Subgroups — C₄×#17984108 #31789559

Order 17536.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984108	C₂	1	1	yes	maximal, characteristic
2	C₄×#14555510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14555500	C₂	1	2	yes	maximal, complemented
4	C₄×#14555497	C₂	1	2	yes	maximal, complemented
5	C₄×#14555498	C₂	1	2	yes	maximal, complemented
6	C₄×#14555495	C₂	1	2	yes	maximal, complemented
7	#22577439	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577399	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577397	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	#17984108	C₄	1	8	yes	direct product factor
14	C₂×#14555500	C₂²	1	1	yes	characteristic
15	C₂×#14555498	C₂²	1	1	yes	characteristic
16	C₂×#14555497	C₂²	1	1	yes	characteristic
17	C₂×#14555495	C₂²	1	1	yes	characteristic
18	C₂²×#11764040	C₂²	1	1	yes	characteristic
19	C₂²×#11764040	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₇₄	C₂²	1	2	yes	—
21	C₄×D₅₄₈	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₃₇	—	2	2	no	—
23	#17984047	C₂²	1	1	yes	characteristic
24	#17984045	—	2	1	no	—
25	C₄²×D₁₃₇	—	2	1	no	—
26	#17984021	C₂²	1	2	yes	—
27	#17984019	C₂²	1	2	yes	—
28	#17984012	C₂²	1	2	yes	—
29	#17984011	C₂²	1	2	yes	—
30	#17984009	C₂²	1	2	yes	—
31	#17984008	C₂²	1	2	yes	—
32	C₄×#11764040	C₂²	1	2	yes	—
33	C₄×#11764040	C₂²	1	2	yes	—
34	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
36	C₄×D₄×C₁₃₇	—	2	2	no	—
37	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
38	#117×C₁₃₇	—	2	1	no	—
39	#117×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	—	2	1	no	—
41	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
42	#14555510	—	2	2	no	—
43	#14555510	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
47	C₂×C₄×D₁₃₇	—	2	1	no	—
48	C₂×C₄×D₁₃₇	—	2	2	no	—
49	#14555500	C₂×C₄	1	4	yes	complemented
50	#14555498	C₂×C₄	1	4	yes	complemented
51	#14555497	C₂×C₄	1	4	yes	complemented
52	#14555495	C₂×C₄	1	4	yes	complemented
53	C₂×#11764040	—	2	2	no	—
54	C₂×#11764040	C₂×C₄	1	4	yes	complemented
55	C₂×#11764040	C₂³	1	2	yes	—
56	C₂×#11764040	C₂³	1	1	yes	characteristic
57	C₂×#11764040	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
59	#14555489	—	2	2	no	—
60	C₂²⋊Dic₁₃₇	—	2	1	no	—
61	#14555487	—	2	2	no	—
62	C₄×Dic₁₃₇	—	2	1	no	—
63	C₄×Dic₁₃₇	—	2	2	no	—
64	#41×C₁₃₇	—	2	2	no	—
65	#41×C₁₃₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₃₇	—	2	1	no	—
68	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₃₇	—	2	2	no	—
74	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄²×C₁₃₇	—	2	1	no	—
78	C₄²×C₁₃₇	—	2	2	no	—
79	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
81	C₄²×C₁₃₇	C₂³	1	2	yes	—
82	C₂²×D₁₃₇	—	2	1	no	—
83	Dic₂₇₄	—	2	1	no	—
84	Dic₂₇₄	C₂²×C₄	1	2	yes	—
85	D₅₄₈	—	2	1	no	—
86	D₅₄₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₃₇	—	2	4	no	—
88	C₂×Dic₁₃₇	—	2	1	no	—
89	C₂×Dic₁₃₇	—	2	2	no	—
90	C₄×D₁₃₇	—	2	4	no	—
91	C₄×D₁₃₇	—	4	4	no	—
92	#11764040	—	2	2	no	—
93	#11764040	C₂²×C₄	1	2	yes	—
94	#11764040	C₂²×C₄	1	2	yes	—
95	C₂³×C₁₃₇	—	2	1	no	—
96	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₃₇	—	2	1	no	—
98	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
99	D₄×C₁₃₇	—	2	4	no	—
100	D₄×C₁₃₇	—	2	1	no	—
101	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	#41	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₃₇	—	2	2	no	—
114	C₂²×C₁₃₇	—	2	2	no	—
115	C₂²×C₁₃₇	—	4	2	no	—
116	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
117	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₃₇	—	2	2	no	—
119	C₂×D₁₃₇	—	4	2	no	—
120	C₄×C₁₃₇	—	2	2	no	—
121	C₄×C₁₃₇	—	2	2	no	—
122	C₄×C₁₃₇	—	4	2	no	—
123	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
124	C₄×C₁₃₇	#135	1	4	yes	complemented
125	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₃₇	—	2	2	no	—
128	Dic₁₃₇	—	4	2	no	—
129	C₂×C₁₃₇	—	2	2	no	—
130	C₂×C₁₃₇	—	4	2	no	—
131	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₃₇	C₂×#135	1	2	yes	—
133	D₁₃₇	—	4	2	no	—
134	C₁₃₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
135	C₄×#135	—	137	1	no	maximal, Sylow
136	C₂×#135	—	137	1	no	—
137	C₄×#41	—	137	1	no	—
138	C₄×#41	D₁₃₇	1	1	yes	characteristic, complemented
139	C₄×Q₁₆	—	137	2	no	—
140	C₄×SD₁₆	—	137	2	no	—
141	C₄×SD₁₆	—	137	2	no	—
142	C₄×D₈	—	137	2	no	—
143	#427	—	137	1	no	—
144	#416	—	137	1	no	—
145	#416	—	137	1	no	—
146	C₂×C₄×C₈	—	137	1	no	—
147	C₂×#41	—	137	1	no	—
148	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
149	#135	—	137	8	no	—
150	C₂×Q₁₆	—	137	1	no	—
151	C₂×SD₁₆	—	137	1	no	—
152	C₂×SD₁₆	—	137	1	no	—
153	C₂×D₈	—	137	1	no	—
154	C₂²×C₈	—	137	1	no	—
155	C₂²×C₈	—	137	1	no	—
156	C₄×Q₈	—	137	2	no	—
157	C₄×Q₈	C₂×D₁₃₇	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	274	2	no	—
160	C₄×D₄	—	137	2	no	—
161	C₄×D₄	C₂×D₁₃₇	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	274	1	no	—
164	#117	—	137	1	no	—
165	#117	C₂×D₁₃₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	274	1	no	—
168	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
169	#107	—	137	2	no	—
170	#106	—	137	2	no	—
171	#103	—	137	2	no	—
172	#103	—	137	2	no	—
173	#102	—	137	2	no	—
174	#102	—	137	2	no	—
175	C₄×C₈	—	137	2	no	—
176	C₄×C₈	—	137	2	no	—
177	#41	—	2	2	no	—
178	#41	—	274	2	no	—
179	#41	—	137	4	no	—
180	#41	C₄×D₁₃₇	1	4	yes	complemented
181	C₂×Q₈	—	137	1	no	—
182	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	274	1	no	—
185	C₂×D₄	—	137	1	no	—
186	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	274	1	no	—
190	C₂²×C₄	—	274	2	no	—
191	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
193	Q₁₆	—	137	4	no	—
194	SD₁₆	—	137	4	no	—
195	SD₁₆	—	137	4	no	—
196	D₈	—	137	4	no	—
197	C₂×C₈	—	274	2	no	—
198	C₂×C₈	—	137	2	no	—
199	C₂×C₈	—	137	4	no	—
200	C₂×C₈	—	137	1	no	—
201	C₂×C₈	—	137	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	274	2	no	—
204	C₄⋊C₄	—	137	2	no	—
205	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	274	1	no	—
209	C₂²⋊C₄	—	274	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	274	1	no	—
213	C₄²	—	274	2	no	—
214	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
216	C₄²	C₂²×D₁₃₇	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	274	1	no	—
219	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	274	1	no	—
222	Q₈	—	137	2	no	—
223	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	274	4	no	—
227	D₄	—	274	1	no	—
228	D₄	—	137	2	no	—
229	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	274	1	no	—
236	C₂×C₄	—	274	4	no	—
237	C₂×C₄	—	274	2	no	—
238	C₂×C₄	—	548	4	no	—
239	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
243	C₂×C₄	#14555510	1	2	yes	—
244	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
245	C₈	—	274	2	no	—
246	C₈	—	137	2	no	—
247	C₈	—	137	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	274	2	no	—
252	C₂²	—	548	2	no	—
253	C₂²	C₄×C₂²⋊D₁₃₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	274	2	no	—
259	C₄	—	548	2	no	—
260	C₄	C₄×C₂²⋊D₁₃₇	1	2	yes	—
261	C₄	#17984108	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
263	C₄	C₂×#14555510	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	548	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
268	C₂	C₂×#17984108	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₄×#41×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#17984108	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₄×#41	D₁₃₇
hypocenter	C₁₃₇	C₄×#135
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14555510	C₂	semidirect
3. C₄×#14555500	C₂	semidirect
4. C₄×#14555497	C₂	semidirect
5. C₄×#14555498	C₂	semidirect
6. C₄×#14555495	C₂	semidirect
7. #22577439	C₂	semidirect
8. #22577399	C₂	semidirect
9. #22577397	C₂	semidirect
10. C₂×C₄×#11764040	C₂	semidirect
11. C₄×#41×C₁₃₇	C₂	semidirect
13. #17984108	C₄	direct (N × H)
43. #14555510	C₂×C₄	semidirect
49. #14555500	C₂×C₄	semidirect
50. #14555498	C₂×C₄	semidirect
51. #14555497	C₂×C₄	semidirect
52. #14555495	C₂×C₄	semidirect
54. C₂×#11764040	C₂×C₄	semidirect
65. #41×C₁₃₇	C₂×C₄	semidirect
124. C₄×C₁₃₇	#135	semidirect
134. C₁₃₇	C₄×#135	semidirect
138. C₄×#41	D₁₃₇	semidirect
180. #41	C₄×D₁₃₇	semidirect
261. C₄	#17984108	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×#17984108	2
C₄×#14555510	2
C₄×#14555500	2
C₄×#14555497	2
C₄×#14555498	2
C₄×#14555495	2
#22577439	2
#22577399	2
#22577397	2
C₂×C₄×#11764040	2
C₄×#41×C₁₃₇	2
C₄×#135	137

Subgroups — C4×#17984108 #31789559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984108 #31789559