Subgroups — C₂×C₃₁×#2148 #21490150

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2148	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#362	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#361	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	—	2	1	no	—
8	C₂×C₃₁×#135	—	2	2	no	—
9	C₂×C₃₁×#135	—	2	1	no	—
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₂×C₃₁×#135	—	2	2	no	—
12	C₂×C₃₁×#135	C₄	1	2	yes	—
13	C₂×C₃₁×#135	C₄	1	4	yes	—
14	C₂×C₃₁×#135	C₄	1	2	yes	—
15	C₂×C₃₁×#135	C₂²	1	4	yes	—
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₃₁	—	2	1	no	—
18	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₃₁	—	2	1	no	—
20	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×M₃₂	—	2	1	no	—
22	C₂×C₃₁×M₃₂	—	2	2	no	—
23	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
24	C₂×C₃₁×M₃₂	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₃₁	—	2	1	no	—
26	C₂×C₃₁×#108	—	2	2	no	—
27	C₂×C₃₁×#108	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	—
29	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
30	C₃₁×#362	C₂²	1	8	yes	complemented
31	C₃₁×#361	C₂²	1	8	yes	complemented
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	—	2	4	no	—
34	C₂×#41×C₃₁	—	2	2	no	—
35	C₂×#41×C₃₁	—	2	4	no	—
36	C₂×#41×C₃₁	—	4	1	no	—
37	C₂×#41×C₃₁	—	4	2	no	—
38	C₂×#41×C₃₁	—	4	1	no	—
39	C₂×#41×C₃₁	—	4	2	no	—
40	C₂²×Q₈×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	—	4	1	no	—
43	C₂³×C₄×C₃₁	—	4	1	no	—
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₃₁×#135	—	2	4	no	—
46	C₃₁×#135	—	2	4	no	—
47	C₃₁×#135	—	2	4	no	—
48	C₃₁×#135	—	2	4	no	—
49	C₃₁×#135	—	2	4	no	—
50	C₃₁×#135	—	2	4	no	—
51	C₃₁×#135	C₂×C₄	1	8	yes	—
52	C₃₁×#135	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₃₁	—	2	1	no	—
54	C₂×Q₁₆×C₃₁	—	2	1	no	—
55	C₂×Q₁₆×C₃₁	—	2	1	no	—
56	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂×SD₁₆×C₃₁	—	2	4	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	4	no	—
63	C₂×D₈×C₃₁	—	2	1	no	—
64	C₂×D₈×C₃₁	—	2	1	no	—
65	C₂×D₈×C₃₁	—	2	1	no	—
66	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×D₈×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	—	2	2	no	—
70	C₂×M₁₆×C₃₁	—	2	4	no	—
71	C₂×M₁₆×C₃₁	—	4	1	no	—
72	C₂×M₁₆×C₃₁	—	4	2	no	—
73	C₂²×C₈×C₃₁	—	4	1	no	—
74	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	D₄	1	2	yes	—
79	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	D₄	1	2	yes	—
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
84	C₃₁×M₃₂	—	2	2	no	—
85	C₃₁×M₃₂	—	2	4	no	—
86	C₃₁×M₃₂	C₂³	1	4	yes	—
87	C₂×C₁₆×C₃₁	—	2	2	no	—
88	C₂×C₁₆×C₃₁	—	2	4	no	—
89	C₃₁×#108	—	2	2	no	—
90	C₃₁×#108	—	2	4	no	—
91	C₃₁×#108	C₂³	1	4	yes	—
92	C₂⁴×C₃₁	—	4	1	no	—
93	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
94	#41×C₃₁	—	2	8	no	—
95	#41×C₃₁	—	2	8	no	—
96	#41×C₃₁	—	4	4	no	—
97	#41×C₃₁	—	4	4	no	—
98	#41×C₃₁	—	4	4	no	—
99	#41×C₃₁	—	4	4	no	—
100	#41×C₃₁	—	4	4	no	—
101	#41×C₃₁	—	4	4	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	4	1	no	—
106	C₂×Q₈×C₃₁	—	4	1	no	—
107	C₂×Q₈×C₃₁	—	4	1	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	—	4	4	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	D₈	1	2	yes	—
128	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
130	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	Q₁₆×C₃₁	—	2	2	no	—
134	Q₁₆×C₃₁	—	2	2	no	—
135	Q₁₆×C₃₁	—	2	2	no	—
136	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
137	SD₁₆×C₃₁	—	2	8	no	—
138	SD₁₆×C₃₁	—	2	8	no	—
139	D₈×C₃₁	—	2	2	no	—
140	D₈×C₃₁	—	2	2	no	—
141	D₈×C₃₁	—	2	2	no	—
142	D₈×C₃₁	C₂²×C₄	1	4	yes	—
143	M₁₆×C₃₁	—	2	4	no	—
144	M₁₆×C₃₁	—	4	2	no	—
145	M₁₆×C₃₁	—	4	4	no	—
146	C₂×C₈×C₃₁	—	4	2	no	—
147	C₂×C₈×C₃₁	—	4	4	no	—
148	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
153	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₁₆×C₃₁	—	2	4	no	—
167	C₂×#2148	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	8	4	no	—
173	C₂³×C₃₁	#102	1	1	yes	characteristic
174	C₂³×C₃₁	#102	1	1	yes	characteristic
175	C₂³×C₃₁	#102	1	2	yes	—
176	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
177	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
178	Q₈×C₃₁	—	2	4	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	Q₈×C₃₁	—	4	2	no	—
181	Q₈×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	4	8	no	—
184	D₄×C₃₁	—	4	8	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	—	4	8	no	—
193	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
194	C₂×C₄×C₃₁	#102	1	2	yes	—
195	C₂×C₄×C₃₁	#102	1	2	yes	—
196	C₂×C₄×C₃₁	#102	1	2	yes	—
197	C₂×C₄×C₃₁	#102	1	2	yes	—
198	C₂×C₄×C₃₁	#102	1	2	yes	—
199	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
206	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₈×C₃₁	—	4	4	no	—
212	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
215	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
216	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
217	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
219	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#2148	C₂×C₃₁	1	8	yes	direct product factor
222	C₂×#362	C₂×C₃₁	1	2	yes	complemented
223	C₂×#361	C₂×C₃₁	1	2	yes	complemented
224	C₂²×C₃₁	—	2	2	no	—
225	C₂²×C₃₁	—	2	2	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	8	4	no	—
231	C₂²×C₃₁	—	8	4	no	—
232	C₂²×C₃₁	—	8	4	no	—
233	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
234	C₂²×C₃₁	C₂×#102	1	2	yes	—
235	C₂²×C₃₁	C₂×#102	1	2	yes	—
236	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
237	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
238	C₄×C₃₁	—	4	4	no	—
239	C₄×C₃₁	C₂×#102	1	2	yes	—
240	C₄×C₃₁	C₂×#102	1	2	yes	—
241	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
242	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
243	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
244	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
245	C₂²×#41	—	2	1	no	—
246	C₂×#135	—	2	2	no	—
247	C₂×#135	—	2	1	no	—
248	C₂×#135	—	2	1	no	—
249	C₂×#135	—	2	2	no	—
250	C₂×#135	C₄×C₃₁	1	2	yes	—
251	C₂×#135	C₄×C₃₁	1	4	yes	—
252	C₂×#135	C₄×C₃₁	1	2	yes	—
253	C₂×#135	C₂²×C₃₁	1	4	yes	—
254	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	C₂²×SD₁₆	—	2	1	no	—
256	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂²×M₁₆	—	2	1	no	—
258	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×M₃₂	—	2	1	no	—
260	C₂×M₃₂	—	2	2	no	—
261	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
262	C₂×M₃₂	C₂²×C₃₁	1	4	yes	complemented
263	C₂²×C₁₆	—	2	1	no	—
264	C₂×#108	—	2	2	no	—
265	C₂×#108	—	2	1	no	—
266	C₂×#108	C₂²×C₃₁	1	2	yes	—
267	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
268	#362	C₂²×C₃₁	1	8	yes	complemented
269	#361	C₂²×C₃₁	1	8	yes	complemented
270	C₂×C₃₁	—	2	1	no	—
271	C₂×C₃₁	—	2	1	no	—
272	C₂×C₃₁	—	2	2	no	—
273	C₂×C₃₁	—	2	2	no	—
274	C₂×C₃₁	—	8	4	no	—
275	C₂×C₃₁	#2148	1	2	yes	direct product factor
276	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
277	C₂×#41	—	2	2	no	—
278	C₂×#41	—	2	4	no	—
279	C₂×#41	—	2	2	no	—
280	C₂×#41	—	2	4	no	—
281	C₂×#41	—	4	1	no	—
282	C₂×#41	—	4	1	no	—
283	C₂×#41	—	4	2	no	—
284	C₂×#41	—	4	2	no	—
285	C₂²×Q₈	—	2	1	no	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	—	4	1	no	—
288	C₂³×C₄	—	4	1	no	—
289	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
290	#135	—	2	4	no	—
291	#135	—	2	4	no	—
292	#135	—	2	4	no	—
293	#135	—	2	4	no	—
294	#135	—	2	4	no	—
295	#135	—	2	4	no	—
296	#135	C₂×C₄×C₃₁	1	8	yes	—
297	#135	C₂×C₄×C₃₁	1	8	yes	—
298	C₂×Q₁₆	—	2	1	no	—
299	C₂×Q₁₆	—	2	1	no	—
300	C₂×Q₁₆	—	2	1	no	—
301	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
302	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
303	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
304	C₂×SD₁₆	—	2	2	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×D₈	—	2	1	no	—
309	C₂×D₈	—	2	1	no	—
310	C₂×D₈	—	2	1	no	—
311	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
312	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
313	C₂×D₈	C₂³×C₃₁	1	2	yes	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	—	2	4	no	—
316	C₂×M₁₆	—	4	1	no	—
317	C₂×M₁₆	—	4	2	no	—
318	C₂²×C₈	—	4	1	no	—
319	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
321	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₈	D₄×C₃₁	1	2	yes	—
324	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₈	D₄×C₃₁	1	2	yes	—
327	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
329	M₃₂	—	2	2	no	—
330	M₃₂	—	2	4	no	—
331	M₃₂	C₂³×C₃₁	1	4	yes	—
332	C₂×C₁₆	—	2	2	no	—
333	C₂×C₁₆	—	2	4	no	—
334	#108	—	2	2	no	—
335	#108	—	2	4	no	—
336	#108	C₂³×C₃₁	1	4	yes	—
337	C₃₁	C₂×#2148	1	1	yes	characteristic, direct product factor, Sylow
338	C₂⁴	—	4	1	no	—
339	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
340	#41	—	2	8	no	—
341	#41	—	2	8	no	—
342	#41	—	4	4	no	—
343	#41	—	4	4	no	—
344	#41	—	4	4	no	—
345	#41	—	4	4	no	—
346	#41	—	4	4	no	—
347	#41	—	4	4	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	4	1	no	—
352	C₂×Q₈	—	4	1	no	—
353	C₂×Q₈	—	4	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	4	4	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	4	1	no	—
364	C₂²×C₄	—	4	1	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	D₈×C₃₁	1	2	yes	—
375	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
379	Q₁₆	—	2	2	no	—
380	Q₁₆	—	2	2	no	—
381	Q₁₆	—	2	2	no	—
382	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
383	SD₁₆	—	2	8	no	—
384	SD₁₆	—	2	8	no	—
385	D₈	—	2	2	no	—
386	D₈	—	2	2	no	—
387	D₈	—	2	2	no	—
388	D₈	C₂²×C₄×C₃₁	1	4	yes	—
389	M₁₆	—	2	4	no	—
390	M₁₆	—	4	2	no	—
391	M₁₆	—	4	4	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	—	4	4	no	—
394	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
395	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
396	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
398	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
399	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
400	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
401	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
404	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
405	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
409	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
411	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₁₆	—	2	4	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	8	4	no	—
418	C₂³	C₃₁×#102	1	1	yes	characteristic
419	C₂³	C₃₁×#102	1	1	yes	characteristic
420	C₂³	C₃₁×#102	1	2	yes	—
421	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
422	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
423	Q₈	—	2	4	no	—
424	Q₈	—	4	2	no	—
425	Q₈	—	4	2	no	—
426	Q₈	—	4	2	no	—
427	D₄	—	2	4	no	—
428	D₄	—	4	8	no	—
429	D₄	—	4	8	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	—	4	8	no	—
438	C₂×C₄	C₃₁×#102	1	2	yes	—
439	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
440	C₂×C₄	C₃₁×#102	1	2	yes	—
441	C₂×C₄	C₃₁×#102	1	2	yes	—
442	C₂×C₄	C₃₁×#102	1	2	yes	—
443	C₂×C₄	C₃₁×#102	1	2	yes	—
444	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
445	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
448	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
449	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
451	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
453	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
454	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	C₈	—	4	4	no	—
457	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
458	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
460	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
461	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
462	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	8	4	no	—
470	C₂²	—	8	4	no	—
471	C₂²	—	8	4	no	—
472	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
473	C₂²	C₂×C₃₁×#102	1	2	yes	—
474	C₂²	C₂×C₃₁×#102	1	2	yes	—
475	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
476	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
477	C₄	—	4	4	no	—
478	C₄	C₂×C₃₁×#102	1	2	yes	—
479	C₄	C₂×C₃₁×#102	1	2	yes	—
480	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
481	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
482	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
484	C₂	—	2	1	no	—
485	C₂	—	2	1	no	—
486	C₂	—	2	2	no	—
487	C₂	—	2	2	no	—
488	C₂	—	8	4	no	—
489	C₂	C₃₁×#2148	1	2	yes	direct product factor
490	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#102
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×M₃₂	C₂	semidirect
3. C₂²×C₃₁×#108	C₂	semidirect
4. C₃₁×#2148	C₂	direct (N × H)
5. C₂×C₃₁×#362	C₂	semidirect
6. C₂×C₃₁×#361	C₂	semidirect
24. C₂×C₃₁×M₃₂	C₂²	semidirect
29. C₂×C₃₁×#108	C₂²	semidirect
30. C₃₁×#362	C₂²	semidirect
31. C₃₁×#361	C₂²	semidirect
167. C₂×#2148	C₃₁	direct (N × H)
219. C₂²×M₃₂	C₂×C₃₁	semidirect
220. C₂²×#108	C₂×C₃₁	semidirect
221. #2148	C₂×C₃₁	direct (N × H)
222. C₂×#362	C₂×C₃₁	semidirect
223. C₂×#361	C₂×C₃₁	semidirect
262. C₂×M₃₂	C₂²×C₃₁	semidirect
267. C₂×#108	C₂²×C₃₁	semidirect
268. #362	C₂²×C₃₁	semidirect
269. #361	C₂²×C₃₁	semidirect
275. C₂×C₃₁	#2148	direct (N × H)
337. C₃₁	C₂×#2148	direct (N × H)
489. C₂	C₃₁×#2148	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂²×C₃₁×M₃₂	2
C₂²×C₃₁×#108	2
C₃₁×#2148	2
C₂×C₃₁×#362	2
C₂×C₃₁×#361	2
C₂×#2148	31

Subgroups — C2×C31×#2148 #21490150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2148 #21490150