Subgroups — #31789790

Order 17536.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577935	C₂	1	1	yes	maximal, characteristic
2	#22577533	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577527	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577481	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577478	C₂	1	2	yes	maximal, complemented
6	#22577474	C₂	1	2	yes	maximal, complemented
7	#22577461	C₂	1	2	yes	maximal, complemented
8	#22577455	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577358	C₂	1	2	yes	maximal, complemented
10	C₄×#14555493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555494	C₂²	1	2	yes	—
13	C₂×#14555493	C₂²	1	1	yes	characteristic
14	C₂×#14555493	C₂²	1	1	yes	characteristic
15	#17984081	C₄	1	2	yes	—
16	#17984079	C₄	1	1	yes	characteristic
17	#17984075	C₄	1	1	yes	characteristic
18	#17984070	C₄	1	2	yes	—
19	#17984068	C₄	1	1	yes	characteristic
20	#17984059	C₄	1	1	yes	characteristic
21	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
23	#17984047	C₂²	1	2	yes	—
24	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11764040	C₂²	1	1	yes	characteristic
26	#17984017	C₂²	1	2	yes	—
27	#17984017	C₂²	1	2	yes	—
28	C₂²⋊#11764040	C₂²	1	2	yes	—
29	#17984000	C₂²	1	2	yes	—
30	#17983998	C₂²	1	1	yes	characteristic
31	#17983998	C₂²	1	2	yes	—
32	#17983998	C₂²	1	1	yes	characteristic
33	C₄×#11764040	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₃₇	C₂²	1	2	yes	—
35	C₈×Dic₁₃₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
46	#14555494	—	2	4	no	—
47	#14555493	—	2	2	no	—
48	#14555493	—	2	2	no	—
49	C₂×#11764040	C₂³	1	2	yes	—
50	C₂×#11764040	C₂³	1	1	yes	characteristic
51	C₂×#11764040	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
55	#14555489	C₂×C₄	1	2	yes	—
56	#14555489	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
60	#14555487	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃₇	C₂³	1	2	yes	—
64	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
73	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
75	Dic₂₇₄	—	2	2	no	—
76	C₂²⋊D₁₃₇	—	4	2	no	—
77	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
82	C₄×D₁₃₇	—	2	2	no	—
83	C₄×D₁₃₇	—	2	1	no	—
84	C₄×D₁₃₇	—	2	1	no	—
85	C₄×D₁₃₇	—	4	1	no	—
86	#11764040	—	2	1	no	—
87	#11764040	—	2	2	no	—
88	#11764040	#41	1	2	yes	—
89	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
90	D₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₃₇	—	2	2	no	—
98	C₈×C₁₃₇	—	2	1	no	—
99	C₈×C₁₃₇	#41	1	2	yes	—
100	C₂²×C₁₃₇	—	2	2	no	—
101	C₂²×C₁₃₇	—	2	2	no	—
102	C₂²×C₁₃₇	—	2	2	no	—
103	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₃₇	—	2	1	no	—
105	C₂×D₁₃₇	—	2	1	no	—
106	C₂×D₁₃₇	—	2	1	no	—
107	C₄×C₁₃₇	—	2	2	no	—
108	C₄×C₁₃₇	—	2	1	no	—
109	C₄×C₁₃₇	#131	1	2	yes	—
110	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
112	Dic₁₃₇	—	2	1	no	—
113	Dic₁₃₇	—	2	1	no	—
114	Dic₁₃₇	—	2	2	no	—
115	Dic₁₃₇	—	2	1	no	—
116	Dic₁₃₇	—	2	2	no	—
117	C₂×C₁₃₇	—	4	2	no	—
118	C₂×C₁₃₇	C₄×#41	1	1	yes	characteristic
119	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
120	C₂×C₁₃₇	#568	1	1	yes	characteristic
121	D₁₃₇	—	4	1	no	—
122	C₁₃₇	#2969	1	1	yes	characteristic, complemented, Sylow
123	#2969	—	137	1	no	maximal, Sylow
124	#529	—	137	1	no	—
125	#446	—	137	1	no	—
126	C₈×Q₈	—	137	1	no	—
127	#436	—	137	1	no	—
128	#436	—	137	2	no	—
129	#435	—	137	2	no	—
130	C₈×D₄	D₁₃₇	1	1	yes	characteristic, complemented
131	#433	—	137	2	no	—
132	#431	—	137	1	no	—
133	#405	—	137	2	no	—
134	C₄×M₁₆	—	137	1	no	—
135	C₂×M₁₆	—	137	2	no	—
136	C₂×M₁₆	—	137	1	no	—
137	C₂×M₁₆	—	137	1	no	—
138	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
139	#126	—	137	2	no	—
140	#125	—	137	1	no	—
141	#124	—	137	1	no	—
142	#123	—	137	2	no	—
143	C₂²⋊Q₈	—	137	1	no	—
144	#121	—	137	1	no	—
145	C₄×Q₈	—	137	1	no	—
146	C₄×D₄	—	137	2	no	—
147	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
148	#117	—	137	2	no	—
149	C₂×C₄²	—	137	1	no	—
150	C₄⋊C₈	—	137	2	no	—
151	C₄⋊C₈	—	137	2	no	—
152	C₄⋊C₈	—	137	1	no	—
153	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
154	C₂²⋊C₈	—	137	2	no	—
155	C₂²⋊C₈	—	137	2	no	—
156	C₂²⋊C₈	C₂×D₁₃₇	1	2	yes	—
157	#97	—	137	1	no	—
158	#97	—	137	1	no	—
159	#97	—	137	2	no	—
160	C₄×C₈	—	137	2	no	—
161	C₄×C₈	—	137	2	no	—
162	C₄×C₈	—	137	1	no	—
163	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
164	C₂×Q₈	—	137	1	no	—
165	C₂×D₄	—	137	2	no	—
166	C₂×D₄	C₄×D₁₃₇	1	1	yes	characteristic
167	C₂²×C₄	—	137	1	no	—
168	C₂²×C₄	—	137	1	no	—
169	C₂²×C₄	—	137	1	no	—
170	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
171	M₁₆	—	274	2	no	—
172	M₁₆	—	274	4	no	—
173	M₁₆	—	274	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	137	2	no	—
176	C₂×C₈	—	137	1	no	—
177	C₂×C₈	—	137	1	no	—
178	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
179	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
181	C₄⋊C₄	—	137	2	no	—
182	C₄⋊C₄	—	137	2	no	—
183	C₄⋊C₄	—	137	2	no	—
184	C₄⋊C₄	—	137	2	no	—
185	C₄⋊C₄	—	137	1	no	—
186	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
187	C₂²⋊C₄	—	137	2	no	—
188	C₂²⋊C₄	—	137	2	no	—
189	C₂²⋊C₄	—	137	2	no	—
190	C₂²⋊C₄	—	137	2	no	—
191	C₂²⋊C₄	C₄×D₁₃₇	1	2	yes	—
192	C₄²	—	137	1	no	—
193	C₄²	—	137	1	no	—
194	C₄²	—	137	1	no	—
195	C₄²	—	137	2	no	—
196	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
197	C₂³	—	137	1	no	—
198	C₂³	C₂×C₄×D₁₃₇	1	2	yes	—
199	Q₈	—	274	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	548	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	274	2	no	—
205	C₂×C₄	—	274	1	no	—
206	C₂×C₄	—	274	1	no	—
207	C₂×C₄	—	548	1	no	—
208	C₂×C₄	—	137	1	no	—
209	C₂×C₄	—	137	1	no	—
210	C₂×C₄	—	137	1	no	—
211	C₂×C₄	—	137	2	no	—
212	C₂×C₄	—	137	2	no	—
213	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	274	2	no	—
220	C₈	—	274	1	no	—
221	C₈	—	137	2	no	—
222	C₈	#14555509	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	274	1	no	—
227	C₂²	—	274	1	no	—
228	C₂²	—	274	1	no	—
229	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	274	1	no	—
233	C₄	—	274	1	no	—
234	C₄	—	274	2	no	—
235	C₄	—	274	1	no	—
236	C₄	—	274	2	no	—
237	C₄	#17984094	1	2	yes	—
238	C₄	#41×D₁₃₇	1	1	yes	characteristic
239	C₄	C₂×#14555509	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	548	1	no	—
242	C₂	C₄×#14555509	1	1	yes	characteristic
243	C₂	C₂×#17984094	1	1	yes	characteristic
244	C₂	#22578168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₈×D₄×C₁₃₇	C₂
prime radical	#17984059	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₈×D₄	D₁₃₇
hypocenter	C₁₃₇	#2969
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #22577533	C₂	semidirect
3. #22577527	C₂	semidirect
4. #22577481	C₂	semidirect
5. #22577478	C₂	semidirect
6. #22577474	C₂	semidirect
7. #22577461	C₂	semidirect
8. #22577455	C₂	semidirect
9. #22577358	C₂	semidirect
10. C₄×#14555493	C₂	semidirect
11. C₈×D₄×C₁₃₇	C₂	semidirect
122. C₁₃₇	#2969	semidirect
130. C₈×D₄	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577935	2
#22577533	2
#22577527	2
#22577481	2
#22577478	2
#22577474	2
#22577461	2
#22577455	2
#22577358	2
C₄×#14555493	2
C₈×D₄×C₁₃₇	2
#2969	137