Subgroups — D₄×#14558746 #34061428

Order 19072.

544 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×#14558746	C₂	1	2	yes	maximal, complemented
2	#23750659	C₂	1	2	yes	maximal, complemented
3	#23750654	C₂	1	4	yes	maximal, complemented
4	D₄×#11924753	C₂	1	2	yes	maximal, complemented
5	#23750629	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750596	C₂	1	2	yes	maximal, complemented
7	C₄×#14558746	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×D₁₄₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14558746	—	2	8	no	—
10	C₂×#14558746	C₂²	1	2	yes	—
11	C₂×#14558746	C₂²	1	1	yes	characteristic
12	C₂×#14558746	C₂²	1	1	yes	characteristic
13	C₂×#14558746	C₂²	1	4	yes	complemented
14	C₂²×#11924753	C₂²	1	4	yes	complemented
15	C₄⋊#11924753	C₂²	1	2	yes	complemented
16	C₄⋊#11924753	C₂²	1	2	yes	complemented
17	#18011968	C₂²	1	4	yes	complemented
18	C₂²⋊#11924753	C₂²	1	4	yes	complemented
19	#18011966	C₂²	1	2	yes	complemented
20	C₄×#11924753	C₂²	1	2	yes	complemented
21	C₂×D₄×D₁₄₉	C₄	1	1	yes	characteristic
22	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
23	C₄×D₅₉₆	C₄	1	1	yes	characteristic
24	D₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
25	#18011873	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₄₉	C₄	1	2	yes	—
27	#18011871	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₄₉	C₄	1	1	yes	characteristic
29	C₂²⋊C₄×D₁₄₉	C₄	1	2	yes	—
30	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₄₉	C₄	1	1	yes	characteristic
32	#14558746	—	2	8	no	—
33	#14558746	—	2	4	no	—
34	#14558746	—	2	2	no	—
35	#14558746	D₄	1	4	yes	direct product factor
36	C₂×#11924753	—	2	8	no	—
37	C₂×#11924753	C₂³	1	2	yes	—
38	C₂×#11924753	C₂³	1	4	yes	—
39	C₂×#11924753	C₂³	1	2	yes	—
40	C₂³×D₁₄₉	C₂×C₄	1	2	yes	—
41	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
43	D₄×D₁₄₉	—	2	4	no	—
44	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
45	C₂×C₄×D₁₄₉	—	2	4	no	—
46	C₂×C₄×D₁₄₉	—	2	4	no	—
47	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
53	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
54	#14558715	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
56	#14558713	C₂×C₄	1	2	yes	—
57	#14558713	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
63	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
65	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
66	#11924753	—	2	4	no	—
67	#11924753	—	2	2	no	—
68	#11924753	C₂×D₄	1	4	yes	complemented
69	C₂²×D₁₄₉	—	2	4	no	—
70	C₂²×D₁₄₉	—	2	4	no	—
71	C₂²×D₁₄₉	—	4	2	no	—
72	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
73	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
74	D₅₉₆	—	2	2	no	—
75	C₂²⋊D₁₄₉	—	2	4	no	—
76	C₂×Dic₁₄₉	—	2	2	no	—
77	C₂×Dic₁₄₉	—	2	2	no	—
78	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
82	C₄×D₁₄₉	—	2	4	no	—
83	C₄×D₁₄₉	—	2	4	no	—
84	C₄×D₁₄₉	—	2	2	no	—
85	C₄×D₁₄₉	—	2	2	no	—
86	C₄×D₁₄₉	—	2	2	no	—
87	C₄×D₁₄₉	—	4	1	no	—
88	C₄×D₁₄₉	C₂×D₄	1	2	yes	—
89	C₄×D₁₄₉	#41	1	2	yes	—
90	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
91	D₄×C₁₄₉	M₁₆	1	4	yes	complemented
92	C₂×C₄×C₁₄₉	—	2	4	no	—
93	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₁₄₉	—	2	2	no	—
98	C₂²×C₁₄₉	—	2	2	no	—
99	C₂²×C₁₄₉	C₂×M₁₆	1	4	yes	complemented
100	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₄₉	—	2	1	no	—
102	C₂×D₁₄₉	—	2	1	no	—
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	2	2	no	—
105	C₂×D₁₄₉	—	2	2	no	—
106	C₂×D₁₄₉	—	4	8	no	—
107	C₂×D₁₄₉	C₄×D₄	1	2	yes	—
108	C₄×C₁₄₉	—	2	2	no	—
109	C₄×C₁₄₉	—	2	1	no	—
110	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
111	C₄×C₁₄₉	C₂×M₁₆	1	2	yes	complemented
112	Dic₁₄₉	—	2	2	no	—
113	Dic₁₄₉	—	2	1	no	—
114	Dic₁₄₉	—	2	1	no	—
115	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
116	Dic₁₄₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₄₉	—	2	4	no	—
118	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₄₉	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₁₄₉	#568	1	1	yes	characteristic
121	D₁₄₉	—	2	2	no	—
122	D₁₄₉	—	4	2	no	—
123	C₁₄₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
124	D₄×M₁₆	—	149	1	no	maximal, Sylow
125	C₂²×M₁₆	—	149	2	no	—
126	C₂×C₄×D₄	—	149	1	no	—
127	#436	—	149	2	no	—
128	#435	—	149	4	no	—
129	C₈×D₄	—	149	2	no	—
130	#423	—	149	1	no	—
131	#407	—	149	2	no	—
132	C₄×M₁₆	—	149	1	no	—
133	C₂²×D₄	—	149	1	no	—
134	C₂³×C₄	—	149	2	no	—
135	C₂×M₁₆	—	298	8	no	—
136	C₂×M₁₆	—	149	2	no	—
137	C₂×M₁₆	—	149	4	no	—
138	C₂×M₁₆	—	149	1	no	—
139	C₂×M₁₆	—	149	1	no	—
140	C₂²×C₈	—	149	4	no	—
141	C₄×D₄	—	149	1	no	—
142	C₄×D₄	—	149	2	no	—
143	C₄×D₄	—	149	1	no	—
144	C₄×D₄	—	149	2	no	—
145	C₄×D₄	—	149	1	no	—
146	C₄×D₄	#10597010	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	149	1	no	—
148	C₂×C₂²⋊C₄	—	149	2	no	—
149	C₂×C₄²	—	149	1	no	—
150	C₄⋊C₈	—	149	2	no	—
151	C₄⋊C₈	—	149	2	no	—
152	C₂²⋊C₈	—	149	4	no	—
153	C₂²⋊C₈	—	149	4	no	—
154	#97	—	149	2	no	—
155	C₄×C₈	—	149	2	no	—
156	C₂⁴	—	149	2	no	—
157	C₂×D₄	—	298	4	no	—
158	C₂×D₄	—	149	2	no	—
159	C₂×D₄	—	149	1	no	—
160	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
161	C₂²×C₄	—	298	4	no	—
162	C₂²×C₄	—	298	4	no	—
163	C₂²×C₄	—	149	2	no	—
164	C₂²×C₄	—	149	1	no	—
165	C₂²×C₄	—	149	2	no	—
166	C₂²×C₄	—	149	2	no	—
167	C₂²×C₄	—	149	2	no	—
168	C₂²×C₄	—	149	1	no	—
169	C₂²×C₄	—	149	1	no	—
170	C₂²×C₄	C₂×#10597010	1	2	yes	—
171	M₁₆	—	298	4	no	—
172	M₁₆	—	298	2	no	—
173	M₁₆	—	298	8	no	—
174	M₁₆	—	149	4	no	—
175	C₂×C₈	—	298	8	no	—
176	C₂×C₈	—	149	2	no	—
177	C₂×C₈	—	149	4	no	—
178	C₂×C₈	—	149	2	no	—
179	C₄⋊C₄	—	149	2	no	—
180	C₄⋊C₄	—	149	1	no	—
181	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
182	C₂²⋊C₄	—	149	2	no	—
183	C₂²⋊C₄	—	149	2	no	—
184	C₂²⋊C₄	—	149	2	no	—
185	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
186	C₄²	—	149	1	no	—
187	C₄²	—	149	1	no	—
188	C₄²	—	149	1	no	—
189	C₄²	C₂×#10597010	1	1	yes	characteristic
190	C₂³	—	298	4	no	—
191	C₂³	—	298	4	no	—
192	C₂³	—	596	2	no	—
193	C₂³	—	149	2	no	—
194	C₂³	—	149	1	no	—
195	C₂³	C₂²×#10597010	1	2	yes	—
196	D₄	—	298	4	no	—
197	D₄	—	298	2	no	—
198	D₄	#14558746	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	298	4	no	—
201	C₂×C₄	—	298	2	no	—
202	C₂×C₄	—	298	2	no	—
203	C₂×C₄	—	298	2	no	—
204	C₂×C₄	—	298	2	no	—
205	C₂×C₄	—	298	4	no	—
206	C₂×C₄	—	298	2	no	—
207	C₂×C₄	—	596	1	no	—
208	C₂×C₄	—	149	1	no	—
209	C₂×C₄	—	149	1	no	—
210	C₂×C₄	—	149	2	no	—
211	C₂×C₄	—	149	1	no	—
212	C₂×C₄	—	149	2	no	—
213	C₂×C₄	—	149	2	no	—
214	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
215	C₂×C₄	C₂²×#10597010	1	2	yes	—
216	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
217	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
218	C₈	—	298	4	no	—
219	C₈	—	298	2	no	—
220	C₈	—	149	4	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	298	1	no	—
224	C₂²	—	298	1	no	—
225	C₂²	—	298	2	no	—
226	C₂²	—	298	2	no	—
227	C₂²	—	298	2	no	—
228	C₂²	—	596	8	no	—
229	C₂²	—	149	2	no	—
230	C₂²	C₂×#14558746	1	4	yes	complemented
231	C₂²	C₂³×#10597010	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	298	2	no	—
235	C₄	—	298	1	no	—
236	C₄	—	298	1	no	—
237	C₄	—	149	1	no	—
238	C₄	—	149	1	no	—
239	C₄	D₄×#10597010	1	2	yes	—
240	C₄	C₂×#14558746	1	2	yes	complemented
241	C₂	—	2	4	no	—
242	C₂	—	298	2	no	—
243	C₂	—	596	2	no	—
244	C₂	C₂×D₄×#10597010	1	1	yes	characteristic
245	C₂	C₂²×#14558746	1	1	yes	characteristic
246	C₂	#23750704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₄×D₄×C₁₄₉	C₄
prime radical	C₂×D₄×D₁₄₉	C₄
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	C₄×D₄	#10597010
hypocenter	C₁₄₉	D₄×M₁₆
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14558746	C₂	semidirect
2. #23750659	C₂	semidirect
3. #23750654	C₂	semidirect
4. D₄×#11924753	C₂	semidirect
5. #23750629	C₂	semidirect
6. #23750596	C₂	semidirect
7. C₄×#14558746	C₂	semidirect
13. C₂×#14558746	C₂²	semidirect
14. C₂²×#11924753	C₂²	semidirect
15. C₄⋊#11924753	C₂²	semidirect
16. C₄⋊#11924753	C₂²	semidirect
17. #18011968	C₂²	semidirect
18. C₂²⋊#11924753	C₂²	semidirect
19. #18011966	C₂²	semidirect
20. C₄×#11924753	C₂²	semidirect
35. #14558746	D₄	direct (N × H)
68. #11924753	C₂×D₄	semidirect
91. D₄×C₁₄₉	M₁₆	semidirect
99. C₂²×C₁₄₉	C₂×M₁₆	semidirect
111. C₄×C₁₄₉	C₂×M₁₆	semidirect
123. C₁₄₉	D₄×M₁₆	semidirect
198. D₄	#14558746	direct (N × H)
230. C₂²	C₂×#14558746	semidirect
240. C₄	C₂×#14558746	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14558746	2
#23750659	2
#23750654	2
D₄×#11924753	2
#23750629	2
#23750596	2
C₄×#14558746	2
C₄×D₄×D₁₄₉	2
D₄×M₁₆	149

Subgroups — D4×#14558746 #34061428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14558746 #34061428