Subgroups — C₂³×C₃₁×#104 #21516849

Order 7936.

5758 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#104	C₂	1	28	yes	maximal, direct product factor
5	C₂²×#41×C₃₁	—	2	7	no	—
6	C₂²×#41×C₃₁	—	2	7	no	—
7	C₂²×#41×C₃₁	C₄	1	14	yes	complemented
8	C₂²×#41×C₃₁	C₂²	1	14	yes	—
9	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₃₁	—	2	1	no	—
11	C₂³×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
12	C₂⁴×C₄×C₃₁	—	2	1	no	—
13	C₂⁴×C₄×C₃₁	—	2	1	no	—
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	—	2	7	no	—
16	C₂²×M₁₆×C₃₁	C₂²	1	14	yes	complemented
17	C₂³×C₈×C₃₁	—	2	1	no	—
18	C₂²×C₄²×C₃₁	—	2	7	no	—
19	C₂²×C₄²×C₃₁	C₂²	1	14	yes	complemented
20	C₂×C₃₁×#104	C₂²	1	112	yes	direct product factor
21	C₂⁵×C₃₁	—	2	1	no	—
22	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
23	C₂×#41×C₃₁	—	2	84	no	—
24	C₂×#41×C₃₁	—	2	42	no	—
25	C₂×#41×C₃₁	—	2	42	no	—
26	C₂×#41×C₃₁	C₂×C₄	1	84	yes	complemented
27	C₂×#41×C₃₁	C₂³	1	28	yes	—
28	C₂²×Q₈×C₃₁	—	2	7	no	—
29	C₂²×Q₈×C₃₁	C₂×C₄	1	14	yes	complemented
30	C₂²×D₄×C₃₁	—	2	14	no	—
31	C₂²×D₄×C₃₁	—	2	14	no	—
32	C₂²×D₄×C₃₁	—	2	7	no	—
33	C₂²×D₄×C₃₁	C₂×C₄	1	14	yes	complemented
34	C₂³×C₄×C₃₁	—	2	16	no	—
35	C₂³×C₄×C₃₁	—	2	1	no	—
36	C₂³×C₄×C₃₁	—	2	14	no	—
37	C₂³×C₄×C₃₁	—	2	14	no	—
38	C₂³×C₄×C₃₁	—	2	14	no	—
39	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂×C₄	1	7	yes	—
41	C₂³×C₄×C₃₁	D₄	1	7	yes	—
42	C₂³×C₄×C₃₁	D₄	1	7	yes	—
43	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	7	yes	—
45	C₂×M₁₆×C₃₁	—	2	42	no	—
46	C₂×M₁₆×C₃₁	C₂³	1	28	yes	complemented
47	C₂²×C₈×C₃₁	—	2	14	no	—
48	C₂×C₄²×C₃₁	—	2	42	no	—
49	C₂×C₄²×C₃₁	C₂³	1	28	yes	complemented
50	C₃₁×#104	C₂³	1	64	yes	direct product factor
51	C₂⁴×C₃₁	—	2	8	no	—
52	C₂⁴×C₃₁	—	2	14	no	—
53	C₂⁴×C₃₁	—	4	8	no	—
54	C₂⁴×C₃₁	C₂²⋊C₄	1	7	yes	—
55	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂⁴×C₃₁	C₂×D₄	1	7	yes	—
57	#41×C₃₁	—	2	168	no	—
58	#41×C₃₁	—	2	28	no	—
59	#41×C₃₁	—	2	28	no	—
60	#41×C₃₁	C₂²×C₄	1	56	yes	complemented
61	#41×C₃₁	C₂⁴	1	8	yes	—
62	C₂×Q₈×C₃₁	—	2	42	no	—
63	C₂×Q₈×C₃₁	C₂²×C₄	1	28	yes	complemented
64	C₂×D₄×C₃₁	—	2	84	no	—
65	C₂×D₄×C₃₁	—	2	28	no	—
66	C₂×D₄×C₃₁	—	2	42	no	—
67	C₂×D₄×C₃₁	C₂²×C₄	1	28	yes	complemented
68	C₂²×C₄×C₃₁	—	2	112	no	—
69	C₂²×C₄×C₃₁	—	2	14	no	—
70	C₂²×C₄×C₃₁	—	2	28	no	—
71	C₂²×C₄×C₃₁	—	2	84	no	—
72	C₂²×C₄×C₃₁	—	2	28	no	—
73	C₂²×C₄×C₃₁	C₂²⋊C₄	1	7	yes	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	7	yes	—
75	C₂²×C₄×C₃₁	C₂²⋊C₄	1	42	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	7	yes	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	21	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	21	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	7	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	21	yes	—
81	C₂²×C₄×C₃₁	C₂⁴	1	7	yes	—
82	M₁₆×C₃₁	—	2	28	no	—
83	M₁₆×C₃₁	C₂⁴	1	8	yes	complemented
84	C₂×C₈×C₃₁	—	2	28	no	—
85	C₄²×C₃₁	—	2	28	no	—
86	C₄²×C₃₁	C₂⁴	1	8	yes	complemented
87	C₂³×#104	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	28	no	—
89	C₂³×C₃₁	—	2	28	no	—
90	C₂³×C₃₁	—	2	28	no	—
91	C₂³×C₃₁	—	4	56	no	—
92	C₂³×C₃₁	#104	1	8	yes	direct product factor
93	C₂³×C₃₁	C₂×C₂²⋊C₄	1	21	yes	—
94	C₂³×C₃₁	C₂×C₂²⋊C₄	1	7	yes	—
95	C₂³×C₃₁	C₂²×D₄	1	7	yes	—
96	Q₈×C₃₁	—	2	28	no	—
97	Q₈×C₃₁	C₂³×C₄	1	8	yes	complemented
98	D₄×C₃₁	—	2	56	no	—
99	D₄×C₃₁	—	2	8	no	—
100	D₄×C₃₁	—	2	28	no	—
101	D₄×C₃₁	C₂³×C₄	1	8	yes	complemented
102	C₂×C₄×C₃₁	—	2	112	no	—
103	C₂×C₄×C₃₁	—	2	28	no	—
104	C₂×C₄×C₃₁	—	2	8	no	—
105	C₂×C₄×C₃₁	—	2	56	no	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	21	yes	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	21	yes	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	42	yes	—
110	C₂×C₄×C₃₁	C₂³×C₄	1	7	yes	—
111	C₂×C₄×C₃₁	C₂³×C₄	1	7	yes	—
112	C₂×C₄×C₃₁	C₂²×D₄	1	7	yes	—
113	C₂×C₄×C₃₁	C₂²×D₄	1	7	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	7	yes	—
115	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
116	C₈×C₃₁	—	2	8	no	—
117	C₂³×#41	C₂×C₃₁	1	1	yes	characteristic
118	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
120	C₂²×#104	C₂×C₃₁	1	28	yes	direct product factor
121	C₂²×C₃₁	—	2	14	no	—
122	C₂²×C₃₁	—	2	42	no	—
123	C₂²×C₃₁	—	2	8	no	—
124	C₂²×C₃₁	—	4	56	no	—
125	C₂²×C₃₁	C₂×#104	1	28	yes	direct product factor
126	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
127	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
128	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
129	C₄×C₃₁	—	2	16	no	—
130	C₄×C₃₁	—	2	8	no	—
131	C₄×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
132	C₄×C₃₁	C₂²×C₂²⋊C₄	1	7	yes	—
133	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
135	C₂²×#41	—	2	7	no	—
136	C₂²×#41	—	2	7	no	—
137	C₂²×#41	C₄×C₃₁	1	14	yes	complemented
138	C₂²×#41	C₂²×C₃₁	1	14	yes	—
139	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
140	C₂³×D₄	—	2	1	no	—
141	C₂³×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
142	C₂⁴×C₄	—	2	1	no	—
143	C₂⁴×C₄	—	2	1	no	—
144	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
145	C₂²×M₁₆	—	2	7	no	—
146	C₂²×M₁₆	C₂²×C₃₁	1	14	yes	complemented
147	C₂³×C₈	—	2	1	no	—
148	C₂²×C₄²	—	2	7	no	—
149	C₂²×C₄²	C₂²×C₃₁	1	14	yes	complemented
150	C₂×#104	C₂²×C₃₁	1	112	yes	direct product factor
151	C₂×C₃₁	—	2	1	no	—
152	C₂×C₃₁	—	2	7	no	—
153	C₂×C₃₁	—	4	8	no	—
154	C₂×C₃₁	C₂²×#104	1	14	yes	direct product factor
155	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
156	C₂⁵	—	2	1	no	—
157	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
158	C₂×#41	—	2	84	no	—
159	C₂×#41	—	2	42	no	—
160	C₂×#41	—	2	42	no	—
161	C₂×#41	C₂×C₄×C₃₁	1	84	yes	complemented
162	C₂×#41	C₂³×C₃₁	1	28	yes	—
163	C₂²×Q₈	—	2	7	no	—
164	C₂²×Q₈	C₂×C₄×C₃₁	1	14	yes	complemented
165	C₂²×D₄	—	2	14	no	—
166	C₂²×D₄	—	2	14	no	—
167	C₂²×D₄	—	2	7	no	—
168	C₂²×D₄	C₂×C₄×C₃₁	1	14	yes	complemented
169	C₂³×C₄	—	2	16	no	—
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	14	no	—
172	C₂³×C₄	—	2	14	no	—
173	C₂³×C₄	—	2	14	no	—
174	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₃₁	1	7	yes	—
176	C₂³×C₄	D₄×C₃₁	1	7	yes	—
177	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
178	C₂³×C₄	D₄×C₃₁	1	7	yes	—
179	C₂³×C₄	C₂³×C₃₁	1	7	yes	—
180	C₂×M₁₆	—	2	42	no	—
181	C₂×M₁₆	C₂³×C₃₁	1	28	yes	complemented
182	C₂²×C₈	—	2	14	no	—
183	C₂×C₄²	—	2	42	no	—
184	C₂×C₄²	C₂³×C₃₁	1	28	yes	complemented
185	#104	C₂³×C₃₁	1	64	yes	direct product factor
186	C₃₁	C₂³×#104	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	8	no	—
188	C₂⁴	—	2	14	no	—
189	C₂⁴	—	4	8	no	—
190	C₂⁴	C₂²⋊C₄×C₃₁	1	7	yes	—
191	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂⁴	C₂×D₄×C₃₁	1	7	yes	—
193	#41	—	2	168	no	—
194	#41	—	2	28	no	—
195	#41	—	2	28	no	—
196	#41	C₂²×C₄×C₃₁	1	56	yes	complemented
197	#41	C₂⁴×C₃₁	1	8	yes	—
198	C₂×Q₈	—	2	42	no	—
199	C₂×Q₈	C₂²×C₄×C₃₁	1	28	yes	complemented
200	C₂×D₄	—	2	84	no	—
201	C₂×D₄	—	2	28	no	—
202	C₂×D₄	—	2	42	no	—
203	C₂×D₄	C₂²×C₄×C₃₁	1	28	yes	complemented
204	C₂²×C₄	—	2	112	no	—
205	C₂²×C₄	—	2	14	no	—
206	C₂²×C₄	—	2	28	no	—
207	C₂²×C₄	—	2	84	no	—
208	C₂²×C₄	—	2	28	no	—
209	C₂²×C₄	C₂²⋊C₄×C₃₁	1	7	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₃₁	1	7	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	42	yes	—
212	C₂²×C₄	C₂²×C₄×C₃₁	1	7	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	21	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	21	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	7	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	21	yes	—
217	C₂²×C₄	C₂⁴×C₃₁	1	7	yes	—
218	M₁₆	—	2	28	no	—
219	M₁₆	C₂⁴×C₃₁	1	8	yes	complemented
220	C₂×C₈	—	2	28	no	—
221	C₄²	—	2	28	no	—
222	C₄²	C₂⁴×C₃₁	1	8	yes	complemented
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	2	28	no	—
226	C₂³	—	4	56	no	—
227	C₂³	C₃₁×#104	1	8	yes	direct product factor
228	C₂³	C₂×C₂²⋊C₄×C₃₁	1	21	yes	—
229	C₂³	C₂×C₂²⋊C₄×C₃₁	1	7	yes	—
230	C₂³	C₂²×D₄×C₃₁	1	7	yes	—
231	Q₈	—	2	28	no	—
232	Q₈	C₂³×C₄×C₃₁	1	8	yes	complemented
233	D₄	—	2	56	no	—
234	D₄	—	2	8	no	—
235	D₄	—	2	28	no	—
236	D₄	C₂³×C₄×C₃₁	1	8	yes	complemented
237	C₂×C₄	—	2	112	no	—
238	C₂×C₄	—	2	28	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	56	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	21	yes	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	21	yes	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	42	yes	—
245	C₂×C₄	C₂³×C₄×C₃₁	1	7	yes	—
246	C₂×C₄	C₂³×C₄×C₃₁	1	7	yes	—
247	C₂×C₄	C₂²×D₄×C₃₁	1	7	yes	—
248	C₂×C₄	C₂²×D₄×C₃₁	1	7	yes	—
249	C₂×C₄	C₂²×D₄×C₃₁	1	7	yes	—
250	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	56	no	—
256	C₂²	C₂×C₃₁×#104	1	28	yes	direct product factor
257	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
258	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
259	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	2	8	no	—
262	C₄	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
263	C₄	C₂²×C₂²⋊C₄×C₃₁	1	7	yes	—
264	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
265	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	4	8	no	—
269	C₂	C₂²×C₃₁×#104	1	14	yes	direct product factor
270	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₃₁	D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂⁴×C₃₁	C₂²⋊C₄
prime radical	C₂³×#41×C₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₃₁	C₂	semidirect
3. C₂³×C₄²×C₃₁	C₂	semidirect
4. C₂²×C₃₁×#104	C₂	direct (N × H)
7. C₂²×#41×C₃₁	C₄	semidirect
9. C₂³×Q₈×C₃₁	C₄	semidirect
11. C₂³×D₄×C₃₁	C₄	semidirect
16. C₂²×M₁₆×C₃₁	C₂²	semidirect
19. C₂²×C₄²×C₃₁	C₂²	semidirect
20. C₂×C₃₁×#104	C₂²	direct (N × H)
26. C₂×#41×C₃₁	C₂×C₄	semidirect
29. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
33. C₂²×D₄×C₃₁	C₂×C₄	semidirect
46. C₂×M₁₆×C₃₁	C₂³	semidirect
49. C₂×C₄²×C₃₁	C₂³	semidirect
50. C₃₁×#104	C₂³	direct (N × H)
60. #41×C₃₁	C₂²×C₄	semidirect
63. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
67. C₂×D₄×C₃₁	C₂²×C₄	semidirect
83. M₁₆×C₃₁	C₂⁴	semidirect
86. C₄²×C₃₁	C₂⁴	semidirect
87. C₂³×#104	C₃₁	direct (N × H)
92. C₂³×C₃₁	#104	direct (N × H)
97. Q₈×C₃₁	C₂³×C₄	semidirect
101. D₄×C₃₁	C₂³×C₄	semidirect
118. C₂³×M₁₆	C₂×C₃₁	semidirect
119. C₂³×C₄²	C₂×C₃₁	semidirect
120. C₂²×#104	C₂×C₃₁	direct (N × H)
125. C₂²×C₃₁	C₂×#104	direct (N × H)
137. C₂²×#41	C₄×C₃₁	semidirect
139. C₂³×Q₈	C₄×C₃₁	semidirect
141. C₂³×D₄	C₄×C₃₁	semidirect
146. C₂²×M₁₆	C₂²×C₃₁	semidirect
149. C₂²×C₄²	C₂²×C₃₁	semidirect
150. C₂×#104	C₂²×C₃₁	direct (N × H)
154. C₂×C₃₁	C₂²×#104	direct (N × H)
161. C₂×#41	C₂×C₄×C₃₁	semidirect
164. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
168. C₂²×D₄	C₂×C₄×C₃₁	semidirect
181. C₂×M₁₆	C₂³×C₃₁	semidirect
184. C₂×C₄²	C₂³×C₃₁	semidirect
185. #104	C₂³×C₃₁	direct (N × H)
186. C₃₁	C₂³×#104	direct (N × H)
196. #41	C₂²×C₄×C₃₁	semidirect
199. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
203. C₂×D₄	C₂²×C₄×C₃₁	semidirect
219. M₁₆	C₂⁴×C₃₁	semidirect
222. C₄²	C₂⁴×C₃₁	semidirect
227. C₂³	C₃₁×#104	direct (N × H)
232. Q₈	C₂³×C₄×C₃₁	semidirect
236. D₄	C₂³×C₄×C₃₁	semidirect
256. C₂²	C₂×C₃₁×#104	direct (N × H)
269. C₂	C₂²×C₃₁×#104	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₃₁	2
C₂³×M₁₆×C₃₁	2
C₂³×C₄²×C₃₁	2
C₂²×C₃₁×#104	2
C₂³×#104	31

Subgroups — C23×C31×#104 #21516849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₁×#104 #21516849