Subgroups — C₂×#14521759 #16859245

Order 3968.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521759	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11758446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758447	C₂	1	2	yes	maximal, complemented
4	C₂×#11758473	C₂	1	4	yes	maximal, complemented
5	C₂×#11758523	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×D₁₂₄	C₂	1	2	yes	maximal
8	C₂×C₄⋊C₄×D₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
13	C₂²×C₄×D₃₁	C₂²	1	4	yes	—
14	C₂²×Dic₆₂	C₂²	1	2	yes	—
15	C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	#11758523	C₂²	1	16	yes	complemented
21	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
22	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
26	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
27	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
28	#11758473	C₂²	1	32	yes	complemented
29	C₄⋊C₄×D₃₁	C₂²	1	16	yes	complemented
30	C₄×D₁₂₄	C₂²	1	16	yes	—
31	C₄×D₁₂₄	C₂²	1	8	yes	—
32	#11758447	C₂²	1	16	yes	complemented
33	#11758446	C₂²	1	8	yes	complemented
34	C₂³×D₃₁	C₂³	1	2	yes	—
35	C₂×Q₈×C₃₁	—	2	4	no	—
36	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
37	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₃₁	C₂³	1	2	yes	—
43	C₂²×Dic₃₁	C₂³	1	4	yes	—
44	C₂×D₁₂₄	Q₈	1	8	yes	complemented
45	C₂×D₁₂₄	C₂³	1	4	yes	—
46	C₂×C₄×D₃₁	—	2	8	no	—
47	C₂×C₄×D₃₁	—	4	8	no	—
48	C₂×C₄×D₃₁	C₂³	1	8	yes	—
49	C₂×C₄×D₃₁	C₂³	1	16	yes	—
50	C₂×Dic₆₂	—	4	4	no	—
51	C₂×Dic₆₂	C₂³	1	8	yes	—
52	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
54	C₄²×C₃₁	C₂³	1	4	yes	—
55	C₄²×C₃₁	C₂³	1	8	yes	—
56	(C₂×D₃₁)⋊C₄	C₂³	1	8	yes	—
57	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
58	C₄⋊Dic₃₁	C₂³	1	8	yes	—
59	C₄⋊Dic₃₁	C₂³	1	8	yes	—
60	C₄⋊Dic₃₁	C₂³	1	4	yes	—
61	Dic₃₁⋊C₄	C₂³	1	16	yes	—
62	Dic₃₁⋊C₄	C₂³	1	16	yes	—
63	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₂²×D₃₁	—	2	4	no	—
65	C₂²×D₃₁	—	2	4	no	—
66	C₂²×D₃₁	C₂×Q₈	1	8	yes	—
67	C₂²×D₃₁	C₂⁴	1	4	yes	—
68	Q₈×C₃₁	—	2	8	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₃₁	#41	1	4	yes	—
74	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
75	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
76	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
77	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
78	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
79	C₂×Dic₃₁	—	2	4	no	—
80	C₂×Dic₃₁	—	2	8	no	—
81	C₂×Dic₃₁	C₂⁴	1	4	yes	—
82	C₂×Dic₃₁	C₂⁴	1	8	yes	—
83	D₁₂₄	C₂×Q₈	1	16	yes	complemented
84	C₄×D₃₁	—	2	16	no	—
85	C₄×D₃₁	—	4	16	no	—
86	Dic₆₂	—	4	8	no	—
87	C₂×#554	—	31	1	no	maximal, Sylow
88	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
89	C₂²×C₃₁	C₂×#41	1	2	yes	—
90	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
91	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
92	C₂×D₃₁	—	2	16	no	—
93	C₂×D₃₁	—	2	4	no	—
94	C₂×D₃₁	—	2	4	no	—
95	C₂×D₃₁	C₂²×Q₈	1	8	yes	—
96	C₄×C₃₁	—	2	4	no	—
97	C₄×C₃₁	—	2	2	no	—
98	C₄×C₃₁	—	2	4	no	—
99	C₄×C₃₁	C₂²×Q₈	1	4	yes	—
100	C₄×C₃₁	C₂×#41	1	4	yes	—
101	Dic₃₁	—	2	4	no	—
102	Dic₃₁	—	2	8	no	—
103	#554	—	31	16	no	—
104	C₂×C₄⋊Q₈	—	31	1	no	—
105	C₂×#125	—	31	2	no	—
106	C₂×C₂²⋊Q₈	—	31	2	no	—
107	C₂×C₂²⋊Q₈	—	31	4	no	—
108	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
109	C₂×C₄×D₄	—	31	2	no	—
110	C₂×C₄×D₄	—	31	1	no	—
111	C₂²×C₄⋊C₄	—	31	2	no	—
112	C₂×C₃₁	#554	1	4	yes	complemented
113	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
114	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
115	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
116	D₃₁	—	2	8	no	—
117	C₂²×Q₈	—	31	2	no	—
118	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
119	C₂²×D₄	—	31	1	no	—
120	C₂³×C₄	—	31	2	no	—
121	C₂³×C₄	—	31	4	no	—
122	C₄⋊Q₈	—	31	8	no	—
123	#125	—	31	16	no	—
124	C₂²⋊Q₈	—	31	16	no	—
125	C₂²⋊Q₈	—	31	32	no	—
126	C₄×Q₈	C₂×D₃₁	1	8	yes	complemented
127	C₄×D₄	—	31	16	no	—
128	C₄×D₄	—	31	8	no	—
129	C₂×C₄⋊C₄	—	31	4	no	—
130	C₂×C₄⋊C₄	—	31	2	no	—
131	C₂×C₄⋊C₄	—	31	1	no	—
132	C₂×C₄⋊C₄	—	31	2	no	—
133	C₂×C₄⋊C₄	—	31	4	no	—
134	C₂×C₄⋊C₄	—	31	16	no	—
135	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
137	C₂×C₂²⋊C₄	—	31	2	no	—
138	C₂×C₂²⋊C₄	—	31	4	no	—
139	C₂×C₄²	C₂×D₃₁	1	2	yes	—
140	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
141	C₃₁	C₂×#554	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	—	31	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	31	8	no	—
145	C₂×Q₈	—	124	4	no	—
146	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
147	C₂×D₄	—	31	8	no	—
148	C₂×D₄	—	31	4	no	—
149	C₂²×C₄	—	31	2	no	—
150	C₂²×C₄	—	31	4	no	—
151	C₂²×C₄	—	31	8	no	—
152	C₂²×C₄	—	31	16	no	—
153	C₂²×C₄	—	62	8	no	—
154	C₂²×C₄	—	124	8	no	—
155	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
156	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
158	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
160	C₄⋊C₄	—	31	16	no	—
161	C₄⋊C₄	—	31	16	no	—
162	C₄⋊C₄	—	31	8	no	—
163	C₄⋊C₄	—	31	4	no	—
164	C₄⋊C₄	—	31	8	no	—
165	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
166	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
167	C₂²⋊C₄	—	31	8	no	—
168	C₂²⋊C₄	—	31	16	no	—
169	C₄²	C₂²×D₃₁	1	8	yes	—
170	C₄²	C₂²×D₃₁	1	4	yes	—
171	C₂³	—	31	8	no	—
172	C₂³	—	31	4	no	—
173	C₂³	—	62	4	no	—
174	C₂³	—	62	4	no	—
175	C₂³	C₂³×D₃₁	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	Q₈	—	124	8	no	—
178	D₄	—	31	16	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	31	4	no	—
183	C₂×C₄	—	31	8	no	—
184	C₂×C₄	—	62	4	no	—
185	C₂×C₄	—	62	8	no	—
186	C₂×C₄	—	62	16	no	—
187	C₂×C₄	—	124	16	no	—
188	C₂×C₄	#92495	1	4	yes	—
189	C₂×C₄	Q₈×D₃₁	1	4	yes	—
190	C₂×C₄	C₂³×D₃₁	1	4	yes	—
191	C₂×C₄	C₂³×D₃₁	1	2	yes	—
192	C₂×C₄	C₂³×D₃₁	1	4	yes	—
193	C₂×C₄	C₂³×D₃₁	1	2	yes	—
194	C₂×C₄	C₂³×D₃₁	1	2	yes	—
195	C₂²	—	31	8	no	—
196	C₂²	—	62	4	no	—
197	C₂²	—	62	4	no	—
198	C₂²	—	62	16	no	—
199	C₂²	C₂×#92495	1	2	yes	—
200	C₂²	C₂×Q₈×D₃₁	1	2	yes	—
201	C₂²	#11758566	1	2	yes	—
202	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	—	62	4	no	—
207	C₄	—	62	8	no	—
208	C₄	C₂×#92495	1	4	yes	—
209	C₄	C₂×Q₈×D₃₁	1	4	yes	—
210	C₂	—	62	8	no	—
211	C₂	#14521759	1	4	yes	direct product factor
212	C₂	C₂²×Q₈×D₃₁	1	1	yes	characteristic
213	C₂	C₂²×#92495	1	1	yes	characteristic
214	C₂	C₂×#11758566	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×D₁₂₄	C₂²
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	C₂×#554
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14521759	C₂	direct (N × H)
2. C₂×#11758446	C₂	semidirect
3. C₂×#11758447	C₂	semidirect
4. C₂×#11758473	C₂	semidirect
5. C₂×#11758523	C₂	semidirect
8. C₂×C₄⋊C₄×D₃₁	C₂	semidirect
9. C₂×C₄×Q₈×C₃₁	C₂	semidirect
15. C₄×Q₈×C₃₁	C₂²	semidirect
20. #11758523	C₂²	semidirect
28. #11758473	C₂²	semidirect
29. C₄⋊C₄×D₃₁	C₂²	semidirect
32. #11758447	C₂²	semidirect
33. #11758446	C₂²	semidirect
44. C₂×D₁₂₄	Q₈	semidirect
83. D₁₂₄	C₂×Q₈	semidirect
108. C₂×C₄×Q₈	D₃₁	semidirect
112. C₂×C₃₁	#554	semidirect
126. C₄×Q₈	C₂×D₃₁	semidirect
141. C₃₁	C₂×#554	semidirect
211. C₂	#14521759	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521759	2
C₂×#11758446	2
C₂×#11758447	2
C₂×#11758473	2
C₂×#11758523	2
C₂×C₄×D₁₂₄	2
C₂×C₄⋊C₄×D₃₁	2
C₂×C₄×Q₈×C₃₁	2
C₂×#554	31

Subgroups — C2×#14521759 #16859245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521759 #16859245