Subgroups — #31789823

Order 17536.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
2	#22577482	C₂	1	2	yes	maximal, complemented
3	#22577473	C₂	1	4	yes	maximal, complemented
4	C₈×D₅₄₈	C₂	1	2	yes	maximal, complemented
5	#22577460	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577365	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
8	#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₄₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
11	C₂×#14555493	C₂²	1	4	yes	complemented
12	C₂×C₈×D₁₃₇	C₂²	1	4	yes	complemented
13	C₄×D₅₄₈	C₄	1	2	yes	—
14	C₄×D₅₄₈	C₄	1	1	yes	characteristic
15	C₄×D₅₄₈	C₄	1	1	yes	characteristic
16	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
17	C₄×D₅₄₈	C₂²	1	2	yes	—
18	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
20	C₂×#14555487	C₄	1	2	yes	—
21	C₄⋊#11764040	C₂²	1	2	yes	complemented
22	C₂²⋊#11764040	C₂²	1	2	yes	—
23	#17984000	C₂²	1	4	yes	complemented
24	#17984000	C₂²	1	4	yes	complemented
25	#17983999	C₂²	1	2	yes	complemented
26	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	complemented
28	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₃₇	C₂²	1	2	yes	complemented
31	C₂³×D₁₃₇	C₂×C₄	1	2	yes	—
32	C₂×D₅₄₈	—	2	4	no	—
33	C₂×D₅₄₈	C₂×C₄	1	2	yes	—
34	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
35	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₃₇	—	4	4	no	—
38	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
42	#14555493	—	4	4	no	—
43	C₂×#11764040	C₂³	1	4	yes	—
44	C₈×D₁₃₇	—	4	4	no	—
45	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
48	#14555487	C₂×C₄	1	2	yes	—
49	#14555487	C₂×C₄	1	2	yes	—
50	#14555487	C₂×C₄	1	2	yes	—
51	#14555487	C₂×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
57	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
61	C₂²×D₁₃₇	—	2	4	no	—
62	C₂²×D₁₃₇	—	2	4	no	—
63	C₂²×D₁₃₇	—	4	2	no	—
64	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
65	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
66	D₅₄₈	—	2	4	no	—
67	D₅₄₈	—	2	2	no	—
68	D₅₄₈	M₁₆	1	4	yes	complemented
69	C₂×Dic₁₃₇	—	4	2	no	—
70	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
72	C₄×D₁₃₇	—	2	4	no	—
73	C₄×D₁₃₇	—	2	4	no	—
74	C₄×D₁₃₇	—	4	2	no	—
75	C₄×D₁₃₇	—	4	2	no	—
76	#11764040	—	2	4	no	—
77	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₃₇	—	2	2	no	—
88	C₈×C₁₃₇	—	2	2	no	—
89	C₂²×C₁₃₇	—	2	1	no	—
90	C₂²×C₁₃₇	—	2	1	no	—
91	C₂²×C₁₃₇	#131	1	2	yes	—
92	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₃₇	—	2	2	no	—
94	C₂×D₁₃₇	—	2	2	no	—
95	C₂×D₁₃₇	—	2	2	no	—
96	C₂×D₁₃₇	—	2	2	no	—
97	C₂×D₁₃₇	—	2	2	no	—
98	C₂×D₁₃₇	—	4	8	no	—
99	C₂×D₁₃₇	C₂×M₁₆	1	4	yes	—
100	C₄×C₁₃₇	—	2	1	no	—
101	C₄×C₁₃₇	—	2	1	no	—
102	C₄×C₁₃₇	—	2	1	no	—
103	C₄×C₁₃₇	—	2	1	no	—
104	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
105	C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
106	Dic₁₃₇	—	2	2	no	—
107	Dic₁₃₇	—	2	2	no	—
108	C₂×C₁₃₇	—	2	2	no	—
109	C₂×C₁₃₇	#518	1	1	yes	characteristic
110	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
112	D₁₃₇	—	2	4	no	—
113	D₁₃₇	—	4	2	no	—
114	C₁₃₇	#2974	1	1	yes	characteristic, complemented, Sylow
115	#2974	—	137	1	no	maximal, Sylow
116	C₂×C₄×D₄	—	137	1	no	—
117	#436	—	137	2	no	—
118	#435	—	137	4	no	—
119	C₈×D₄	—	137	2	no	—
120	#432	—	137	1	no	—
121	#431	D₁₃₇	1	1	yes	characteristic, complemented
122	#407	—	137	2	no	—
123	C₂×C₂²⋊C₈	—	137	2	no	—
124	C₂²×D₄	—	137	1	no	—
125	C₂³×C₄	—	137	2	no	—
126	C₂×M₁₆	—	137	4	no	—
127	C₂²×C₈	—	137	4	no	—
128	C₄×D₄	—	137	2	no	—
129	C₄×D₄	—	137	1	no	—
130	C₄×D₄	—	137	1	no	—
131	C₄×D₄	—	137	1	no	—
132	C₄×D₄	—	137	2	no	—
133	C₄×D₄	—	137	1	no	—
134	C₂×C₄⋊C₄	—	137	1	no	—
135	C₂×C₂²⋊C₄	—	137	2	no	—
136	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
137	C₄⋊C₈	—	137	2	no	—
138	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	complemented
139	C₂²⋊C₈	—	137	2	no	—
140	C₂²⋊C₈	—	137	4	no	—
141	C₂²⋊C₈	—	137	4	no	—
142	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
143	C₂²⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
144	#97	—	137	2	no	—
145	C₄×C₈	C₂×D₁₃₇	1	2	yes	complemented
146	C₂⁴	—	137	2	no	—
147	C₂×D₄	—	274	4	no	—
148	C₂×D₄	—	137	2	no	—
149	C₂×D₄	—	137	1	no	—
150	C₂×D₄	—	137	1	no	—
151	C₂²×C₄	—	548	4	no	—
152	C₂²×C₄	—	137	2	no	—
153	C₂²×C₄	—	137	2	no	—
154	C₂²×C₄	—	137	2	no	—
155	C₂²×C₄	—	137	2	no	—
156	C₂²×C₄	—	137	2	no	—
157	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
160	M₁₆	—	548	4	no	—
161	C₂×C₈	—	548	4	no	—
162	C₂×C₈	—	137	4	no	—
163	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
164	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
165	C₄⋊C₄	—	137	2	no	—
166	C₄⋊C₄	—	137	1	no	—
167	C₄⋊C₄	—	137	1	no	—
168	C₂²⋊C₄	—	137	2	no	—
169	C₂²⋊C₄	—	137	2	no	—
170	C₂²⋊C₄	—	137	2	no	—
171	C₂²⋊C₄	—	137	2	no	—
172	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
173	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
174	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
175	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
176	C₂³	—	274	4	no	—
177	C₂³	—	274	4	no	—
178	C₂³	—	548	2	no	—
179	C₂³	—	137	2	no	—
180	C₂³	—	137	2	no	—
181	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
182	D₄	—	274	4	no	—
183	D₄	—	274	2	no	—
184	D₄	—	137	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	274	4	no	—
189	C₂×C₄	—	274	4	no	—
190	C₂×C₄	—	548	2	no	—
191	C₂×C₄	—	548	2	no	—
192	C₂×C₄	—	548	2	no	—
193	C₂×C₄	—	137	2	no	—
194	C₂×C₄	—	137	2	no	—
195	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	274	4	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	274	2	no	—
207	C₂²	—	274	2	no	—
208	C₂²	—	274	2	no	—
209	C₂²	—	274	2	no	—
210	C₂²	—	274	2	no	—
211	C₂²	—	548	8	no	—
212	C₂²	—	137	4	no	—
213	C₂²	#17984094	1	2	yes	—
214	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	274	2	no	—
220	C₄	—	274	2	no	—
221	C₄	M₁₆×D₁₃₇	1	2	yes	—
222	C₄	#17984134	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	274	4	no	—
225	C₂	—	548	2	no	—
226	C₂	#22577877	1	1	yes	characteristic
227	C₂	C₂×M₁₆×D₁₃₇	1	1	yes	characteristic
228	C₂	C₂×#17984094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#431×C₁₃₇	C₂
prime radical	C₂²×D₅₄₈	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#431	D₁₃₇
hypocenter	C₁₃₇	#2974
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #22577482	C₂	semidirect
3. #22577473	C₂	semidirect
4. C₈×D₅₄₈	C₂	semidirect
5. #22577460	C₂	semidirect
6. #22577365	C₂	semidirect
7. C₂²⋊C₈×D₁₃₇	C₂	semidirect
8. #431×C₁₃₇	C₂	semidirect
11. C₂×#14555493	C₂²	semidirect
12. C₂×C₈×D₁₃₇	C₂²	semidirect
21. C₄⋊#11764040	C₂²	semidirect
23. #17984000	C₂²	semidirect
24. #17984000	C₂²	semidirect
25. #17983999	C₂²	semidirect
27. C₄⋊C₈×C₁₃₇	C₂²	semidirect
30. C₄×C₈×C₁₃₇	C₂²	semidirect
68. D₅₄₈	M₁₆	semidirect
114. C₁₃₇	#2974	semidirect
121. #431	D₁₃₇	semidirect
138. C₄⋊C₈	C₂×D₁₃₇	semidirect
145. C₄×C₈	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₄₈	2
#22577482	2
#22577473	2
C₈×D₅₄₈	2
#22577460	2
#22577365	2
C₂²⋊C₈×D₁₃₇	2
#431×C₁₃₇	2
#2974	137