Subgroups — C₁₃₇×#9802 #46859188

Order 35072.

610 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1965×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1567×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1545×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₇	C₄	1	4	yes	—
7	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
8	#493×C₁₃₇	—	2	1	no	—
9	#493×C₁₃₇	—	2	1	no	—
10	#493×C₁₃₇	—	2	2	no	—
11	#431×C₁₃₇	C₂²	1	1	yes	characteristic
12	#423×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#100×C₁₃₇	—	2	4	no	—
14	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#335×C₁₃₇	—	2	2	no	—
17	#331×C₁₃₇	—	2	4	no	—
18	#322×C₁₃₇	—	2	1	no	—
19	C₈²×C₁₃₇	—	2	1	no	—
20	C₂²×D₄×C₁₃₇	—	2	2	no	—
21	C₂²×D₄×C₁₃₇	—	2	2	no	—
22	C₂²×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂×D₈×C₁₃₇	—	2	4	no	—
24	C₂×D₈×C₁₃₇	—	2	4	no	—
25	C₂×D₈×C₁₃₇	—	4	8	no	—
26	C₂×M₁₆×C₁₃₇	—	2	2	no	—
27	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	#127×C₁₃₇	—	2	2	no	—
30	#127×C₁₃₇	—	2	2	no	—
31	#127×C₁₃₇	—	2	2	no	—
32	#127×C₁₃₇	—	2	2	no	—
33	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₇	—	2	2	no	—
35	C₄⋊C₈×C₁₃₇	—	2	2	no	—
36	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
37	#100×C₁₃₇	—	4	8	no	—
38	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
39	#97×C₁₃₇	—	2	1	no	—
40	C₄×C₈×C₁₃₇	—	2	1	no	—
41	C₄×C₈×C₁₃₇	—	2	1	no	—
42	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
45	C₂⁴×C₁₃₇	—	2	4	no	—
46	C₂×D₄×C₁₃₇	—	2	4	no	—
47	C₂×D₄×C₁₃₇	—	2	4	no	—
48	C₂×D₄×C₁₃₇	—	4	8	no	—
49	C₂×D₄×C₁₃₇	—	4	2	no	—
50	C₂×D₄×C₁₃₇	—	4	2	no	—
51	C₂×D₄×C₁₃₇	—	4	2	no	—
52	C₂×D₄×C₁₃₇	—	4	2	no	—
53	C₂×D₄×C₁₃₇	—	4	4	no	—
54	C₂×D₄×C₁₃₇	—	4	4	no	—
55	C₂×D₄×C₁₃₇	—	8	4	no	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
59	D₈×C₁₃₇	—	4	8	no	—
60	D₈×C₁₃₇	—	4	8	no	—
61	M₁₆×C₁₃₇	—	2	2	no	—
62	M₁₆×C₁₃₇	—	4	4	no	—
63	M₁₆×C₁₃₇	D₈	1	4	yes	complemented
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	4	no	—
66	C₂×C₈×C₁₃₇	—	2	2	no	—
67	C₂×C₈×C₁₃₇	—	4	2	no	—
68	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₃₇	#41	1	1	yes	characteristic
74	C₂³×C₁₃₇	—	4	4	no	—
75	C₂³×C₁₃₇	—	4	8	no	—
76	C₂³×C₁₃₇	—	8	4	no	—
77	C₂³×C₁₃₇	—	8	4	no	—
78	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
79	D₄×C₁₃₇	—	4	8	no	—
80	D₄×C₁₃₇	—	8	4	no	—
81	D₄×C₁₃₇	—	8	4	no	—
82	D₄×C₁₃₇	—	8	4	no	—
83	D₄×C₁₃₇	—	8	4	no	—
84	D₄×C₁₃₇	—	8	2	no	—
85	D₄×C₁₃₇	—	8	2	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃₇	#135	1	2	yes	—
96	C₈×C₁₃₇	—	2	2	no	—
97	C₈×C₁₃₇	—	2	4	no	—
98	C₈×C₁₃₇	—	4	2	no	—
99	C₈×C₁₃₇	—	8	2	no	—
100	C₈×C₁₃₇	#100	1	4	yes	complemented
101	C₂²×C₁₃₇	—	2	1	no	—
102	C₂²×C₁₃₇	—	2	1	no	—
103	C₂²×C₁₃₇	—	8	4	no	—
104	C₂²×C₁₃₇	—	8	4	no	—
105	C₂²×C₁₃₇	—	8	4	no	—
106	C₂²×C₁₃₇	—	16	8	no	—
107	C₂²×C₁₃₇	#390	1	1	yes	characteristic
108	C₂²×C₁₃₇	C₄×D₈	1	2	yes	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	C₂×#100	1	2	yes	—
112	C₄×C₁₃₇	#466	1	2	yes	—
113	C₄×C₁₃₇	#472	1	2	yes	—
114	C₄×C₁₃₇	#493	1	1	yes	characteristic
115	C₄×C₁₃₇	#495	1	1	yes	characteristic
116	C₂×C₁₃₇	—	2	2	no	—
117	C₂×C₁₃₇	—	16	4	no	—
118	C₂×C₁₃₇	#1965	1	1	yes	characteristic
119	C₂×C₁₃₇	#1970	1	1	yes	characteristic
120	C₂×C₁₃₇	#1978	1	1	yes	characteristic
121	#9802	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₇	#9802	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic
124	#1965	C₂×C₁₃₇	1	2	yes	complemented
125	#1567	C₂×C₁₃₇	1	1	yes	characteristic, complemented
126	#1545	C₂×C₁₃₇	1	2	yes	complemented
127	C₈×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
128	C₂²×D₈	C₄×C₁₃₇	1	4	yes	—
129	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
130	#493	—	2	1	no	—
131	#493	—	2	1	no	—
132	#493	—	2	2	no	—
133	#431	C₂²×C₁₃₇	1	1	yes	characteristic
134	#423	C₂²×C₁₃₇	1	2	yes	—
135	C₂×#100	—	2	4	no	—
136	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
138	#335	—	2	2	no	—
139	#331	—	2	4	no	—
140	#322	—	2	1	no	—
141	C₈²	—	2	1	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	—	4	8	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
151	#127	—	2	2	no	—
152	#127	—	2	2	no	—
153	#127	—	2	2	no	—
154	#127	—	2	2	no	—
155	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
159	#100	—	4	8	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	#97	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
165	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
166	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
167	C₂⁴	—	2	4	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	4	8	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	8	4	no	—
178	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
181	D₈	—	4	8	no	—
182	D₈	—	4	8	no	—
183	M₁₆	—	2	2	no	—
184	M₁₆	—	4	4	no	—
185	M₁₆	D₈×C₁₃₇	1	4	yes	complemented
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
191	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
192	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
195	C₄²	#41×C₁₃₇	1	1	yes	characteristic
196	C₂³	—	4	4	no	—
197	C₂³	—	4	8	no	—
198	C₂³	—	8	4	no	—
199	C₂³	—	8	4	no	—
200	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
201	D₄	—	4	8	no	—
202	D₄	—	8	4	no	—
203	D₄	—	8	4	no	—
204	D₄	—	8	4	no	—
205	D₄	—	8	4	no	—
206	D₄	—	8	2	no	—
207	D₄	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
213	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
214	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
217	C₂×C₄	#135×C₁₃₇	1	2	yes	—
218	C₈	—	2	2	no	—
219	C₈	—	2	4	no	—
220	C₈	—	4	2	no	—
221	C₈	—	8	2	no	—
222	C₈	#100×C₁₃₇	1	4	yes	complemented
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	8	4	no	—
226	C₂²	—	8	4	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	16	8	no	—
229	C₂²	#390×C₁₃₇	1	1	yes	characteristic
230	C₂²	C₄×D₈×C₁₃₇	1	2	yes	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	C₂×#100×C₁₃₇	1	2	yes	—
234	C₄	#466×C₁₃₇	1	2	yes	—
235	C₄	#472×C₁₃₇	1	2	yes	—
236	C₄	#493×C₁₃₇	1	1	yes	characteristic
237	C₄	#495×C₁₃₇	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	16	4	no	—
240	C₂	#1965×C₁₃₇	1	1	yes	characteristic
241	C₂	#1970×C₁₃₇	1	1	yes	characteristic
242	C₂	#1978×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂×#493×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1965×C₁₃₇	C₂	semidirect
3. #1567×C₁₃₇	C₂	semidirect
4. #1545×C₁₃₇	C₂	semidirect
5. C₈×M₁₆×C₁₃₇	C₂	semidirect
63. M₁₆×C₁₃₇	D₈	semidirect
100. C₈×C₁₃₇	#100	semidirect
121. #9802	C₁₃₇	direct (N × H)
122. C₁₃₇	#9802	direct (N × H)
124. #1965	C₂×C₁₃₇	semidirect
125. #1567	C₂×C₁₃₇	semidirect
126. #1545	C₂×C₁₃₇	semidirect
127. C₈×M₁₆	C₂×C₁₃₇	semidirect
185. M₁₆	D₈×C₁₃₇	semidirect
222. C₈	#100×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₇	2
#1965×C₁₃₇	2
#1567×C₁₃₇	2
#1545×C₁₃₇	2
C₈×M₁₆×C₁₃₇	2
#9802	137

Subgroups — C137×#9802 #46859188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9802 #46859188