Subgroups — C₂²×C₄×Dic₂₃₆ #29452706

Order 15104.

2830 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₁₈	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₃₆	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14273652	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₁₈	C₂²	1	2	yes	—
8	C₂²×Dic₂₃₆	C₄	1	16	yes	direct product factor
9	C₂²×Dic₂₃₆	C₂²	1	12	yes	—
10	C₂×C₄×Dic₁₁₈	—	2	12	no	—
11	C₂×C₄×Dic₁₁₈	C₂²	1	24	yes	—
12	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
13	C₂²×Dic₅₉⋊C₄	—	2	2	no	—
14	C₂²×C₄×Dic₅₉	—	2	2	no	—
15	C₄×Dic₂₃₆	C₂²	1	64	yes	direct product factor
16	C₂×#14273661	C₂²	1	12	yes	—
17	C₂×#14273652	C₂²	1	24	yes	—
18	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	12	yes	—
22	C₂²×Dic₁₁₈	—	2	6	no	—
23	C₂²×Dic₁₁₈	—	2	8	no	—
24	C₂²×Dic₁₁₈	C₂×C₄	1	16	yes	—
25	C₂²×Dic₁₁₈	C₂³	1	12	yes	—
26	C₂³×Dic₅₉	—	2	2	no	—
27	C₂³×Dic₅₉	—	2	2	no	—
28	C₂×Dic₂₃₆	C₂×C₄	1	96	yes	direct product factor
29	C₂×Dic₂₃₆	C₂³	1	16	yes	—
30	C₄×Dic₁₁₈	—	2	48	no	—
31	C₄×Dic₁₁₈	C₂³	1	32	yes	—
32	C₂×C₄⋊Dic₅₉	—	2	6	no	—
33	C₂×C₄⋊Dic₅₉	C₂³	1	12	yes	—
34	C₂×Dic₅₉⋊C₄	—	2	24	no	—
35	C₂×C₄×Dic₅₉	—	2	24	no	—
36	#14273661	C₂³	1	16	yes	—
37	#14273652	C₂³	1	32	yes	—
38	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	—	2	4	no	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
44	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
45	C₂×C₄²×C₅₉	D₄	1	6	yes	—
46	C₂×C₄²×C₅₉	C₂³	1	6	yes	—
47	C₄×C₈×C₅₉	C₂³	1	16	yes	—
48	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	—	2	4	no	—
50	C₂²×C₄×C₅₉	Q₁₆	1	8	yes	complemented
51	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
55	C₂²×C₄×C₅₉	#41	1	4	yes	—
56	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
57	C₂²×Dic₅₉	—	2	16	no	—
58	C₂²×Dic₅₉	—	2	12	no	—
59	C₂²×Dic₅₉	—	2	12	no	—
60	C₂²×Dic₅₉	—	4	8	no	—
61	C₂×Dic₁₁₈	—	2	48	no	—
62	C₂×Dic₁₁₈	—	2	12	no	—
63	C₂×Dic₁₁₈	—	2	24	no	—
64	C₂×Dic₁₁₈	C₂²×C₄	1	48	yes	—
65	C₂×Dic₁₁₈	C₂⁴	1	8	yes	—
66	C₂×C₈×C₅₉	—	2	12	no	—
67	C₂×C₈×C₅₉	C₂²×C₄	1	24	yes	—
68	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
69	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
70	C₄²×C₅₉	C₂×D₄	1	12	yes	—
71	C₄²×C₅₉	C₂⁴	1	4	yes	—
72	C₄⋊Dic₅₉	—	2	12	no	—
73	C₄⋊Dic₅₉	C₂⁴	1	8	yes	—
74	Dic₅₉⋊C₄	—	2	32	no	—
75	C₄×Dic₅₉	—	2	32	no	—
76	Dic₂₃₆	C₂²×C₄	1	64	yes	direct product factor
77	C₂³×C₅₉	C₄×D₄	1	4	yes	—
78	C₂³×C₅₉	C₂×Q₁₆	1	4	yes	—
79	C₂³×C₅₉	#135	1	4	yes	—
80	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₅₉	—	2	12	no	—
82	C₂×C₄×C₅₉	C₄×D₄	1	12	yes	—
83	C₂×C₄×C₅₉	C₂×Q₁₆	1	24	yes	complemented
84	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
85	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
86	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
87	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
88	C₂×C₄×C₅₉	C₂×#41	1	6	yes	—
89	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₅₉	—	2	48	no	—
91	C₂×Dic₅₉	—	2	8	no	—
92	C₂×Dic₅₉	—	2	8	no	—
93	C₂×Dic₅₉	—	4	24	no	—
94	Dic₁₁₈	—	2	48	no	—
95	Dic₁₁₈	—	2	8	no	—
96	Dic₁₁₈	C₂³×C₄	1	16	yes	—
97	C₈×C₅₉	—	2	4	no	—
98	C₈×C₅₉	C₂³×C₄	1	8	yes	—
99	C₂²×C₄×Q₁₆	—	59	1	no	maximal, Sylow
100	C₂²×C₅₉	C₄×Q₁₆	1	16	yes	complemented
101	C₂²×C₅₉	C₂×C₄×D₄	1	6	yes	—
102	C₂²×C₅₉	C₂²×Q₁₆	1	6	yes	—
103	C₂²×C₅₉	C₂×#135	1	6	yes	—
104	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
105	C₄×C₅₉	—	2	4	no	—
106	C₄×C₅₉	C₂×C₄×D₄	1	6	yes	—
107	C₄×C₅₉	C₂²×Q₁₆	1	8	yes	complemented
108	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₅₉	C₂²×#41	1	1	yes	characteristic
110	Dic₅₉	—	2	16	no	—
111	Dic₅₉	—	4	8	no	—
112	C₂³×Q₁₆	—	59	1	no	—
113	C₂²×C₄×Q₈	—	59	2	no	—
114	C₂×C₄×Q₁₆	—	59	24	no	—
115	C₂²×#107	—	59	1	no	—
116	C₂²×#103	—	59	2	no	—
117	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
118	C₂×C₅₉	C₂×C₄×Q₁₆	1	12	yes	complemented
119	C₂×C₅₉	C₂²×C₄×D₄	1	1	yes	characteristic
120	C₂×C₅₉	C₂³×Q₁₆	1	1	yes	characteristic
121	C₂×C₅₉	C₂²×#135	1	1	yes	characteristic
122	C₂³×Q₈	—	59	2	no	—
123	C₂²×Q₁₆	—	59	16	no	—
124	C₂²×Q₁₆	—	59	12	no	—
125	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
127	C₂×C₄×Q₈	—	59	24	no	—
128	C₂×C₄×Q₈	—	118	12	no	—
129	C₂²×C₄⋊C₄	—	59	2	no	—
130	C₂²×C₄⋊C₄	—	118	2	no	—
131	C₂²×C₄²	—	118	2	no	—
132	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
133	C₄×Q₁₆	—	59	64	no	—
134	C₂×#107	—	59	12	no	—
135	C₂×#103	—	59	24	no	—
136	C₂×C₄×C₈	C₂×D₅₉	1	12	yes	—
137	C₅₉	C₂²×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	59	16	no	—
139	C₂²×Q₈	—	59	12	no	—
140	C₂²×Q₈	—	118	6	no	—
141	C₂²×Q₈	—	118	8	no	—
142	C₂³×C₄	—	118	2	no	—
143	C₂³×C₄	—	118	2	no	—
144	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
145	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
146	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
147	C₂×Q₁₆	—	59	96	no	—
148	C₂×Q₁₆	—	59	16	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₄×D₅₉	1	8	yes	—
151	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
152	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
153	C₄×Q₈	—	59	32	no	—
154	C₄×Q₈	—	118	48	no	—
155	C₂×C₄⋊C₄	—	59	12	no	—
156	C₂×C₄⋊C₄	—	118	6	no	—
157	C₂×C₄⋊C₄	—	118	24	no	—
158	C₂×C₄²	—	118	24	no	—
159	C₂×C₄²	D₂₃₆	1	6	yes	—
160	C₂×C₄²	C₂²×D₅₉	1	6	yes	—
161	#107	—	59	16	no	—
162	#103	—	59	32	no	—
163	C₄×C₈	C₂²×D₅₉	1	16	yes	—
164	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
165	C₂×Q₈	—	59	48	no	—
166	C₂×Q₈	—	59	8	no	—
167	C₂×Q₈	—	118	48	no	—
168	C₂×Q₈	—	118	12	no	—
169	C₂×Q₈	—	118	24	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	118	16	no	—
172	C₂²×C₄	—	118	12	no	—
173	C₂²×C₄	—	118	12	no	—
174	C₂²×C₄	—	236	8	no	—
175	C₂²×C₄	Dic₂₃₆	1	8	yes	direct product factor
176	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
177	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
178	C₂²×C₄	C₂×D₂₃₆	1	3	yes	—
179	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
180	C₂²×C₄	#11745685	1	4	yes	—
181	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
182	Q₁₆	—	59	64	no	—
183	C₂×C₈	—	2	12	no	—
184	C₂×C₈	C₂×C₄×D₅₉	1	24	yes	—
185	C₂×C₈	C₂³×D₅₉	1	4	yes	—
186	C₂×C₈	C₂³×D₅₉	1	4	yes	—
187	C₄⋊C₄	—	59	8	no	—
188	C₄⋊C₄	—	118	12	no	—
189	C₄⋊C₄	—	118	32	no	—
190	C₄²	—	118	32	no	—
191	C₄²	C₂×D₂₃₆	1	12	yes	—
192	C₄²	C₂³×D₅₉	1	4	yes	—
193	C₂³	C₄×D₂₃₆	1	4	yes	—
194	C₂³	C₂×Dic₂₃₆	1	4	yes	—
195	C₂³	#14273749	1	4	yes	—
196	C₂³	C₂²×D₂₃₆	1	3	yes	—
197	Q₈	—	59	16	no	—
198	Q₈	—	118	8	no	—
199	Q₈	—	118	48	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	118	8	no	—
202	C₂×C₄	—	118	48	no	—
203	C₂×C₄	—	118	8	no	—
204	C₂×C₄	—	236	24	no	—
205	C₂×C₄	C₄×D₂₃₆	1	12	yes	—
206	C₂×C₄	C₂×Dic₂₃₆	1	24	yes	direct product factor
207	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
208	C₂×C₄	C₂²×D₂₃₆	1	4	yes	—
209	C₂×C₄	C₂²×D₂₃₆	1	4	yes	—
210	C₂×C₄	C₂²×D₂₃₆	1	3	yes	—
211	C₂×C₄	C₂×#11745685	1	6	yes	—
212	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂²×C₄×D₅₉	1	8	yes	—
215	C₂²	C₄×Dic₂₃₆	1	16	yes	direct product factor
216	C₂²	C₂×C₄×D₂₃₆	1	6	yes	—
217	C₂²	C₂²×Dic₂₃₆	1	6	yes	—
218	C₂²	C₂×#14273749	1	6	yes	—
219	C₂²	C₂³×D₂₃₆	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	118	16	no	—
222	C₄	—	236	8	no	—
223	C₄	C₂×C₄×D₂₃₆	1	6	yes	—
224	C₄	C₂²×Dic₂₃₆	1	8	yes	direct product factor
225	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
226	C₄	C₂²×#11745685	1	1	yes	characteristic
227	C₂	C₂×C₄×Dic₂₃₆	1	12	yes	direct product factor
228	C₂	C₂²×C₄×D₂₃₆	1	1	yes	characteristic
229	C₂	C₂³×Dic₂₃₆	1	1	yes	characteristic
230	C₂	C₂²×#14273749	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₃₆
commutator	C₄×C₅₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂×C₄×C₅₉	C₂⁵
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	C₂²×C₄×Q₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₃₆	C₂	direct (N × H)
8. C₂²×Dic₂₃₆	C₄	direct (N × H)
15. C₄×Dic₂₃₆	C₂²	direct (N × H)
28. C₂×Dic₂₃₆	C₂×C₄	direct (N × H)
50. C₂²×C₄×C₅₉	Q₁₆	semidirect
76. Dic₂₃₆	C₂²×C₄	direct (N × H)
83. C₂×C₄×C₅₉	C₂×Q₁₆	semidirect
100. C₂²×C₅₉	C₄×Q₁₆	semidirect
107. C₄×C₅₉	C₂²×Q₁₆	semidirect
118. C₂×C₅₉	C₂×C₄×Q₁₆	semidirect
137. C₅₉	C₂²×C₄×Q₁₆	semidirect
175. C₂²×C₄	Dic₂₃₆	direct (N × H)
206. C₂×C₄	C₂×Dic₂₃₆	direct (N × H)
215. C₂²	C₄×Dic₂₃₆	direct (N × H)
224. C₄	C₂²×Dic₂₃₆	direct (N × H)
227. C₂	C₂×C₄×Dic₂₃₆	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×Dic₂₃₆	2
C₂²×C₄×Dic₁₁₈	2
C₂×C₄×Dic₂₃₆	2
C₂²×#14273661	2
C₂²×#14273652	2
C₂²×C₄×C₈×C₅₉	2
C₂²×C₄×Q₁₆	59

Subgroups — C22×C4×Dic236 #29452706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Dic₂₃₆ #29452706