Group #21469496 — subgroups

Subgroups — C₃₁×#12737 #21469496

Order 7936.

906 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#362	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₃₁	—	2	1	no	—
6	C₂²×Q₁₆×C₃₁	—	2	8	no	—
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	—	2	2	no	—
9	C₂²×Q₁₆×C₃₁	C₄	1	4	yes	—
10	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₂×C₃₁×#108	C₂²	1	2	yes	—
14	C₃₁×#407	—	2	1	no	—
15	C₃₁×#362	—	2	8	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
17	C₃₁×#341	—	2	1	no	—
18	C₂²×Q₈×C₃₁	—	2	8	no	—
19	C₂²×Q₈×C₃₁	—	2	4	no	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×Q₈×C₃₁	—	4	4	no	—
22	C₂²×Q₈×C₃₁	—	4	2	no	—
23	C₂²×Q₈×C₃₁	—	4	1	no	—
24	C₂³×C₄×C₃₁	—	4	1	no	—
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂×Q₁₆×C₃₁	—	2	8	no	—
27	C₂×Q₁₆×C₃₁	—	2	16	no	—
28	C₂×Q₁₆×C₃₁	—	2	16	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	2	4	no	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×Q₁₆×C₃₁	D₄	1	8	yes	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×M₃₂	—	2	4	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#108	—	2	4	no	—
43	C₃₁×C₂²⋊C₈	—	4	2	no	—
44	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
45	C₂×Q₈×C₃₁	—	2	16	no	—
46	C₂×Q₈×C₃₁	—	2	4	no	—
47	C₂×Q₈×C₃₁	—	2	8	no	—
48	C₂×Q₈×C₃₁	—	4	8	no	—
49	C₂×Q₈×C₃₁	—	4	8	no	—
50	C₂×Q₈×C₃₁	—	4	16	no	—
51	C₂×Q₈×C₃₁	—	4	4	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	—	4	4	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	4	8	no	—
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	D₈	1	2	yes	—
61	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	16	no	—
64	Q₁₆×C₃₁	—	2	4	no	—
65	Q₁₆×C₃₁	—	2	8	no	—
66	Q₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
67	M₁₆×C₃₁	—	4	2	no	—
68	M₁₆×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₁₆×C₃₁	—	4	2	no	—
78	#12737	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	—	2	4	no	—
80	C₂³×C₃₁	—	2	2	no	—
81	C₂³×C₃₁	#102	1	2	yes	—
82	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	8	no	—
84	Q₈×C₃₁	—	4	16	no	—
85	Q₈×C₃₁	—	4	4	no	—
86	Q₈×C₃₁	—	4	8	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	4	16	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	—	4	8	no	—
94	C₂×C₄×C₃₁	#102	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	#120	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
99	C₈×C₃₁	—	2	2	no	—
100	C₈×C₃₁	—	8	2	no	—
101	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
102	C₈×C₃₁	#120	1	2	yes	—
103	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
104	C₂×#362	C₂×C₃₁	1	4	yes	complemented
105	#2112	C₂×C₃₁	1	1	yes	characteristic
106	#1810	C₂×C₃₁	1	1	yes	characteristic
107	C₂²×C₃₁	—	2	8	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	#362	1	4	yes	complemented
112	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
113	C₂²×C₃₁	#450	1	2	yes	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	4	8	no	—
116	C₄×C₃₁	#379	1	1	yes	characteristic
117	C₄×C₃₁	#418	1	1	yes	characteristic
118	C₄×C₃₁	#447	1	2	yes	—
119	C₂³×Q₈	—	2	1	no	—
120	C₂²×Q₁₆	—	2	8	no	—
121	C₂²×Q₁₆	—	2	1	no	—
122	C₂²×Q₁₆	—	2	2	no	—
123	C₂²×Q₁₆	C₄×C₃₁	1	4	yes	—
124	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
125	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
127	C₂×#108	C₂²×C₃₁	1	2	yes	—
128	#407	—	2	1	no	—
129	#362	—	2	8	no	—
130	C₂²⋊C₁₆	—	2	2	no	—
131	#341	—	2	1	no	—
132	C₂×C₃₁	—	2	4	no	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	#1786	1	1	yes	characteristic
135	C₂×C₃₁	C₂×#362	1	2	yes	—
136	C₂²×Q₈	—	2	8	no	—
137	C₂²×Q₈	—	2	4	no	—
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×Q₈	—	4	4	no	—
140	C₂²×Q₈	—	4	2	no	—
141	C₂²×Q₈	—	4	1	no	—
142	C₂³×C₄	—	4	1	no	—
143	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
144	C₂×Q₁₆	—	2	8	no	—
145	C₂×Q₁₆	—	2	16	no	—
146	C₂×Q₁₆	—	2	16	no	—
147	C₂×Q₁₆	—	2	4	no	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
151	C₂×Q₁₆	D₄×C₃₁	1	8	yes	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
155	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₈	D₄×C₃₁	1	2	yes	—
157	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	M₃₂	—	2	4	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#108	—	2	4	no	—
161	C₂²⋊C₈	—	4	2	no	—
162	C₃₁	#12737	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
164	C₂×Q₈	—	2	16	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	8	no	—
167	C₂×Q₈	—	4	8	no	—
168	C₂×Q₈	—	4	8	no	—
169	C₂×Q₈	—	4	16	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	8	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	D₈×C₃₁	1	2	yes	—
180	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
182	Q₁₆	—	2	16	no	—
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	2	8	no	—
185	Q₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
195	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₁₆	—	4	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	4	no	—
199	C₂³	C₃₁×#102	1	2	yes	—
200	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
201	Q₈	—	2	8	no	—
202	Q₈	—	4	16	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	4	8	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	16	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	C₃₁×#102	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₃₁×#120	1	2	yes	—
215	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
217	C₈	—	2	2	no	—
218	C₈	—	8	2	no	—
219	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
220	C₈	C₃₁×#120	1	2	yes	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₃₁×#362	1	4	yes	complemented
226	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
227	C₂²	C₃₁×#450	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	4	8	no	—
230	C₄	C₃₁×#379	1	1	yes	characteristic
231	C₄	C₃₁×#418	1	1	yes	characteristic
232	C₄	C₃₁×#447	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₃₁×#1786	1	1	yes	characteristic
236	C₂	C₂×C₃₁×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×#362

C₂

yes

maximal, complemented

C₃₁×#2112

C₂

yes

maximal, characteristic

C₃₁×#1810

C₂

yes

maximal, characteristic

C₂³×Q₈×C₃₁

—

C₂²×Q₁₆×C₃₁

—

C₂²×Q₁₆×C₃₁

—

C₂²×Q₁₆×C₃₁

—

C₂²×Q₁₆×C₃₁

C₄

yes

—

C₂²×Q₁₆×C₃₁

C₂²

yes

—

C₂³×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₃₁×M₃₂

C₂²

yes

—

C₂×C₃₁×#108

C₂²

yes

—

C₃₁×#407

—

C₃₁×#362

—

C₃₁×C₂²⋊C₁₆

—

C₃₁×#341

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂³×C₄×C₃₁

—

C₂³×C₄×C₃₁

D₄

yes

characteristic

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

C₂×C₄

yes

—

C₂×Q₁₆×C₃₁

D₄

yes

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₃₁×#108

—

C₃₁×C₂²⋊C₈

—

C₂⁴×C₃₁

SD₁₆

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

D₈

yes

—

C₂²×C₄×C₃₁

SD₁₆

yes

characteristic

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

C₂²⋊C₄

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₁₆×C₃₁

—

#12737

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

#102

yes

—

C₂³×C₃₁

C₂×SD₁₆

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#102

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#120

yes

—

C₂×C₄×C₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₈×C₃₁

—

100

C₈×C₃₁

—

101

C₈×C₃₁

C₂×C₂²⋊C₄

yes

—

102

C₈×C₃₁

#120

yes

—

103

C₂³×Q₁₆

C₂×C₃₁

yes

characteristic

104

C₂×#362

C₂×C₃₁

yes

complemented

105

#2112

C₂×C₃₁

yes

characteristic

106

#1810

C₂×C₃₁

yes

characteristic

107

C₂²×C₃₁

—

108

C₂²×C₃₁

—

109

C₂²×C₃₁

—

110

C₂²×C₃₁

—

111

C₂²×C₃₁

#362

yes

complemented

112

C₂²×C₃₁

C₂×#102

yes

characteristic

113

C₂²×C₃₁

#450

yes

—

114

C₄×C₃₁

—

115

C₄×C₃₁

—

116

C₄×C₃₁

#379

yes

characteristic

117

C₄×C₃₁

#418

yes

characteristic

118

C₄×C₃₁

#447

yes

—

119

C₂³×Q₈

—

120

C₂²×Q₁₆

—

121

C₂²×Q₁₆

—

122

C₂²×Q₁₆

—

123

C₂²×Q₁₆

C₄×C₃₁

yes

—

124

C₂²×Q₁₆

C₂²×C₃₁

yes

—

125

C₂³×C₈

C₂²×C₃₁

yes

characteristic

126

C₂×M₃₂

C₂²×C₃₁

yes

—

127

C₂×#108

C₂²×C₃₁

yes

—

128

#407

—

129

#362

—

130

C₂²⋊C₁₆

—

131

#341

—

132

C₂×C₃₁

—

133

C₂×C₃₁

—

134

C₂×C₃₁

#1786

yes

characteristic

135

C₂×C₃₁

C₂×#362

yes

—

136

C₂²×Q₈

—

137

C₂²×Q₈

—

138

C₂²×Q₈

—

139

C₂²×Q₈

—

140

C₂²×Q₈

—

141

C₂²×Q₈

—

142

C₂³×C₄

—

143

C₂³×C₄

D₄×C₃₁

yes

characteristic

144

C₂×Q₁₆

—

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

148

C₂×Q₁₆

—

149

C₂×Q₁₆

—

150

C₂×Q₁₆

C₂×C₄×C₃₁

yes

—

151

C₂×Q₁₆

D₄×C₃₁

yes

—

152

C₂×M₁₆

—

153

C₂²×C₈

—

154

C₂²×C₈

C₂×C₄×C₃₁

yes

—

155

C₂²×C₈

D₄×C₃₁

yes

characteristic

156

C₂²×C₈

D₄×C₃₁

yes

—

157

C₂²×C₈

C₂³×C₃₁

yes

characteristic

158

M₃₂

—

159

C₂×C₁₆

—

160

#108

—

161

C₂²⋊C₈

—

162

C₃₁

#12737

yes

characteristic, direct product factor, Sylow

163

C₂⁴

SD₁₆×C₃₁

yes

characteristic

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂²×C₄

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

179

C₂²×C₄

D₈×C₃₁

yes

—

180

C₂²×C₄

SD₁₆×C₃₁

yes

characteristic

181

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

182

Q₁₆

—

183

Q₁₆

—

184

Q₁₆

—

185

Q₁₆

C₂²⋊C₄×C₃₁

yes

—

186

M₁₆

—

187

M₁₆

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

193

C₂×C₈

C₂²×C₄×C₃₁

yes

characteristic

194

C₂×C₈

C₂×D₄×C₃₁

yes

—

195

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

196

C₁₆

—

197

C₂³

—

198

C₂³

—

199

C₂³

C₃₁×#102

yes

—

200

C₂³

C₂×SD₁₆×C₃₁

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

C₃₁×#102

yes

—

213

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

214

C₂×C₄

C₃₁×#120

yes

—

215

C₂×C₄

C₂×D₈×C₃₁

yes

characteristic

216

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

217

C₈

—

218

C₈

—

219

C₈

C₂×C₂²⋊C₄×C₃₁

yes

—

220

C₈

C₃₁×#120

yes

—

221

C₂²

—

222

C₂²

—

223

C₂²

—

224

C₂²

—

225

C₂²

C₃₁×#362

yes

complemented

226

C₂²

C₂×C₃₁×#102

yes

characteristic

227

C₂²

C₃₁×#450

yes

—

228

C₄

—

229

C₄

—

230

C₄

C₃₁×#379

yes

characteristic

231

C₄

C₃₁×#418

yes

characteristic

232

C₄

C₃₁×#447

yes

—

233

C₂

—

234

C₂

—

235

C₂

C₃₁×#1786

yes

characteristic

236

C₂

C₂×C₃₁×#362

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

prime radical

C₂⁴×C₃₁

SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₂×C₃₁×#362	C₂	semidirect
78. #12737	C₃₁	direct (N × H)
104. C₂×#362	C₂×C₃₁	semidirect
111. C₂²×C₃₁	#362	semidirect
162. C₃₁	#12737	direct (N × H)
225. C₂²	C₃₁×#362	semidirect

H ≅ G/N

type

2. C₂×C₃₁×#362

C₂

semidirect

78. #12737

C₃₁

direct (N × H)

104. C₂×#362

C₂×C₃₁

semidirect

111. C₂²×C₃₁

#362

semidirect

162. C₃₁

#12737

direct (N × H)

225. C₂²

C₃₁×#362

semidirect

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#362	2
C₃₁×#2112	2
C₃₁×#1810	2
#12737	31

subgroup

index

C₂³×Q₁₆×C₃₁

C₂×C₃₁×#362

C₃₁×#2112

C₃₁×#1810

#12737

Subgroups — C₃₁×#12737 #21469496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12737 #21469496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12737 #21469496