Subgroups — C₄×#17984111 #31789561

Order 17536.

500 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×#17984111	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555511	C₂	1	2	yes	maximal, complemented
3	Q₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14555497	C₂	1	2	yes	maximal, complemented
5	#22577488	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577485	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577440	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577398	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555511	C₂²	1	2	yes	—
13	#17984111	C₄	1	8	yes	direct product factor
14	C₂×#14555497	C₂²	1	2	yes	—
15	C₂×D₁₀₉₆	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₃₇	C₂²	1	2	yes	—
19	C₄×D₅₄₈	—	2	2	no	—
20	C₄×D₅₄₈	C₂²	1	2	yes	—
21	#17984050	—	2	2	no	—
22	#17984050	C₂²	1	2	yes	—
23	#17984046	—	2	2	no	—
24	#17984046	C₂²	1	2	yes	—
25	C₄²×D₁₃₇	—	2	2	no	—
26	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
27	#17984022	C₂²	1	1	yes	characteristic
28	#17984019	C₂²	1	2	yes	—
29	#17984012	C₂²	1	2	yes	—
30	#17984010	C₂²	1	2	yes	—
31	#17984008	C₂²	1	2	yes	—
32	C₄×#11764040	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
37	#107×C₁₃₇	C₂²	1	1	yes	characteristic
38	#103×C₁₃₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
40	#14555511	—	2	4	no	—
41	#14555511	C₂×C₄	1	8	yes	complemented
42	C₂×D₅₄₈	—	2	2	no	—
43	C₂×D₅₄₈	C₂³	1	2	yes	—
44	C₂×C₄×D₁₃₇	—	2	2	no	—
45	C₂×C₄×D₁₃₇	—	2	2	no	—
46	C₂×C₄×D₁₃₇	—	2	2	no	—
47	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
50	#14555497	C₂×C₄	1	8	yes	complemented
51	D₁₀₉₆	C₂×C₄	1	4	yes	complemented
52	C₂×#11764040	C₂³	1	1	yes	characteristic
53	C₂×#11764040	C₂³	1	1	yes	characteristic
54	C₈×D₁₃₇	—	2	2	no	—
55	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
56	C₄⋊Dic₁₃₇	—	2	2	no	—
57	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
58	#14555487	—	2	2	no	—
59	#14555487	—	2	2	no	—
60	#14555487	—	2	2	no	—
61	C₄×Dic₁₃₇	—	2	2	no	—
62	C₄×Dic₁₃₇	—	2	2	no	—
63	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
64	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
65	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
67	Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
68	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
72	C₄²×C₁₃₇	—	2	2	no	—
73	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₃₇	—	2	2	no	—
75	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
76	D₅₄₈	—	2	8	no	—
77	D₅₄₈	—	2	2	no	—
78	D₅₄₈	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₃₇	—	2	2	no	—
80	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄×D₁₃₇	—	2	2	no	—
84	C₄×D₁₃₇	—	2	8	no	—
85	C₄×D₁₃₇	—	2	2	no	—
86	C₄×D₁₃₇	—	4	4	no	—
87	C₄×D₁₃₇	—	4	4	no	—
88	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
89	C₄×D₁₃₇	#41	1	2	yes	—
90	#11764040	—	2	1	no	—
91	#11764040	C₂²×C₄	1	2	yes	—
92	Q₈×C₁₃₇	—	2	2	no	—
93	Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₃₇	—	2	1	no	—
100	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₁₃₇	—	2	1	no	—
103	C₂×D₁₃₇	—	2	1	no	—
104	C₂×D₁₃₇	—	2	4	no	—
105	C₂×D₁₃₇	—	4	2	no	—
106	C₂×D₁₃₇	—	4	2	no	—
107	C₂×D₁₃₇	C₄×D₄	1	2	yes	—
108	C₄×C₁₃₇	—	2	4	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	4	2	no	—
111	C₄×C₁₃₇	#135	1	2	yes	complemented
112	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
114	Dic₁₃₇	—	2	1	no	—
115	Dic₁₃₇	—	4	2	no	—
116	Dic₁₃₇	C₄×D₄	1	2	yes	—
117	Dic₁₃₇	#135	1	2	yes	complemented
118	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
120	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
121	D₁₃₇	—	2	2	no	—
122	D₁₃₇	—	4	4	no	—
123	C₁₃₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
124	C₄×#135	—	137	1	no	maximal, Sylow
125	C₂×#135	—	137	1	no	—
126	C₄×#41	—	137	2	no	—
127	C₄×Q₁₆	—	137	1	no	—
128	C₄×Q₁₆	D₁₃₇	1	1	yes	characteristic, complemented
129	C₄×SD₁₆	—	137	2	no	—
130	C₄×SD₁₆	—	137	2	no	—
131	C₄×D₈	—	137	1	no	—
132	C₄×D₈	—	137	1	no	—
133	#427	—	137	1	no	—
134	#416	—	137	2	no	—
135	C₂×C₄×C₈	—	137	1	no	—
136	C₂×#41	—	137	2	no	—
137	#135	—	137	8	no	—
138	C₂×Q₁₆	C₂×D₁₃₇	1	1	yes	characteristic
139	C₂×SD₁₆	—	137	2	no	—
140	C₂×D₈	—	137	1	no	—
141	C₂²×C₈	—	137	1	no	—
142	C₂²×C₈	—	137	1	no	—
143	C₄×Q₈	—	137	2	no	—
144	C₄×Q₈	C₂×D₁₃₇	1	2	yes	—
145	C₄×D₄	—	274	2	no	—
146	C₄×D₄	—	274	2	no	—
147	C₄×D₄	—	137	2	no	—
148	C₄×D₄	—	137	2	no	—
149	#117	—	274	2	no	—
150	#117	—	137	2	no	—
151	C₂×C₄²	—	274	2	no	—
152	C₂×C₄²	—	137	1	no	—
153	#107	—	137	1	no	—
154	#107	C₂×D₁₃₇	1	1	yes	characteristic
155	#106	—	137	2	no	—
156	#103	—	137	2	no	—
157	#103	C₂×D₁₃₇	1	2	yes	—
158	#102	—	137	2	no	—
159	#102	—	137	2	no	—
160	C₄×C₈	—	137	1	no	—
161	C₄×C₈	—	137	1	no	—
162	C₄×C₈	—	137	1	no	—
163	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
164	#41	—	274	4	no	—
165	#41	—	137	8	no	—
166	C₂×Q₈	C₂²×D₁₃₇	1	2	yes	—
167	C₂×D₄	—	274	2	no	—
168	C₂×D₄	—	137	2	no	—
169	C₂²×C₄	—	274	2	no	—
170	C₂²×C₄	—	274	2	no	—
171	C₂²×C₄	—	274	2	no	—
172	C₂²×C₄	—	137	1	no	—
173	C₂²×C₄	—	137	1	no	—
174	C₂²×C₄	—	137	1	no	—
175	Q₁₆	C₄×D₁₃₇	1	4	yes	complemented
176	SD₁₆	—	137	8	no	—
177	D₈	—	137	4	no	—
178	C₂×C₈	—	274	2	no	—
179	C₂×C₈	—	137	1	no	—
180	C₂×C₈	—	137	1	no	—
181	C₂×C₈	—	137	4	no	—
182	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	274	2	no	—
186	C₄⋊C₄	—	137	2	no	—
187	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
188	C₂²⋊C₄	—	274	2	no	—
189	C₂²⋊C₄	—	274	2	no	—
190	C₂²⋊C₄	—	274	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	274	2	no	—
193	C₄²	—	274	2	no	—
194	C₄²	—	137	1	no	—
195	C₄²	—	137	1	no	—
196	C₄²	—	137	1	no	—
197	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
198	C₂³	—	274	2	no	—
199	C₂³	—	137	1	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×C₄×D₁₃₇	1	4	yes	—
202	D₄	—	274	8	no	—
203	D₄	—	274	2	no	—
204	D₄	—	137	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	274	8	no	—
208	C₂×C₄	—	274	2	no	—
209	C₂×C₄	—	274	2	no	—
210	C₂×C₄	—	274	2	no	—
211	C₂×C₄	—	548	4	no	—
212	C₂×C₄	—	548	4	no	—
213	C₂×C₄	—	137	1	no	—
214	C₂×C₄	—	137	1	no	—
215	C₂×C₄	—	137	1	no	—
216	C₂×C₄	—	137	2	no	—
217	C₂×C₄	—	137	2	no	—
218	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
219	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	274	1	no	—
223	C₈	—	137	2	no	—
224	C₈	C₂×C₄×D₁₃₇	1	2	yes	—
225	C₂²	—	274	1	no	—
226	C₂²	—	274	1	no	—
227	C₂²	—	274	4	no	—
228	C₂²	—	548	2	no	—
229	C₂²	—	548	2	no	—
230	C₂²	—	137	2	no	—
231	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	274	1	no	—
236	C₄	—	548	2	no	—
237	C₄	—	137	2	no	—
238	C₄	—	137	2	no	—
239	C₄	#17984111	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
241	C₄	#41×D₁₃₇	1	1	yes	characteristic
242	C₂	—	274	2	no	—
243	C₂	—	548	4	no	—
244	C₂	C₄×D₄×D₁₃₇	1	1	yes	characteristic
245	C₂	#135×D₁₃₇	1	1	yes	characteristic
246	C₂	C₂×#17984111	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₄×Q₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984111	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃₇
hypocenter	C₁₃₇	C₄×#135
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14555511	C₂	semidirect
3. Q₁₆×Dic₁₃₇	C₂	semidirect
4. C₄×#14555497	C₂	semidirect
5. #22577488	C₂	semidirect
6. C₄×D₁₀₉₆	C₂	semidirect
7. #22577485	C₂	semidirect
8. #22577440	C₂	semidirect
9. #22577398	C₂	semidirect
10. C₄×C₈×D₁₃₇	C₂	semidirect
11. C₄×Q₁₆×C₁₃₇	C₂	semidirect
13. #17984111	C₄	direct (N × H)
41. #14555511	C₂×C₄	semidirect
50. #14555497	C₂×C₄	semidirect
51. D₁₀₉₆	C₂×C₄	semidirect
55. C₈×D₁₃₇	C₂×C₄	semidirect
67. Q₁₆×C₁₃₇	C₂×C₄	semidirect
111. C₄×C₁₃₇	#135	semidirect
117. Dic₁₃₇	#135	semidirect
123. C₁₃₇	C₄×#135	semidirect
128. C₄×Q₁₆	D₁₃₇	semidirect
175. Q₁₆	C₄×D₁₃₇	semidirect
239. C₄	#17984111	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984111	2
C₄×#14555511	2
Q₁₆×Dic₁₃₇	2
C₄×#14555497	2
#22577488	2
C₄×D₁₀₉₆	2
#22577485	2
#22577440	2
#22577398	2
C₄×C₈×D₁₃₇	2
C₄×Q₁₆×C₁₃₇	2
C₄×#135	137

Subgroups — C4×#17984111 #31789561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984111 #31789561