—
194	C₂×C₄	—	149	8	no	—
195	C₂×C₄	—	149	8	no	—
196	C₂×C₄	—	149	4	no	—
197	C₂×C₄	D₄×D₁₄₉	1	4	yes	—
198	C₂×C₄	C₂×Dic₂₉₈	1	4	yes	complemented
199	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
200	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
201	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
202	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	16	no	—
206	C₂²	C₂×D₄×D₁₄₉	1	2	yes	—
207	C₂²	C₂²×Dic₂₉₈	1	2	yes	—
208	C₂²	C₂²×Dic₂₉₈	1	8	yes	complemented
209	C₂²	#18011958	1	2	yes	—
210	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	298	8	no	—
214	C₄	—	298	4	no	—
215	C₄	—	149	8	no	—
216	C₄	C₂×D₄×D₁₄₉	1	4	yes	—
217	C₄	C₂²×Dic₂₉₈	1	4	yes	complemented
218	C₂	—	2	8	no	—
219	C₂	D₄×Dic₂₉₈	1	4	yes	direct product factor
220	C₂	C₂²×D₄×D₁₄₉	1	1	yes	characteristic
221	C₂	C₂³×Dic₂₉₈	1	1	yes	characteristic
222	C₂	C₂×#18011958	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting	C₂×C₄×D₄×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂²×D₄×C₁₄₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₄₉
hypocenter	C₁₄₉	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₉₈	C₂	semidirect
2. D₄×Dic₂₉₈	C₂	direct (N × H)
3. C₂×#18011911	C₂	semidirect
4. C₂×#18011910	C₂	semidirect
5. C₂×#18011890	C₂	semidirect
6. C₂×#18011888	C₂	semidirect
7. C₂×C₄×Dic₂₉₈	C₂	semidirect
14. C₂²×Dic₂₉₈	C₂²	semidirect
16. #18011911	C₂²	semidirect
17. #18011910	C₂²	semidirect
18. #18011890	C₂²	semidirect
19. #18011888	C₂²	semidirect
20. C₄×Dic₂₉₈	C₂²	semidirect
38. C₂×Dic₂₉₈	D₄	direct (N × H)
55. C₂×D₄×C₁₄₉	Q₈	semidirect
68. Dic₂₉₈	C₂×D₄	direct (N × H)
78. C₂³×C₁₄₉	C₂×Q₈	semidirect
81. D₄×C₁₄₉	C₂×Q₈	semidirect
86. C₂×C₄×C₁₄₉	C₂×Q₈	semidirect
96. C₂²×C₁₄₉	C₂²×Q₈	semidirect
102. C₄×C₁₄₉	C₂²×Q₈	semidirect
107. C₂×C₁₄₉	D₄×Q₈	semidirect
111. C₁₄₉	C₂×D₄×Q₈	semidirect
155. C₂×D₄	Dic₂₉₈	direct (N × H)
180. C₂³	C₂×Dic₂₉₈	semidirect
186. D₄	C₂×Dic₂₉₈	direct (N × H)
198. C₂×C₄	C₂×Dic₂₉₈	semidirect
208. C₂²	C₂²×Dic₂₉₈	semidirect
217. C₄	C₂²×Dic₂₉₈	semidirect
219. C₂	D₄×Dic₂₉₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₉₈	2
D₄×Dic₂₉₈	2
C₂×#18011911	2
C₂×#18011910	2
C₂×#18011890	2
C₂×#18011888	2
C₂×C₄×Dic₂₉₈	2
C₂×D₄×Dic₁₄₉	2
C₂×C₄×D₄×C₁₄₉	2
C₂×D₄×Q₈	149

Subgroups — C2×D4×Dic298 #34058165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Dic₂₉₈ #34058165