Subgroups — M₁₆×D₅₄₈ #31789527

Order 17536.

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	C₂×M₁₆×D₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
3	#22577482	C₂	1	2	yes	maximal, complemented
4	#22577476	C₂	1	4	yes	maximal, complemented
5	C₈×D₅₄₈	C₂	1	2	yes	maximal, complemented
6	#22577427	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577366	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₄₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
11	C₂×#14555494	C₂²	1	2	yes	—
12	C₂×#14555493	C₂²	1	4	yes	complemented
13	M₁₆×D₁₃₇	—	2	8	no	—
14	C₂×C₈×D₁₃₇	C₂²	1	4	yes	complemented
15	C₄×D₅₄₈	C₄	1	2	yes	—
16	C₄×D₅₄₈	C₄	1	1	yes	characteristic
17	C₄×D₅₄₈	C₄	1	1	yes	characteristic
18	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
19	C₄×D₅₄₈	C₂²	1	2	yes	—
20	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
22	C₂×#14555487	C₄	1	2	yes	—
23	C₄⋊#11764040	C₂²	1	2	yes	complemented
24	C₄⋊#11764040	C₂²	1	2	yes	complemented
25	#17984000	C₂²	1	4	yes	complemented
26	#17984000	C₂²	1	4	yes	complemented
27	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
30	#97×C₁₃₇	C₂²	1	2	yes	complemented
31	C₄×C₈×C₁₃₇	C₂²	1	2	yes	complemented
32	C₂³×D₁₃₇	C₂×C₄	1	2	yes	—
33	C₂×D₅₄₈	—	2	4	no	—
34	C₂×D₅₄₈	C₂×C₄	1	2	yes	—
35	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
36	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₃₇	—	2	4	no	—
39	C₂×C₄×D₁₃₇	—	2	4	no	—
40	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
44	#14555494	—	2	4	no	—
45	#14555493	—	2	8	no	—
46	C₂×#11764040	C₂³	1	4	yes	—
47	C₈×D₁₃₇	—	2	8	no	—
48	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
51	#14555487	C₂×C₄	1	2	yes	—
52	#14555487	C₂×C₄	1	2	yes	—
53	#14555487	C₂×C₄	1	2	yes	—
54	#14555487	C₂×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₃₇	—	2	2	no	—
59	M₁₆×C₁₃₇	D₄	1	4	yes	complemented
60	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
62	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₂²×D₁₃₇	—	2	4	no	—
67	C₂²×D₁₃₇	—	2	4	no	—
68	C₂²×D₁₃₇	—	4	2	no	—
69	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
71	D₅₄₈	—	2	4	no	—
72	D₅₄₈	—	2	2	no	—
73	D₅₄₈	M₁₆	1	4	yes	direct product factor
74	C₂×Dic₁₃₇	—	2	4	no	—
75	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
77	C₄×D₁₃₇	—	2	4	no	—
78	C₄×D₁₃₇	—	2	4	no	—
79	C₄×D₁₃₇	—	2	2	no	—
80	C₄×D₁₃₇	—	2	2	no	—
81	C₄×D₁₃₇	—	2	2	no	—
82	C₄×D₁₃₇	—	2	2	no	—
83	#11764040	—	2	4	no	—
84	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	4	1	no	—
87	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃₇	#41	1	2	yes	—
94	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃₇	—	2	2	no	—
96	C₈×C₁₃₇	C₂×D₄	1	4	yes	complemented
97	C₂²×C₁₃₇	—	2	1	no	—
98	C₂²×C₁₃₇	—	2	1	no	—
99	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
100	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₃₇	—	2	2	no	—
102	C₂×D₁₃₇	—	2	2	no	—
103	C₂×D₁₃₇	—	2	2	no	—
104	C₂×D₁₃₇	—	2	2	no	—
105	C₂×D₁₃₇	—	2	2	no	—
106	C₂×D₁₃₇	—	4	8	no	—
107	C₂×D₁₃₇	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₁₃₇	—	2	1	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
112	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
115	Dic₁₃₇	—	2	2	no	—
116	Dic₁₃₇	—	2	2	no	—
117	C₂×C₁₃₇	—	2	2	no	—
118	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₁₃₇	#568	1	1	yes	characteristic
121	D₁₃₇	—	2	4	no	—
122	D₁₃₇	—	4	2	no	—
123	C₁₃₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
124	D₄×M₁₆	—	137	1	no	maximal, Sylow
125	C₂²×M₁₆	—	137	2	no	—
126	C₂×C₄×D₄	—	137	1	no	—
127	#436	—	137	2	no	—
128	#435	—	137	4	no	—
129	C₈×D₄	—	137	2	no	—
130	#423	—	137	1	no	—
131	#407	—	137	2	no	—
132	C₄×M₁₆	D₁₃₇	1	1	yes	characteristic, complemented
133	C₂²×D₄	—	137	1	no	—
134	C₂³×C₄	—	137	2	no	—
135	C₂×M₁₆	—	274	8	no	—
136	C₂×M₁₆	—	137	2	no	—
137	C₂×M₁₆	—	137	4	no	—
138	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
139	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
140	C₂²×C₈	—	137	4	no	—
141	C₄×D₄	—	137	2	no	—
142	C₄×D₄	—	137	1	no	—
143	C₄×D₄	—	137	1	no	—
144	C₄×D₄	—	137	1	no	—
145	C₄×D₄	—	137	2	no	—
146	C₄×D₄	—	137	1	no	—
147	C₂×C₄⋊C₄	—	137	1	no	—
148	C₂×C₂²⋊C₄	—	137	2	no	—
149	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
150	C₄⋊C₈	—	137	2	no	—
151	C₄⋊C₈	—	137	2	no	—
152	C₂²⋊C₈	—	137	4	no	—
153	C₂²⋊C₈	—	137	4	no	—
154	#97	C₂×D₁₃₇	1	2	yes	complemented
155	C₄×C₈	C₂×D₁₃₇	1	2	yes	complemented
156	C₂⁴	—	137	2	no	—
157	C₂×D₄	—	274	4	no	—
158	C₂×D₄	—	137	2	no	—
159	C₂×D₄	—	137	1	no	—
160	C₂×D₄	—	137	1	no	—
161	C₂²×C₄	—	274	4	no	—
162	C₂²×C₄	—	274	4	no	—
163	C₂²×C₄	—	137	2	no	—
164	C₂²×C₄	—	137	2	no	—
165	C₂²×C₄	—	137	2	no	—
166	C₂²×C₄	—	137	2	no	—
167	C₂²×C₄	—	137	2	no	—
168	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	274	4	no	—
173	M₁₆	—	274	8	no	—
174	M₁₆	D₅₄₈	1	4	yes	direct product factor
175	C₂×C₈	—	274	8	no	—
176	C₂×C₈	—	137	4	no	—
177	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
178	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
179	C₄⋊C₄	—	137	2	no	—
180	C₄⋊C₄	—	137	1	no	—
181	C₄⋊C₄	—	137	1	no	—
182	C₂²⋊C₄	—	137	2	no	—
183	C₂²⋊C₄	—	137	2	no	—
184	C₂²⋊C₄	—	137	2	no	—
185	C₂²⋊C₄	—	137	2	no	—
186	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
187	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
188	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
189	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
190	C₂³	—	274	4	no	—
191	C₂³	—	274	4	no	—
192	C₂³	—	548	2	no	—
193	C₂³	—	137	2	no	—
194	C₂³	—	137	2	no	—
195	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
196	D₄	—	274	4	no	—
197	D₄	—	274	2	no	—
198	D₄	—	137	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	274	4	no	—
202	C₂×C₄	—	274	4	no	—
203	C₂×C₄	—	274	4	no	—
204	C₂×C₄	—	274	2	no	—
205	C₂×C₄	—	274	2	no	—
206	C₂×C₄	—	274	2	no	—
207	C₂×C₄	—	274	2	no	—
208	C₂×C₄	—	137	2	no	—
209	C₂×C₄	—	137	2	no	—
210	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
215	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
216	C₂×C₄	#14555510	1	2	yes	—
217	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	274	4	no	—
220	C₈	C₂×D₅₄₈	1	4	yes	complemented
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	274	2	no	—
224	C₂²	—	274	2	no	—
225	C₂²	—	274	2	no	—
226	C₂²	—	274	2	no	—
227	C₂²	—	274	2	no	—
228	C₂²	—	548	8	no	—
229	C₂²	—	137	4	no	—
230	C₂²	C₄×D₅₄₈	1	2	yes	—
231	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	274	2	no	—
236	C₄	—	274	2	no	—
237	C₄	C₄×D₅₄₈	1	2	yes	—
238	C₄	M₁₆×D₁₃₇	1	2	yes	complemented
239	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
240	C₄	C₂×#14555510	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	274	4	no	—
243	C₂	—	548	2	no	—
244	C₂	C₂×C₄×D₅₄₈	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₁₃₇	1	1	yes	characteristic
246	C₂	#22578169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₄×M₁₆×C₁₃₇	C₂
prime radical	C₂²×D₅₄₈	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₃₇
hypocenter	C₁₃₇	D₄×M₁₆
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₃₇	C₂	semidirect
3. #22577482	C₂	semidirect
4. #22577476	C₂	semidirect
5. C₈×D₅₄₈	C₂	semidirect
6. #22577427	C₂	semidirect
7. #22577366	C₂	semidirect
8. C₄×M₁₆×C₁₃₇	C₂	semidirect
12. C₂×#14555493	C₂²	semidirect
14. C₂×C₈×D₁₃₇	C₂²	semidirect
23. C₄⋊#11764040	C₂²	semidirect
24. C₄⋊#11764040	C₂²	semidirect
25. #17984000	C₂²	semidirect
26. #17984000	C₂²	semidirect
30. #97×C₁₃₇	C₂²	semidirect
31. C₄×C₈×C₁₃₇	C₂²	semidirect
59. M₁₆×C₁₃₇	D₄	semidirect
73. D₅₄₈	M₁₆	direct (N × H)
96. C₈×C₁₃₇	C₂×D₄	semidirect
107. C₂×D₁₃₇	C₂×M₁₆	semidirect
112. C₄×C₁₃₇	C₂×M₁₆	semidirect
123. C₁₃₇	D₄×M₁₆	semidirect
132. C₄×M₁₆	D₁₃₇	semidirect
154. #97	C₂×D₁₃₇	semidirect
155. C₄×C₈	C₂×D₁₃₇	semidirect
174. M₁₆	D₅₄₈	direct (N × H)
220. C₈	C₂×D₅₄₈	semidirect
238. C₄	M₁₆×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₃₇	2
C₂×C₄×D₅₄₈	2
#22577482	2
#22577476	2
C₈×D₅₄₈	2
#22577427	2
#22577366	2
C₄×M₁₆×C₁₃₇	2
D₄×M₁₆	137

Subgroups — M16×D548 #31789527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×D₅₄₈ #31789527