Subgroups — C₂×#1827×C₁₃₇ #46870093

Order 35072.

1354 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#410×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1827×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#339×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×H(3,ℤ/4)×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₄×#41×C₁₃₇	—	2	2	no	—
10	C₄×#41×C₁₃₇	—	2	2	no	—
11	C₄×#41×C₁₃₇	C₄	1	4	yes	complemented
12	C₄×#41×C₁₃₇	C₂²	1	4	yes	—
13	C₂×C₄×Q₈×C₁₃₇	—	2	1	no	—
14	C₂×C₄×D₄×C₁₃₇	—	2	2	no	—
15	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
16	C₂×#117×C₁₃₇	C₄	1	2	yes	complemented
17	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₁₃₇	C₄	1	2	yes	complemented
20	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
21	#413×C₁₃₇	—	2	2	no	—
22	#413×C₁₃₇	C₂²	1	4	yes	—
23	C₂×#101×C₁₃₇	—	2	1	no	—
24	C₂×#100×C₁₃₇	—	2	1	no	—
25	#410×C₁₃₇	—	2	2	no	—
26	#410×C₁₃₇	C₂²	1	4	yes	—
27	C₂×#99×C₁₃₇	—	2	2	no	—
28	#339×C₁₃₇	C₂²	1	8	yes	complemented
29	H(3,ℤ/4)×C₁₃₇	C₂²	1	8	yes	complemented
30	C₂×#41×C₁₃₇	—	2	2	no	—
31	C₂×#41×C₁₃₇	—	2	2	no	—
32	C₂×#41×C₁₃₇	—	2	2	no	—
33	C₂×#41×C₁₃₇	—	2	2	no	—
34	C₂×#41×C₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×#41×C₁₃₇	D₄	1	2	yes	—
36	C₂×#41×C₁₃₇	D₄	1	2	yes	—
37	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂²×D₄×C₁₃₇	D₄	1	2	yes	—
40	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₃₇	—	2	2	no	—
42	C₂³×C₄×C₁₃₇	—	2	1	no	—
43	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₁₃₇	—	2	2	no	—
46	C₂×M₁₆×C₁₃₇	—	2	2	no	—
47	C₂×M₁₆×C₁₃₇	—	2	2	no	—
48	C₂×M₁₆×C₁₃₇	—	2	2	no	—
49	C₂×M₁₆×C₁₃₇	—	2	2	no	—
50	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₁₃₇	—	2	2	no	—
52	C₄×Q₈×C₁₃₇	—	2	4	no	—
53	C₄×Q₈×C₁₃₇	—	2	4	no	—
54	C₄×D₄×C₁₃₇	—	2	4	no	—
55	C₄×D₄×C₁₃₇	—	2	8	no	—
56	C₄×D₄×C₁₃₇	—	2	8	no	—
57	C₄×D₄×C₁₃₇	—	2	4	no	—
58	#117×C₁₃₇	—	2	4	no	—
59	#117×C₁₃₇	—	2	1	no	—
60	#117×C₁₃₇	—	2	1	no	—
61	#117×C₁₃₇	—	2	2	no	—
62	#117×C₁₃₇	—	2	2	no	—
63	#117×C₁₃₇	—	2	2	no	—
64	#117×C₁₃₇	C₂×C₄	1	2	yes	—
65	#117×C₁₃₇	C₂×C₄	1	8	yes	complemented
66	#117×C₁₃₇	C₂³	1	4	yes	—
67	#117×C₁₃₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄²×C₁₃₇	—	2	2	no	—
75	C₂×C₄²×C₁₃₇	—	2	4	no	—
76	C₂×C₄²×C₁₃₇	—	2	1	no	—
77	C₂×C₄²×C₁₃₇	—	2	1	no	—
78	C₂×C₄²×C₁₃₇	—	2	1	no	—
79	C₂×C₄²×C₁₃₇	—	2	2	no	—
80	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
82	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
84	C₂×C₄²×C₁₃₇	C₂×C₄	1	8	yes	complemented
85	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
86	#101×C₁₃₇	—	2	4	no	—
87	#100×C₁₃₇	—	2	4	no	—
88	#99×C₁₃₇	—	2	8	no	—
89	C₂⁴×C₁₃₇	C₂²⋊C₄	1	2	yes	—
90	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	#41×C₁₃₇	—	2	4	no	—
92	#41×C₁₃₇	—	2	4	no	—
93	#41×C₁₃₇	—	2	4	no	—
94	#41×C₁₃₇	—	2	4	no	—
95	#41×C₁₃₇	—	2	4	no	—
96	#41×C₁₃₇	—	2	4	no	—
97	#41×C₁₃₇	—	2	4	no	—
98	#41×C₁₃₇	—	2	4	no	—
99	C₂×Q₈×C₁₃₇	—	2	2	no	—
100	C₂×Q₈×C₁₃₇	—	2	2	no	—
101	C₂×Q₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
102	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
103	C₂×D₄×C₁₃₇	—	2	4	no	—
104	C₂×D₄×C₁₃₇	—	2	4	no	—
105	C₂×D₄×C₁₃₇	—	2	2	no	—
106	C₂×D₄×C₁₃₇	—	2	2	no	—
107	C₂×D₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
108	C₂×D₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
109	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
110	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₁₃₇	—	2	1	no	—
112	C₂²×C₄×C₁₃₇	—	2	2	no	—
113	C₂²×C₄×C₁₃₇	—	2	2	no	—
114	C₂²×C₄×C₁₃₇	—	2	2	no	—
115	C₂²×C₄×C₁₃₇	—	2	2	no	—
116	C₂²×C₄×C₁₃₇	—	2	2	no	—
117	C₂²×C₄×C₁₃₇	—	2	1	no	—
118	C₂²×C₄×C₁₃₇	—	2	2	no	—
119	C₂²×C₄×C₁₃₇	—	2	8	no	—
120	C₂²×C₄×C₁₃₇	—	2	2	no	—
121	C₂²×C₄×C₁₃₇	—	4	4	no	—
122	C₂²×C₄×C₁₃₇	—	4	8	no	—
123	C₂²×C₄×C₁₃₇	—	4	2	no	—
124	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
125	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
126	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
133	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
134	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
135	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
136	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
137	M₁₆×C₁₃₇	—	2	4	no	—
138	M₁₆×C₁₃₇	—	2	4	no	—
139	M₁₆×C₁₃₇	—	4	2	no	—
140	M₁₆×C₁₃₇	—	4	2	no	—
141	C₂×C₈×C₁₃₇	—	2	4	no	—
142	C₂×C₈×C₁₃₇	—	4	2	no	—
143	C₂×C₈×C₁₃₇	—	4	2	no	—
144	C₄⋊C₄×C₁₃₇	—	2	2	no	—
145	C₄⋊C₄×C₁₃₇	—	2	8	no	—
146	C₄⋊C₄×C₁₃₇	—	2	2	no	—
147	C₄⋊C₄×C₁₃₇	—	2	4	no	—
148	C₄⋊C₄×C₁₃₇	—	4	2	no	—
149	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
150	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
151	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
152	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
153	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
154	C₄²×C₁₃₇	—	2	1	no	—
155	C₄²×C₁₃₇	—	2	2	no	—
156	C₄²×C₁₃₇	—	2	1	no	—
157	C₄²×C₁₃₇	—	2	2	no	—
158	C₄²×C₁₃₇	—	2	8	no	—
159	C₄²×C₁₃₇	—	2	2	no	—
160	C₄²×C₁₃₇	—	2	4	no	—
161	C₄²×C₁₃₇	—	4	2	no	—
162	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
163	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
164	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
165	C₂³×C₁₃₇	—	2	2	no	—
166	C₂³×C₁₃₇	—	2	2	no	—
167	C₂³×C₁₃₇	—	4	2	no	—
168	C₂³×C₁₃₇	—	4	4	no	—
169	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
171	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂³×C₁₃₇	#123	1	2	yes	—
173	C₂³×C₁₃₇	#123	1	2	yes	—
174	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₁₃₇	—	2	4	no	—
176	Q₈×C₁₃₇	—	2	4	no	—
177	D₄×C₁₃₇	—	2	8	no	—
178	D₄×C₁₃₇	—	2	8	no	—
179	D₄×C₁₃₇	—	2	4	no	—
180	D₄×C₁₃₇	—	2	4	no	—
181	C₂×C₄×C₁₃₇	—	2	4	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	2	no	—
184	C₂×C₄×C₁₃₇	—	2	2	no	—
185	C₂×C₄×C₁₃₇	—	2	4	no	—
186	C₂×C₄×C₁₃₇	—	2	4	no	—
187	C₂×C₄×C₁₃₇	—	2	4	no	—
188	C₂×C₄×C₁₃₇	—	2	4	no	—
189	C₂×C₄×C₁₃₇	—	2	2	no	—
190	C₂×C₄×C₁₃₇	—	2	4	no	—
191	C₂×C₄×C₁₃₇	—	4	4	no	—
192	C₂×C₄×C₁₃₇	—	4	4	no	—
193	C₂×C₄×C₁₃₇	—	4	16	no	—
194	C₂×C₄×C₁₃₇	—	4	4	no	—
195	C₂×C₄×C₁₃₇	—	4	2	no	—
196	C₂×C₄×C₁₃₇	—	4	4	no	—
197	C₂×C₄×C₁₃₇	—	8	4	no	—
198	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
199	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
202	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
203	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
204	C₂×C₄×C₁₃₇	#117	1	2	yes	—
205	C₂×C₄×C₁₃₇	#117	1	2	yes	—
206	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
207	C₂×C₄×C₁₃₇	#123	1	2	yes	—
208	C₂×C₄×C₁₃₇	#123	1	2	yes	—
209	C₂×C₄×C₁₃₇	#123	1	2	yes	—
210	C₂×C₄×C₁₃₇	#123	1	2	yes	—
211	C₂×C₄×C₁₃₇	#123	1	2	yes	—
212	C₂×C₄×C₁₃₇	#123	1	2	yes	—
213	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
216	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
217	C₈×C₁₃₇	—	4	4	no	—
218	C₂²×C₁₃₇	—	2	2	no	—
219	C₂²×C₁₃₇	—	2	4	no	—
220	C₂²×C₁₃₇	—	2	4	no	—
221	C₂²×C₁₃₇	—	4	1	no	—
222	C₂²×C₁₃₇	—	4	1	no	—
223	C₂²×C₁₃₇	—	4	2	no	—
224	C₂²×C₁₃₇	—	4	4	no	—
225	C₂²×C₁₃₇	—	4	4	no	—
226	C₂²×C₁₃₇	—	4	4	no	—
227	C₂²×C₁₃₇	#381	1	1	yes	characteristic
228	C₂²×C₁₃₇	#381	1	2	yes	—
229	C₂²×C₁₃₇	#381	1	1	yes	characteristic
230	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₁₃₇	C₂×#123	1	2	yes	—
232	C₄×C₁₃₇	—	2	4	no	—
233	C₄×C₁₃₇	—	2	4	no	—
234	C₄×C₁₃₇	—	4	2	no	—
235	C₄×C₁₃₇	—	4	4	no	—
236	C₄×C₁₃₇	—	8	4	no	—
237	C₄×C₁₃₇	#381	1	1	yes	characteristic
238	C₄×C₁₃₇	#381	1	1	yes	characteristic
239	C₄×C₁₃₇	#381	1	2	yes	—
240	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
241	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
242	C₄×C₁₃₇	C₂×#123	1	2	yes	—
243	C₂×C₁₃₇	—	2	1	no	—
244	C₂×C₁₃₇	—	2	2	no	—
245	C₂×C₁₃₇	—	2	1	no	—
246	C₂×C₁₃₇	—	2	2	no	—
247	C₂×C₁₃₇	—	4	4	no	—
248	C₂×C₁₃₇	#1827	1	2	yes	direct product factor
249	C₂×C₁₃₇	C₂×#381	1	1	yes	characteristic
250	C₂×#1827	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₁₃₇	C₂×#1827	1	1	yes	characteristic, direct product factor, Sylow
252	C₂×C₄×#41	C₂×C₁₃₇	1	1	yes	characteristic
253	C₂×#413	C₂×C₁₃₇	1	1	yes	characteristic
254	C₂×#410	C₂×C₁₃₇	1	1	yes	characteristic
255	#1827	C₂×C₁₃₇	1	8	yes	direct product factor
256	C₂×#339	C₂×C₁₃₇	1	2	yes	complemented
257	C₂×H(3,ℤ/4)	C₂×C₁₃₇	1	2	yes	complemented
258	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
259	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
260	C₄×#41	—	2	2	no	—
261	C₄×#41	—	2	2	no	—
262	C₄×#41	C₄×C₁₃₇	1	4	yes	complemented
263	C₄×#41	C₂²×C₁₃₇	1	4	yes	—
264	C₂×C₄×Q₈	—	2	1	no	—
265	C₂×C₄×D₄	—	2	2	no	—
266	C₂×C₄×D₄	—	2	1	no	—
267	C₂×#117	C₄×C₁₃₇	1	2	yes	complemented
268	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
269	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
270	C₂²×C₄²	C₄×C₁₃₇	1	2	yes	complemented
271	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
272	#413	—	2	2	no	—
273	#413	C₂²×C₁₃₇	1	4	yes	—
274	C₂×#101	—	2	1	no	—
275	C₂×#100	—	2	1	no	—
276	#410	—	2	2	no	—
277	#410	C₂²×C₁₃₇	1	4	yes	—
278	C₂×#99	—	2	2	no	—
279	#339	C₂²×C₁₃₇	1	8	yes	complemented
280	H(3,ℤ/4)	C₂²×C₁₃₇	1	8	yes	complemented
281	C₂×#41	—	2	2	no	—
282	C₂×#41	—	2	2	no	—
283	C₂×#41	—	2	2	no	—
284	C₂×#41	—	2	2	no	—
285	C₂×#41	C₂×C₄×C₁₃₇	1	2	yes	—
286	C₂×#41	D₄×C₁₃₇	1	2	yes	—
287	C₂×#41	D₄×C₁₃₇	1	2	yes	—
288	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
289	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
290	C₂²×D₄	D₄×C₁₃₇	1	2	yes	—
291	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
292	C₂³×C₄	—	2	2	no	—
293	C₂³×C₄	—	2	1	no	—
294	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
295	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
296	C₂×M₁₆	—	2	2	no	—
297	C₂×M₁₆	—	2	2	no	—
298	C₂×M₁₆	—	2	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
302	C₂²×C₈	—	2	2	no	—
303	C₄×Q₈	—	2	4	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×D₄	—	2	4	no	—
306	C₄×D₄	—	2	8	no	—
307	C₄×D₄	—	2	8	no	—
308	C₄×D₄	—	2	4	no	—
309	#117	—	2	4	no	—
310	#117	—	2	1	no	—
311	#117	—	2	1	no	—
312	#117	—	2	2	no	—
313	#117	—	2	2	no	—
314	#117	—	2	2	no	—
315	#117	C₂×C₄×C₁₃₇	1	2	yes	—
316	#117	C₂×C₄×C₁₃₇	1	8	yes	complemented
317	#117	C₂³×C₁₃₇	1	4	yes	—
318	#117	C₂³×C₁₃₇	1	2	yes	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₄²	—	2	2	no	—
326	C₂×C₄²	—	2	4	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
332	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
333	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
334	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
335	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	complemented
336	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
337	#101	—	2	4	no	—
338	#100	—	2	4	no	—
339	#99	—	2	8	no	—
340	C₂⁴	C₂²⋊C₄×C₁₃₇	1	2	yes	—
341	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
342	#41	—	2	4	no	—
343	#41	—	2	4	no	—
344	#41	—	2	4	no	—
345	#41	—	2	4	no	—
346	#41	—	2	4	no	—
347	#41	—	2	4	no	—
348	#41	—	2	4	no	—
349	#41	—	2	4	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
353	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	4	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
359	C₂×D₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
360	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
361	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	8	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
376	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
377	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
378	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
383	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
384	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
385	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
386	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
387	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
388	M₁₆	—	2	4	no	—
389	M₁₆	—	2	4	no	—
390	M₁₆	—	4	2	no	—
391	M₁₆	—	4	2	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	4	2	no	—
394	C₂×C₈	—	4	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	8	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	4	no	—
412	C₄²	—	4	2	no	—
413	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
414	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
415	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	4	no	—
420	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
421	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
422	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
423	C₂³	#123×C₁₃₇	1	2	yes	—
424	C₂³	#123×C₁₃₇	1	2	yes	—
425	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
426	Q₈	—	2	4	no	—
427	Q₈	—	2	4	no	—
428	D₄	—	2	8	no	—
429	D₄	—	2	8	no	—
430	D₄	—	2	4	no	—
431	D₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	16	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	8	4	no	—
449	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
450	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
451	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
452	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
453	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
454	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
455	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
456	C₂×C₄	#117×C₁₃₇	1	2	yes	—
457	C₂×C₄	#117×C₁₃₇	1	2	yes	—
458	C₂×C₄	#123×C₁₃₇	1	2	yes	—
459	C₂×C₄	#123×C₁₃₇	1	2	yes	—
460	C₂×C₄	#123×C₁₃₇	1	2	yes	—
461	C₂×C₄	#123×C₁₃₇	1	2	yes	—
462	C₂×C₄	#123×C₁₃₇	1	2	yes	—
463	C₂×C₄	#123×C₁₃₇	1	2	yes	—
464	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
467	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
468	C₈	—	4	4	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	#381×C₁₃₇	1	1	yes	characteristic
479	C₂²	#381×C₁₃₇	1	2	yes	—
480	C₂²	#381×C₁₃₇	1	1	yes	characteristic
481	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
482	C₂²	C₂×#123×C₁₃₇	1	2	yes	—
483	C₄	—	2	4	no	—
484	C₄	—	2	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	8	4	no	—
488	C₄	#381×C₁₃₇	1	1	yes	characteristic
489	C₄	#381×C₁₃₇	1	1	yes	characteristic
490	C₄	#381×C₁₃₇	1	2	yes	—
491	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
492	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
493	C₄	C₂×#123×C₁₃₇	1	2	yes	—
494	C₂	—	2	1	no	—
495	C₂	—	2	2	no	—
496	C₂	—	2	1	no	—
497	C₂	—	2	2	no	—
498	C₂	—	4	4	no	—
499	C₂	#1827×C₁₃₇	1	2	yes	direct product factor
500	C₂	C₂×#381×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#117
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	#381
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1827×C₁₃₇	C₂	direct (N × H)
5. C₂×#339×C₁₃₇	C₂	semidirect
6. C₂×H(3,ℤ/4)×C₁₃₇	C₂	semidirect
11. C₄×#41×C₁₃₇	C₄	semidirect
16. C₂×#117×C₁₃₇	C₄	semidirect
19. C₂²×C₄²×C₁₃₇	C₄	semidirect
28. #339×C₁₃₇	C₂²	semidirect
29. H(3,ℤ/4)×C₁₃₇	C₂²	semidirect
65. #117×C₁₃₇	C₂×C₄	semidirect
84. C₂×C₄²×C₁₃₇	C₂×C₄	semidirect
248. C₂×C₁₃₇	#1827	direct (N × H)
250. C₂×#1827	C₁₃₇	direct (N × H)
251. C₁₃₇	C₂×#1827	direct (N × H)
255. #1827	C₂×C₁₃₇	direct (N × H)
256. C₂×#339	C₂×C₁₃₇	semidirect
257. C₂×H(3,ℤ/4)	C₂×C₁₃₇	semidirect
262. C₄×#41	C₄×C₁₃₇	semidirect
267. C₂×#117	C₄×C₁₃₇	semidirect
270. C₂²×C₄²	C₄×C₁₃₇	semidirect
279. #339	C₂²×C₁₃₇	semidirect
280. H(3,ℤ/4)	C₂²×C₁₃₇	semidirect
316. #117	C₂×C₄×C₁₃₇	semidirect
335. C₂×C₄²	C₂×C₄×C₁₃₇	semidirect
499. C₂	#1827×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₁₃₇	2
C₂×#413×C₁₃₇	2
C₂×#410×C₁₃₇	2
#1827×C₁₃₇	2
C₂×#339×C₁₃₇	2
C₂×H(3,ℤ/4)×C₁₃₇	2
C₂×#1827	137

Subgroups — C2×#1827×C137 #46870093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1827×C₁₃₇ #46870093