Subgroups — #31789630

Order 17536.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984121	C₂	1	1	yes	maximal, characteristic
2	#22577884	C₂	1	1	yes	maximal, characteristic
3	#22577880	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577509	C₂	1	2	yes	maximal, complemented
5	#22577502	C₂	1	2	yes	maximal, complemented
6	#22577497	C₂	1	2	yes	maximal, complemented
7	#22577490	C₂	1	2	yes	maximal, complemented
8	#22577397	C₂	1	1	yes	maximal, characteristic
9	C₂×#17984011	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577361	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
14	#17984121	C₄	1	8	yes	complemented
15	C₂×Dic₅₄₈	C₂²	1	2	yes	—
16	C₂×#14555499	C₂²	1	2	yes	—
17	C₂²×#11764040	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₇₄	—	2	1	no	—
19	C₄×Dic₂₇₄	—	2	1	no	—
20	#17984053	C₂²	1	2	yes	—
21	#17984053	C₂²	1	2	yes	—
22	#17984053	C₂²	1	2	yes	—
23	#17984050	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₁₃₇	—	2	2	no	—
25	#17984047	C₂²	1	1	yes	characteristic
26	#17984046	—	2	1	no	—
27	#17984044	—	2	2	no	—
28	C₄⋊C₄×D₁₃₇	—	2	1	no	—
29	#17984011	C₂²	1	2	yes	—
30	#17984011	C₂²	1	1	yes	characteristic
31	#17984011	C₂²	1	2	yes	—
32	#17984011	C₂²	1	1	yes	characteristic
33	#17984008	C₂²	1	1	yes	characteristic
34	#17984008	C₂²	1	1	yes	characteristic
35	#17983998	C₂²	1	2	yes	—
36	C₈×Dic₁₃₇	C₂²	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
38	#117×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
40	#107×C₁₃₇	C₂²	1	2	yes	—
41	#106×C₁₃₇	C₂²	1	2	yes	—
42	#14555510	—	2	2	no	—
43	#14555510	C₂×C₄	1	4	yes	—
44	C₂×Dic₂₇₄	—	2	1	no	—
45	C₂×Dic₂₇₄	—	2	2	no	—
46	C₂×Dic₂₇₄	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₇₄	C₂³	1	2	yes	—
48	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
49	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
51	C₂²×Dic₁₃₇	—	2	1	no	—
52	C₂×C₄×D₁₃₇	—	2	2	no	—
53	C₂×C₄×D₁₃₇	—	2	1	no	—
54	Dic₅₄₈	C₂×C₄	1	8	yes	complemented
55	#14555499	C₂×C₄	1	8	yes	complemented
56	C₂×#11764040	—	2	2	no	—
57	C₂×#11764040	C₂³	1	1	yes	characteristic
58	C₂×#11764040	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₃₇	D₄	1	1	yes	characteristic
60	C₄⋊Dic₁₃₇	D₄	1	1	yes	characteristic
61	#14555489	—	2	2	no	—
62	#14555489	—	2	2	no	—
63	#14555489	—	2	2	no	—
64	C₂²⋊Dic₁₃₇	D₄	1	2	yes	—
65	#14555487	—	2	2	no	—
66	C₄×Dic₁₃₇	—	2	2	no	—
67	C₄×Dic₁₃₇	—	2	2	no	—
68	C₄×Dic₁₃₇	—	2	2	no	—
69	C₄×Dic₁₃₇	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₃₇	—	2	1	no	—
71	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
72	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₃₇	—	2	1	no	—
75	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
79	C₄²×C₁₃₇	—	2	1	no	—
80	C₂²×D₁₃₇	—	2	1	no	—
81	Dic₂₇₄	—	2	4	no	—
82	Dic₂₇₄	—	2	2	no	—
83	Dic₂₇₄	—	2	1	no	—
84	Dic₂₇₄	C₂²×C₄	1	4	yes	—
85	Dic₂₇₄	C₂²×C₄	1	2	yes	—
86	D₅₄₈	—	2	1	no	—
87	D₅₄₈	C₂²×C₄	1	2	yes	—
88	C₂²⋊D₁₃₇	—	2	4	no	—
89	C₂×Dic₁₃₇	—	2	4	no	—
90	C₂×Dic₁₃₇	—	2	1	no	—
91	C₂×Dic₁₃₇	—	2	2	no	—
92	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
93	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
94	C₄×D₁₃₇	—	2	4	no	—
95	C₄×D₁₃₇	—	8	2	no	—
96	#11764040	—	2	1	no	—
97	#11764040	—	2	1	no	—
98	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	—	2	1	no	—
100	C₂×C₄×C₁₃₇	—	2	1	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	8	1	no	—
103	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	#41	1	2	yes	—
106	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
108	C₂²×C₁₃₇	—	2	1	no	—
109	C₂²×C₁₃₇	—	2	1	no	—
110	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
111	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₁₃₇	—	2	2	no	—
113	C₂×D₁₃₇	—	4	1	no	—
114	C₂×D₁₃₇	—	4	1	no	—
115	C₄×C₁₃₇	—	4	1	no	—
116	C₄×C₁₃₇	—	4	2	no	—
117	C₄×C₁₃₇	—	4	1	no	—
118	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
119	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
120	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
121	Dic₁₃₇	—	2	2	no	—
122	Dic₁₃₇	—	2	4	no	—
123	Dic₁₃₇	—	2	2	no	—
124	Dic₁₃₇	—	2	2	no	—
125	C₂×C₁₃₇	—	2	2	no	—
126	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₁₃₇	#577	1	1	yes	characteristic
128	C₂×C₁₃₇	#578	1	1	yes	characteristic
129	D₁₃₇	—	4	2	no	—
130	C₁₃₇	#2947	1	1	yes	characteristic, complemented, Sylow
131	#2947	—	137	1	no	maximal, Sylow
132	C₂×#137	—	137	1	no	—
133	#520	—	137	1	no	—
134	#519	—	137	1	no	—
135	#441	—	137	2	no	—
136	#440	—	137	2	no	—
137	C₄×Q₁₆	—	137	2	no	—
138	C₄×SD₁₆	—	137	2	no	—
139	#428	D₁₃₇	1	1	yes	characteristic, complemented
140	#416	—	137	1	no	—
141	C₂×#103	—	137	1	no	—
142	#405	—	137	1	no	—
143	C₂×#41	—	137	1	no	—
144	C₂²×Q₈	—	137	1	no	—
145	#137	—	137	8	no	—
146	C₂×Q₁₆	—	137	2	no	—
147	C₂×SD₁₆	—	137	2	no	—
148	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
149	C₂²×C₈	—	137	1	no	—
150	C₄×Q₈	—	274	1	no	—
151	C₄×Q₈	—	274	1	no	—
152	C₄×Q₈	—	137	2	no	—
153	C₄×Q₈	—	137	2	no	—
154	C₄×Q₈	—	137	2	no	—
155	C₄×D₄	—	274	2	no	—
156	C₄×D₄	—	137	2	no	—
157	#117	—	274	2	no	—
158	#117	—	274	1	no	—
159	#117	—	137	1	no	—
160	#117	C₂×D₁₃₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	274	1	no	—
162	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
163	#107	C₂×D₁₃₇	1	2	yes	—
164	#106	C₂×D₁₃₇	1	2	yes	—
165	#103	—	137	2	no	—
166	#103	—	137	2	no	—
167	#103	—	137	1	no	—
168	#103	—	137	1	no	—
169	#102	—	137	1	no	—
170	#102	—	137	1	no	—
171	#97	—	137	2	no	—
172	C₄×C₈	—	137	2	no	—
173	#41	—	274	2	no	—
174	#41	—	137	4	no	—
175	C₂×Q₈	—	274	1	no	—
176	C₂×Q₈	—	274	2	no	—
177	C₂×Q₈	—	137	4	no	—
178	C₂×Q₈	—	137	2	no	—
179	C₂×Q₈	—	137	1	no	—
180	C₂×D₄	—	274	1	no	—
181	C₂×D₄	—	137	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	274	2	no	—
184	C₂²×C₄	—	274	1	no	—
185	C₂²×C₄	—	274	1	no	—
186	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
187	Q₁₆	—	137	8	no	—
188	SD₁₆	—	137	8	no	—
189	M₁₆	C₄×D₁₃₇	1	4	yes	complemented
190	C₂×C₈	—	274	2	no	—
191	C₂×C₈	—	137	1	no	—
192	C₂×C₈	—	137	1	no	—
193	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	274	2	no	—
196	C₄⋊C₄	—	274	2	no	—
197	C₄⋊C₄	—	274	2	no	—
198	C₄⋊C₄	—	137	1	no	—
199	C₄⋊C₄	—	137	1	no	—
200	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	274	2	no	—
205	C₂²⋊C₄	—	137	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	274	2	no	—
208	C₄²	—	274	2	no	—
209	C₄²	—	274	2	no	—
210	C₄²	—	137	2	no	—
211	C₂³	—	274	1	no	—
212	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
213	Q₈	—	274	4	no	—
214	Q₈	—	274	2	no	—
215	Q₈	—	274	1	no	—
216	Q₈	—	137	4	no	—
217	Q₈	—	137	2	no	—
218	D₄	—	274	4	no	—
219	D₄	—	274	1	no	—
220	D₄	—	137	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	274	4	no	—
226	C₂×C₄	—	274	1	no	—
227	C₂×C₄	—	274	2	no	—
228	C₂×C₄	—	274	4	no	—
229	C₂×C₄	—	1096	2	no	—
230	C₂×C₄	—	137	2	no	—
231	C₂×C₄	—	137	2	no	—
232	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
233	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
234	C₂×C₄	#14555511	1	2	yes	—
235	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
236	C₈	—	274	1	no	—
237	C₈	—	274	1	no	—
238	C₈	C₂×C₄×D₁₃₇	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	274	2	no	—
242	C₂²	—	548	1	no	—
243	C₂²	—	548	1	no	—
244	C₂²	#17984050	1	2	yes	—
245	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	274	2	no	—
250	C₄	—	274	2	no	—
251	C₄	—	274	2	no	—
252	C₄	—	274	4	no	—
253	C₄	#17984050	1	2	yes	—
254	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
255	C₄	C₂×#14555511	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	548	2	no	—
258	C₂	C₂×#17984050	1	1	yes	characteristic
259	C₂	#22578213	1	1	yes	characteristic
260	C₂	#22578219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#428×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#14555510	C₂²
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#428	D₁₃₇
hypocenter	C₁₃₇	#2947
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. #22577880	C₂	semidirect
4. #22577509	C₂	semidirect
5. #22577502	C₂	semidirect
6. #22577497	C₂	semidirect
7. #22577490	C₂	semidirect
9. C₂×#17984011	C₂	semidirect
10. #22577361	C₂	semidirect
11. #428×C₁₃₇	C₂	semidirect
14. #17984121	C₄	semidirect
46. C₂×Dic₂₇₄	C₂×C₄	semidirect
54. Dic₅₄₈	C₂×C₄	semidirect
55. #14555499	C₂×C₄	semidirect
72. M₁₆×C₁₃₇	C₂×C₄	semidirect
130. C₁₃₇	#2947	semidirect
139. #428	D₁₃₇	semidirect
189. M₁₆	C₄×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984121	2
#22577884	2
#22577880	2
#22577509	2
#22577502	2
#22577497	2
#22577490	2
#22577397	2
C₂×#17984011	2
#22577361	2
#428×C₁₃₇	2
#2947	137