Subgroups — D₄×Q₈×D₁₃₇ #31794259

Order 17536.

1568 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
2	Q₈×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₇₄	C₂	1	3	yes	maximal, complemented
4	#22578068	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₁₃₇	C₂	1	2	yes	maximal, complemented
6	#22578066	C₂	1	6	yes	maximal, complemented
7	C₄⋊Q₈×D₁₃₇	C₂	1	3	yes	maximal, complemented
8	C₂²⋊Q₈×D₁₃₇	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₃₇	C₂	1	3	yes	maximal
11	D₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₂₇₄	C₂²	1	6	yes	complemented
13	C₂×Q₈×D₁₃₇	—	2	16	no	—
14	C₂×Q₈×D₁₃₇	C₂²	1	6	yes	—
15	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×D₁₃₇	C₂²	1	6	yes	complemented
17	C₂×Q₈×D₁₃₇	C₂²	1	2	yes	complemented
18	C₂×D₄×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₃₇	C₂²	1	6	yes	—
20	#17984088	C₂²	1	3	yes	complemented
21	#17984087	C₂²	1	6	yes	complemented
22	#17984087	C₂²	1	6	yes	complemented
23	#17984086	C₂²	1	6	yes	complemented
24	#17984068	C₂²	1	6	yes	complemented
25	#17984067	C₂²	1	6	yes	complemented
26	#17984066	C₂²	1	12	yes	complemented
27	#17984065	C₂²	1	12	yes	complemented
28	#17984064	C₂²	1	6	yes	complemented
29	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×Dic₂₇₄	C₂²	1	3	yes	complemented
31	#17984053	C₂²	1	3	yes	complemented
32	C₄×D₅₄₈	C₂²	1	3	yes	—
33	D₄×Dic₁₃₇	C₂²	1	3	yes	—
34	#17984050	C₂²	1	3	yes	—
35	C₄×C₂²⋊D₁₃₇	C₂²	1	6	yes	—
36	#17984048	C₂²	1	6	yes	—
37	C₄⋊C₄×D₁₃₇	C₂²	1	3	yes	—
38	C₄⋊C₄×D₁₃₇	C₂²	1	6	yes	—
39	C₄⋊C₄×D₁₃₇	C₂²	1	3	yes	—
40	C₂²⋊C₄×D₁₃₇	C₂²	1	6	yes	—
41	C₄²×D₁₃₇	C₂²	1	3	yes	—
42	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₁₃₇	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₁₃₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₃₇	C₂²	1	3	yes	—
47	C₂³×D₁₃₇	C₂³	1	2	yes	—
48	C₂×Dic₂₇₄	—	2	24	no	—
49	C₂×Dic₂₇₄	C₂³	1	12	yes	—
50	C₂×Dic₂₇₄	C₂³	1	6	yes	—
51	C₂×Dic₂₇₄	C₂³	1	3	yes	—
52	Q₈×D₁₃₇	—	2	24	no	—
53	Q₈×D₁₃₇	—	2	24	no	—
54	Q₈×D₁₃₇	—	2	8	no	—
55	Q₈×D₁₃₇	D₄	1	8	yes	direct product factor
56	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃₇	C₂³	1	2	yes	—
58	D₄×D₁₃₇	Q₈	1	8	yes	direct product factor
59	C₂²×Dic₁₃₇	C₂³	1	6	yes	—
60	C₂×C₄×D₁₃₇	—	2	24	no	—
61	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
62	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
63	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
64	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
65	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
66	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₃₇	C₂³	1	3	yes	—
68	C₄⋊Dic₁₃₇	C₂³	1	3	yes	—
69	C₄⋊Dic₁₃₇	C₂³	1	6	yes	—
70	#14555489	C₂³	1	6	yes	—
71	#14555489	C₂³	1	6	yes	—
72	#14555489	C₂³	1	12	yes	—
73	C₂²⋊Dic₁₃₇	C₂³	1	6	yes	—
74	#14555487	C₂³	1	6	yes	—
75	#14555487	C₂³	1	6	yes	—
76	C₄×Dic₁₃₇	C₂³	1	3	yes	—
77	C₄×Dic₁₃₇	C₂³	1	3	yes	—
78	C₄×Dic₁₃₇	C₂³	1	3	yes	—
79	C₂×Q₈×C₁₃₇	—	2	8	no	—
80	C₂×Q₈×C₁₃₇	C₂³	1	6	yes	—
81	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
85	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
86	C₄⋊C₄×C₁₃₇	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₁₃₇	C₂³	1	6	yes	—
88	C₄²×C₁₃₇	C₂³	1	3	yes	—
89	C₂²×D₁₃₇	—	2	4	no	—
90	C₂²×D₁₃₇	—	2	4	no	—
91	C₂²×D₁₃₇	C₂×Q₈	1	8	yes	complemented
92	C₂²×D₁₃₇	C₂⁴	1	2	yes	—
93	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
94	Dic₂₇₄	—	2	12	no	—
95	Dic₂₇₄	—	2	24	no	—
96	Dic₂₇₄	C₂×D₄	1	12	yes	complemented
97	D₅₄₈	C₂×Q₈	1	4	yes	complemented
98	C₂²⋊D₁₃₇	C₂×Q₈	1	8	yes	complemented
99	C₂×Dic₁₃₇	—	2	12	no	—
100	C₂×Dic₁₃₇	C₂⁴	1	3	yes	—
101	C₂×Dic₁₃₇	C₂⁴	1	6	yes	—
102	C₂×Dic₁₃₇	C₂⁴	1	3	yes	—
103	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₄×D₁₃₇	—	2	12	no	—
105	C₄×D₁₃₇	—	2	6	no	—
106	C₄×D₁₃₇	—	2	12	no	—
107	C₄×D₁₃₇	—	2	12	no	—
108	C₄×D₁₃₇	C₂×D₄	1	12	yes	—
109	C₄×D₁₃₇	C₂×Q₈	1	4	yes	complemented
110	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
111	Q₈×C₁₃₇	—	2	12	no	—
112	Q₈×C₁₃₇	C₂×D₄	1	4	yes	complemented
113	D₄×C₁₃₇	C₂×Q₈	1	4	yes	complemented
114	C₂×C₄×C₁₃₇	—	2	12	no	—
115	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
116	C₂×C₄×C₁₃₇	C₂⁴	1	6	yes	—
117	C₂×C₄×C₁₃₇	C₂⁴	1	3	yes	—
118	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	C₂²×Q₈	1	4	yes	complemented
122	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
123	C₂×D₁₃₇	—	2	2	no	—
124	C₂×D₁₃₇	—	2	2	no	—
125	C₂×D₁₃₇	—	2	16	no	—
126	C₂×D₁₃₇	C₂²×D₄	1	2	yes	—
127	C₂×D₁₃₇	C₂²×Q₈	1	2	yes	—
128	C₂×D₁₃₇	C₂²×Q₈	1	4	yes	complemented
129	C₂×D₁₃₇	2₋¹⁺⁴	1	2	yes	—
130	C₄×C₁₃₇	—	2	6	no	—
131	C₄×C₁₃₇	—	2	3	no	—
132	C₄×C₁₃₇	C₂²×D₄	1	6	yes	—
133	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	complemented
134	Dic₁₃₇	—	2	6	no	—
135	Dic₁₃₇	—	2	3	no	—
136	Dic₁₃₇	C₂²×D₄	1	6	yes	—
137	Dic₁₃₇	C₂²×Q₈	1	2	yes	complemented
138	C₂×C₁₃₇	—	2	4	no	—
139	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
140	C₂×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
141	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
142	D₁₃₇	—	2	4	no	—
143	D₁₃₇	D₄×Q₈	1	4	yes	direct product factor
144	C₁₃₇	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
145	C₂×D₄×Q₈	—	137	1	no	maximal, Sylow
146	C₂³×Q₈	—	137	2	no	—
147	D₄×Q₈	—	137	2	no	—
148	D₄×Q₈	—	137	3	no	—
149	D₄×Q₈	—	137	6	no	—
150	D₄×Q₈	—	137	1	no	—
151	D₄×Q₈	—	137	3	no	—
152	D₄×Q₈	D₁₃₇	1	1	yes	characteristic, direct product factor
153	C₂×C₄⋊Q₈	—	137	3	no	—
154	C₂×C₂²⋊Q₈	—	137	6	no	—
155	C₂×C₄×Q₈	—	137	1	no	—
156	C₂×C₄×D₄	—	137	3	no	—
157	C₂²×Q₈	—	274	16	no	—
158	C₂²×Q₈	—	137	6	no	—
159	C₂²×Q₈	—	137	6	no	—
160	C₂²×Q₈	—	137	2	no	—
161	C₂²×Q₈	—	137	1	no	—
162	C₂²×Q₈	—	137	6	no	—
163	C₂²×Q₈	C₂×D₁₃₇	1	2	yes	complemented
164	C₂²×D₄	—	137	1	no	—
165	C₂³×C₄	—	137	6	no	—
166	C₄⋊Q₈	—	137	6	no	—
167	C₄⋊Q₈	—	137	6	no	—
168	C₄⋊Q₈	—	137	6	no	—
169	C₄⋊Q₈	—	137	3	no	—
170	C₄⋊Q₈	C₂×D₁₃₇	1	3	yes	complemented
171	C₂²⋊Q₈	—	137	6	no	—
172	C₂²⋊Q₈	—	137	6	no	—
173	C₂²⋊Q₈	—	137	12	no	—
174	C₂²⋊Q₈	—	137	6	no	—
175	C₂²⋊Q₈	—	137	12	no	—
176	C₂²⋊Q₈	C₂×D₁₃₇	1	6	yes	complemented
177	C₄×Q₈	—	137	1	no	—
178	C₄×Q₈	—	137	3	no	—
179	C₄×Q₈	—	137	3	no	—
180	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
181	C₄×D₄	—	137	3	no	—
182	C₄×D₄	—	137	6	no	—
183	C₄×D₄	—	137	3	no	—
184	C₄×D₄	—	137	6	no	—
185	C₄×D₄	—	137	3	no	—
186	C₄×D₄	C₂×D₁₃₇	1	3	yes	—
187	C₂×C₄⋊C₄	—	137	3	no	—
188	C₂×C₄⋊C₄	—	137	3	no	—
189	C₂×C₄⋊C₄	—	137	6	no	—
190	C₂×C₂²⋊C₄	—	137	6	no	—
191	C₂×C₄²	—	137	3	no	—
192	C₂⁴	—	137	2	no	—
193	C₂×Q₈	—	2	8	no	—
194	C₂×Q₈	—	274	24	no	—
195	C₂×Q₈	—	274	24	no	—
196	C₂×Q₈	—	274	8	no	—
197	C₂×Q₈	—	274	24	no	—
198	C₂×Q₈	—	137	8	no	—
199	C₂×Q₈	—	137	6	no	—
200	C₂×Q₈	—	137	12	no	—
201	C₂×Q₈	—	137	3	no	—
202	C₂×Q₈	C₂²×D₁₃₇	1	6	yes	—
203	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
204	C₂×D₄	—	137	2	no	—
205	C₂×D₄	—	137	8	no	—
206	C₂×D₄	—	137	1	no	—
207	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	—	274	24	no	—
209	C₂²×C₄	—	137	6	no	—
210	C₂²×C₄	—	137	3	no	—
211	C₂²×C₄	—	137	6	no	—
212	C₂²×C₄	—	137	6	no	—
213	C₂²×C₄	—	137	6	no	—
214	C₂²×C₄	—	137	1	no	—
215	C₂²×C₄	—	137	3	no	—
216	C₂²×C₄	C₂²×D₁₃₇	1	6	yes	—
217	C₄⋊C₄	—	137	6	no	—
218	C₄⋊C₄	—	137	6	no	—
219	C₄⋊C₄	—	137	12	no	—
220	C₄⋊C₄	—	137	3	no	—
221	C₄⋊C₄	—	137	3	no	—
222	C₄⋊C₄	—	137	6	no	—
223	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
225	C₄⋊C₄	C₂²×D₁₃₇	1	6	yes	—
226	C₂²⋊C₄	—	137	6	no	—
227	C₂²⋊C₄	—	137	6	no	—
228	C₂²⋊C₄	—	137	6	no	—
229	C₂²⋊C₄	C₂²×D₁₃₇	1	6	yes	—
230	C₄²	—	137	3	no	—
231	C₄²	—	137	3	no	—
232	C₄²	—	137	3	no	—
233	C₄²	C₂²×D₁₃₇	1	3	yes	—
234	C₂³	—	274	4	no	—
235	C₂³	—	274	4	no	—
236	C₂³	—	137	8	no	—
237	C₂³	—	137	2	no	—
238	C₂³	—	137	1	no	—
239	C₂³	C₂³×D₁₃₇	1	2	yes	—
240	Q₈	—	2	12	no	—
241	Q₈	—	274	24	no	—
242	Q₈	—	274	12	no	—
243	Q₈	—	137	12	no	—
244	Q₈	D₄×D₁₃₇	1	4	yes	direct product factor
245	D₄	—	137	8	no	—
246	D₄	—	137	4	no	—
247	D₄	Q₈×D₁₃₇	1	4	yes	direct product factor
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	274	12	no	—
250	C₂×C₄	—	274	6	no	—
251	C₂×C₄	—	274	12	no	—
252	C₂×C₄	—	274	12	no	—
253	C₂×C₄	—	274	12	no	—
254	C₂×C₄	—	137	3	no	—
255	C₂×C₄	—	137	6	no	—
256	C₂×C₄	—	137	1	no	—
257	C₂×C₄	—	137	3	no	—
258	C₂×C₄	—	137	12	no	—
259	C₂×C₄	—	137	4	no	—
260	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
261	C₂×C₄	C₂³×D₁₃₇	1	6	yes	—
262	C₂×C₄	C₂³×D₁₃₇	1	3	yes	—
263	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	274	16	no	—
267	C₂²	—	274	2	no	—
268	C₂²	—	274	2	no	—
269	C₂²	—	137	4	no	—
270	C₂²	—	137	2	no	—
271	C₂²	—	137	2	no	—
272	C₂²	—	137	2	no	—
273	C₂²	C₂×Q₈×D₁₃₇	1	4	yes	complemented
274	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
275	C₄	—	2	6	no	—
276	C₄	—	2	3	no	—
277	C₄	—	274	6	no	—
278	C₄	—	274	3	no	—
279	C₄	—	137	6	no	—
280	C₄	—	137	2	no	—
281	C₄	C₂×D₄×D₁₃₇	1	6	yes	—
282	C₄	C₂×Q₈×D₁₃₇	1	2	yes	complemented
283	C₂	—	2	4	no	—
284	C₂	—	274	4	no	—
285	C₂	—	137	4	no	—
286	C₂	C₂²×D₄×D₁₃₇	1	1	yes	characteristic
287	C₂	C₂²×Q₈×D₁₃₇	1	1	yes	characteristic
288	C₂	2₋¹⁺⁴×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Fitting	D₄×Q₈×C₁₃₇	C₂
prime radical	C₂×D₄×D₁₃₇	C₂²
hypercenter	D₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₃₇	C₂	semidirect
2. Q₈×D₅₄₈	C₂	semidirect
3. D₄×Dic₂₇₄	C₂	semidirect
4. #22578068	C₂	semidirect
5. Q₈×C₂²⋊D₁₃₇	C₂	semidirect
6. #22578066	C₂	semidirect
7. C₄⋊Q₈×D₁₃₇	C₂	semidirect
8. C₂²⋊Q₈×D₁₃₇	C₂	semidirect
9. C₄×Q₈×D₁₃₇	C₂	semidirect
11. D₄×Q₈×C₁₃₇	C₂	semidirect
12. C₂²×Dic₂₇₄	C₂²	semidirect
16. C₂×Q₈×D₁₃₇	C₂²	semidirect
17. C₂×Q₈×D₁₃₇	C₂²	semidirect
20. #17984088	C₂²	semidirect
21. #17984087	C₂²	semidirect
22. #17984087	C₂²	semidirect
23. #17984086	C₂²	semidirect
24. #17984068	C₂²	semidirect
25. #17984067	C₂²	semidirect
26. #17984066	C₂²	semidirect
27. #17984065	C₂²	semidirect
28. #17984064	C₂²	semidirect
29. Q₈×Dic₁₃₇	C₂²	semidirect
30. C₄×Dic₂₇₄	C₂²	semidirect
31. #17984053	C₂²	semidirect
42. C₂²×Q₈×C₁₃₇	C₂²	semidirect
43. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
44. C₂²⋊Q₈×C₁₃₇	C₂²	semidirect
45. C₄×Q₈×C₁₃₇	C₂²	semidirect
55. Q₈×D₁₃₇	D₄	direct (N × H)
58. D₄×D₁₃₇	Q₈	direct (N × H)
91. C₂²×D₁₃₇	C₂×Q₈	semidirect
96. Dic₂₇₄	C₂×D₄	semidirect
97. D₅₄₈	C₂×Q₈	semidirect
98. C₂²⋊D₁₃₇	C₂×Q₈	semidirect
109. C₄×D₁₃₇	C₂×Q₈	semidirect
112. Q₈×C₁₃₇	C₂×D₄	semidirect
113. D₄×C₁₃₇	C₂×Q₈	semidirect
121. C₂²×C₁₃₇	C₂²×Q₈	semidirect
128. C₂×D₁₃₇	C₂²×Q₈	semidirect
133. C₄×C₁₃₇	C₂²×Q₈	semidirect
137. Dic₁₃₇	C₂²×Q₈	semidirect
143. D₁₃₇	D₄×Q₈	direct (N × H)
144. C₁₃₇	C₂×D₄×Q₈	semidirect
152. D₄×Q₈	D₁₃₇	direct (N × H)
163. C₂²×Q₈	C₂×D₁₃₇	semidirect
170. C₄⋊Q₈	C₂×D₁₃₇	semidirect
176. C₂²⋊Q₈	C₂×D₁₃₇	semidirect
180. C₄×Q₈	C₂×D₁₃₇	semidirect
244. Q₈	D₄×D₁₃₇	direct (N × H)
247. D₄	Q₈×D₁₃₇	direct (N × H)
273. C₂²	C₂×Q₈×D₁₃₇	semidirect
282. C₄	C₂×Q₈×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₃₇	2
Q₈×D₅₄₈	2
D₄×Dic₂₇₄	2
#22578068	2
Q₈×C₂²⋊D₁₃₇	2
#22578066	2
C₄⋊Q₈×D₁₃₇	2
C₂²⋊Q₈×D₁₃₇	2
C₄×Q₈×D₁₃₇	2
C₄×D₄×D₁₃₇	2
D₄×Q₈×C₁₃₇	2
C₂×D₄×Q₈	137

Subgroups — D4×Q8×D137 #31794259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₈×D₁₃₇ #31794259