Subgroups — C₂×C₄⋊C₄×#10597010 #34061122

Order 19072.

928 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10597010	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#10597010	C₂	1	2	yes	maximal
3	C₂²×C₄×#10597010	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#10597010	C₂	1	8	yes	maximal, direct product factor
5	C₂×(C₂×#10597010)⋊C₄	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₄×D₁₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10597010	C₂²	1	2	yes	—
8	C₂³×#10597010	C₂²	1	1	yes	characteristic
9	C₂³×#10597010	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#10597010	C₄	1	8	yes	complemented
11	C₂×C₄⋊#10597010	C₂²	1	4	yes	—
12	C₂×C₄×#10597010	—	2	8	no	—
13	C₂×C₄×#10597010	C₄	1	8	yes	complemented
14	C₂×C₄×#10597010	C₂²	1	8	yes	—
15	C₂×C₄×#10597010	C₂²	1	4	yes	—
16	(C₂×#10597010)⋊C₄	C₂²	1	8	yes	—
17	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₄₉	C₄	1	1	yes	characteristic, complemented
20	C₂×#14558715	C₄	1	2	yes	complemented
21	C₄⋊C₄×D₁₄₉	C₄	1	4	yes	complemented
22	C₄⋊C₄×D₁₄₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₄₉	C₄	1	1	yes	characteristic, complemented
24	C₂²×#10597010	—	2	4	no	—
25	C₂²×#10597010	—	2	2	no	—
26	C₂²×#10597010	—	2	4	no	—
27	C₂²×#10597010	—	2	2	no	—
28	C₂²×#10597010	C₂×C₄	1	4	yes	—
29	C₂²×#10597010	C₂×C₄	1	8	yes	—
30	C₂²×#10597010	C₂×C₄	1	4	yes	—
31	C₂²×#10597010	D₄	1	4	yes	—
32	C₂²×#10597010	Q₈	1	4	yes	—
33	C₂²×#10597010	C₂³	1	2	yes	—
34	C₂²×#10597010	C₂³	1	4	yes	—
35	C₂²×#10597010	C₂³	1	2	yes	—
36	C₄⋊#10597010	C₂×C₄	1	16	yes	complemented
37	C₄×#10597010	—	2	16	no	—
38	C₄×#10597010	C₂×C₄	1	16	yes	complemented
39	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
41	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₄₉	—	2	4	no	—
43	C₂×C₄×D₁₄₉	—	2	4	no	—
44	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	—
46	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
49	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
50	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	complemented
51	#14558715	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
55	C₂×#10597010	—	2	8	no	—
56	C₂×#10597010	—	2	16	no	—
57	C₂×#10597010	—	2	4	no	—
58	C₂×#10597010	—	2	2	no	—
59	C₂×#10597010	—	2	2	no	—
60	C₂×#10597010	—	2	4	no	—
61	C₂×#10597010	C₄⋊C₄	1	16	yes	direct product factor
62	C₂×#10597010	C₂²×C₄	1	4	yes	—
63	C₂×#10597010	C₂²×C₄	1	8	yes	—
64	C₂×#10597010	C₂²×C₄	1	4	yes	—
65	C₂×#10597010	C₂×D₄	1	4	yes	—
66	C₂×#10597010	C₂×Q₈	1	4	yes	—
67	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₄₉	C₂×D₄	1	2	yes	—
71	C₂²×D₁₄₉	C₂×Q₈	1	2	yes	—
72	C₂²×D₁₄₉	#41	1	2	yes	—
73	C₂²×D₁₄₉	#41	1	2	yes	—
74	C₂²×D₁₄₉	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₁₄₉	—	2	4	no	—
76	C₂×Dic₁₄₉	C₄²	1	4	yes	complemented
77	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
79	C₄×D₁₄₉	—	2	8	no	—
80	C₄×D₁₄₉	—	2	4	no	—
81	C₄×D₁₄₉	—	2	4	no	—
82	C₄×D₁₄₉	C₄²	1	8	yes	complemented
83	C₄×D₁₄₉	C₂²×C₄	1	4	yes	—
84	C₄×D₁₄₉	C₂²×C₄	1	4	yes	—
85	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₄₉	—	2	4	no	—
87	C₂×C₄×C₁₄₉	C₄²	1	4	yes	complemented
88	C₂×C₄×C₁₄₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₁₄₉	C₂×C₄²	1	2	yes	—
91	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
92	C₂²×C₁₄₉	C₄×Q₈	1	2	yes	—
93	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₁₄₉	C₂×C₄²	1	2	yes	—
95	C₂×D₁₄₉	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×D₁₄₉	#117	1	4	yes	—
97	C₂×D₁₄₉	C₄×D₄	1	4	yes	—
98	C₂×D₁₄₉	C₄×D₄	1	2	yes	—
99	C₂×D₁₄₉	C₄×Q₈	1	4	yes	—
100	C₂×D₁₄₉	C₄×Q₈	1	2	yes	—
101	C₂×D₁₄₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₄₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₄₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₄₉	C₂²×Q₈	1	1	yes	characteristic
105	C₂×D₁₄₉	C₂×#41	1	1	yes	characteristic
106	C₂×D₁₄₉	C₂×#41	1	1	yes	characteristic
107	#10597010	—	2	4	no	—
108	#10597010	—	2	8	no	—
109	#10597010	C₂×C₄⋊C₄	1	8	yes	direct product factor
110	C₄×C₁₄₉	—	2	4	no	—
111	C₄×C₁₄₉	C₂×C₄²	1	4	yes	complemented
112	Dic₁₄₉	—	2	4	no	—
113	Dic₁₄₉	C₂×C₄²	1	4	yes	complemented
114	C₂×C₁₄₉	C₄×C₄⋊C₄	1	4	yes	complemented
115	C₂×C₁₄₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂×C₁₄₉	C₂×C₄×Q₈	1	1	yes	characteristic
118	D₁₄₉	C₄×C₄⋊C₄	1	4	yes	complemented
119	D₁₄₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	D₁₄₉	C₂×#117	1	1	yes	characteristic
121	D₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
122	D₁₄₉	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₁₄₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
124	C₂×C₄×C₄⋊C₄	—	149	1	no	maximal, Sylow
125	C₂²×C₄⋊C₄	—	149	1	no	—
126	C₂²×C₄⋊C₄	—	149	1	no	—
127	C₂²×C₄²	—	149	2	no	—
128	C₂²×C₄²	—	149	1	no	—
129	C₄×C₄⋊C₄	—	149	8	no	—
130	C₂×#95	—	149	2	no	—
131	C₂³×C₄	—	149	2	no	—
132	C₂³×C₄	—	149	1	no	—
133	C₂³×C₄	—	149	2	no	—
134	C₂³×C₄	—	149	1	no	—
135	C₂³×C₄	—	149	1	no	—
136	C₂×C₄⋊C₄	—	149	8	no	—
137	C₂×C₄⋊C₄	—	149	4	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	4	no	—
142	C₂×C₄⋊C₄	#10597010	1	1	yes	characteristic, direct product factor
143	C₂×C₄²	—	298	8	no	—
144	C₂×C₄²	—	149	8	no	—
145	C₂×C₄²	—	149	8	no	—
146	C₂×C₄²	—	149	4	no	—
147	#95	—	149	8	no	—
148	C₂⁴	—	149	1	no	—
149	C₂²×C₄	—	298	4	no	—
150	C₂²×C₄	—	298	4	no	—
151	C₂²×C₄	—	298	4	no	—
152	C₂²×C₄	—	298	2	no	—
153	C₂²×C₄	—	298	4	no	—
154	C₂²×C₄	—	298	2	no	—
155	C₂²×C₄	—	149	4	no	—
156	C₂²×C₄	—	149	8	no	—
157	C₂²×C₄	—	149	2	no	—
158	C₂²×C₄	—	149	4	no	—
159	C₂²×C₄	—	149	2	no	—
160	C₂²×C₄	—	149	4	no	—
161	C₂²×C₄	—	149	4	no	—
162	C₂²×C₄	—	149	4	no	—
163	C₂²×C₄	—	149	4	no	—
164	C₂²×C₄	—	149	4	no	—
165	C₂²×C₄	—	149	2	no	—
166	C₂²×C₄	—	149	1	no	—
167	C₂²×C₄	—	149	4	no	—
168	C₂²×C₄	—	149	4	no	—
169	C₂²×C₄	—	149	2	no	—
170	C₂²×C₄	—	149	2	no	—
171	C₂²×C₄	C₂×#10597010	1	2	yes	—
172	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
173	C₄⋊C₄	—	149	16	no	—
174	C₄⋊C₄	—	149	8	no	—
175	C₄⋊C₄	—	149	4	no	—
176	C₄⋊C₄	C₂×#10597010	1	4	yes	direct product factor
177	C₄²	—	298	16	no	—
178	C₄²	—	149	16	no	—
179	C₂³	—	149	2	no	—
180	C₂³	—	149	2	no	—
181	C₂³	—	149	2	no	—
182	C₂³	—	149	2	no	—
183	C₂³	—	149	2	no	—
184	C₂³	—	149	2	no	—
185	C₂³	—	149	1	no	—
186	C₂³	—	149	1	no	—
187	C₂³	C₂²×#10597010	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	298	4	no	—
190	C₂×C₄	—	298	4	no	—
191	C₂×C₄	—	298	8	no	—
192	C₂×C₄	—	298	4	no	—
193	C₂×C₄	—	298	2	no	—
194	C₂×C₄	—	298	2	no	—
195	C₂×C₄	—	298	4	no	—
196	C₂×C₄	—	298	16	no	—
197	C₂×C₄	—	298	8	no	—
198	C₂×C₄	—	298	4	no	—
199	C₂×C₄	—	149	4	no	—
200	C₂×C₄	—	149	8	no	—
201	C₂×C₄	—	149	4	no	—
202	C₂×C₄	—	149	16	no	—
203	C₂×C₄	—	149	4	no	—
204	C₂×C₄	—	149	4	no	—
205	C₂×C₄	—	149	4	no	—
206	C₂×C₄	—	149	4	no	—
207	C₂×C₄	—	149	2	no	—
208	C₂×C₄	—	149	8	no	—
209	C₂×C₄	—	149	4	no	—
210	C₂×C₄	—	149	4	no	—
211	C₂×C₄	C₄×#10597010	1	4	yes	complemented
212	C₂×C₄	C₂²×#10597010	1	4	yes	—
213	C₂×C₄	C₂²×#10597010	1	2	yes	—
214	C₂²	—	149	1	no	—
215	C₂²	—	149	1	no	—
216	C₂²	—	149	2	no	—
217	C₂²	—	149	1	no	—
218	C₂²	—	149	1	no	—
219	C₂²	—	149	2	no	—
220	C₂²	—	149	1	no	—
221	C₂²	—	149	1	no	—
222	C₂²	—	149	2	no	—
223	C₂²	—	149	4	no	—
224	C₂²	—	149	4	no	—
225	C₂²	—	149	4	no	—
226	C₂²	—	149	4	no	—
227	C₂²	C₂×C₄×#10597010	1	2	yes	—
228	C₂²	D₄×#10597010	1	2	yes	—
229	C₂²	Q₈×#10597010	1	2	yes	—
230	C₂²	C₂³×#10597010	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	298	8	no	—
233	C₄	—	298	4	no	—
234	C₄	—	298	4	no	—
235	C₄	—	149	8	no	—
236	C₄	—	149	4	no	—
237	C₄	C₂×C₄×#10597010	1	4	yes	complemented
238	C₂	—	149	1	no	—
239	C₂	—	149	1	no	—
240	C₂	—	149	1	no	—
241	C₂	—	149	1	no	—
242	C₂	—	149	4	no	—
243	C₂	C₄⋊C₄×#10597010	1	4	yes	direct product factor
244	C₂	C₂²×C₄×#10597010	1	1	yes	characteristic
245	C₂	C₂×D₄×#10597010	1	1	yes	characteristic
246	C₂	C₂×Q₈×#10597010	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#10597010
commutator	C₂×C₁₄₉	C₂²×C₄²
Frattini	C₂²	C₂³×#10597010
Fitting	C₂×C₄⋊C₄×C₁₄₉	C₄
socle	C₂³×C₁₄₉	C₂²×C₄
prime radical	C₂³×D₁₄₉	C₂³
cosocle	C₂²×D₁₄₉	C₂⁴
hypercenter	C₂×C₄⋊C₄	#10597010
hypocenter	C₁₄₉	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×#10597010	C₂	direct (N × H)
10. C₂×C₄⋊#10597010	C₄	semidirect
13. C₂×C₄×#10597010	C₄	semidirect
19. C₂×C₄⋊Dic₁₄₉	C₄	semidirect
20. C₂×#14558715	C₄	semidirect
21. C₄⋊C₄×D₁₄₉	C₄	semidirect
23. C₂×C₄⋊C₄×C₁₄₉	C₄	semidirect
36. C₄⋊#10597010	C₂×C₄	semidirect
38. C₄×#10597010	C₂×C₄	semidirect
50. C₄⋊Dic₁₄₉	C₂×C₄	semidirect
51. #14558715	C₂×C₄	semidirect
54. C₄⋊C₄×C₁₄₉	C₂×C₄	semidirect
61. C₂×#10597010	C₄⋊C₄	direct (N × H)
76. C₂×Dic₁₄₉	C₄²	semidirect
82. C₄×D₁₄₉	C₄²	semidirect
87. C₂×C₄×C₁₄₉	C₄²	semidirect
109. #10597010	C₂×C₄⋊C₄	direct (N × H)
111. C₄×C₁₄₉	C₂×C₄²	semidirect
113. Dic₁₄₉	C₂×C₄²	semidirect
114. C₂×C₁₄₉	C₄×C₄⋊C₄	semidirect
118. D₁₄₉	C₄×C₄⋊C₄	semidirect
123. C₁₄₉	C₂×C₄×C₄⋊C₄	semidirect
142. C₂×C₄⋊C₄	#10597010	direct (N × H)
176. C₄⋊C₄	C₂×#10597010	direct (N × H)
211. C₂×C₄	C₄×#10597010	semidirect
237. C₄	C₂×C₄×#10597010	semidirect
243. C₂	C₄⋊C₄×#10597010	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×C₄⋊#10597010	2
C₂²×C₄×#10597010	2
C₄⋊C₄×#10597010	2
C₂×(C₂×#10597010)⋊C₄	2
C₂×C₄⋊C₄×D₁₄₉	2
C₂×C₄×C₄⋊C₄	149

Subgroups — C2×C4⋊C4×#10597010 #34061122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×#10597010 #34061122