Subgroups — #18011398

Order 4736.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558585	C₂	1	4	yes	maximal, complemented
2	#14558584	C₂	1	2	yes	maximal, complemented
3	#14558582	C₂	1	2	yes	maximal, complemented
4	#14558538	C₂	1	2	yes	maximal
5	#14558537	C₂	1	1	yes	maximal, characteristic
6	C₄×#10596953	C₂	1	1	yes	maximal, characteristic
7	C₄×#10596953	C₂	1	2	yes	maximal
8	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#10596953	C₂²	1	2	yes	—
10	C₂×#10596953	C₂²	1	1	yes	characteristic
11	C₂×#10596953	C₂²	1	1	yes	characteristic
12	#11924603	C₂²	1	4	yes	—
13	#11924603	C₂²	1	4	yes	—
14	C₄⋊#64438	C₂²	1	2	yes	—
15	C₄⋊#64438	C₂²	1	2	yes	—
16	#11924591	C₂²	1	4	yes	—
17	#11924591	C₂²	1	2	yes	—
18	C₄×#64438	C₂²	1	2	yes	—
19	C₄×#64438	C₂²	1	4	yes	—
20	C₂×Q₈×D₃₇	C₄	1	1	yes	characteristic
21	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₇₄	C₄	1	2	yes	—
23	C₄×Dic₇₄	C₄	1	1	yes	characteristic
24	#11924501	C₂²	1	1	yes	characteristic
25	#11924501	C₂²	1	2	yes	—
26	C₄⋊C₄×D₃₇	C₄	1	1	yes	characteristic
27	C₄⋊C₄×D₃₇	C₄	1	2	yes	—
28	C₄²×D₃₇	C₂²	1	1	yes	characteristic
29	C₄²×D₃₇	C₂²	1	2	yes	—
30	C₄×Q₈×C₃₇	C₄	1	1	yes	characteristic
31	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	Q₈×D₃₇	—	2	4	no	—
33	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
39	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
40	C₂×#64438	C₂³	1	4	yes	—
41	C₂×#64438	C₂³	1	2	yes	—
42	C₂×#64438	C₂³	1	2	yes	—
43	#10596953	—	2	4	no	—
44	#10596953	—	2	2	no	—
45	#10596953	Q₈	1	4	yes	complemented
46	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₃₇	C₂×C₄	1	2	yes	—
49	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
52	#10596936	C₂×C₄	1	2	yes	—
53	#10596936	C₂×C₄	1	2	yes	—
54	#10596936	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₇	C₂³	1	2	yes	—
58	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
59	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
60	C₄×Dic₃₇	C₂³	1	2	yes	—
61	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₃₇	—	2	2	no	—
63	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
70	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
73	C₄×D₃₇	—	2	2	no	—
74	C₄×D₃₇	—	2	4	no	—
75	C₄×D₃₇	—	4	1	no	—
76	C₄×D₃₇	—	4	2	no	—
77	C₄×D₃₇	C₂×Q₈	1	2	yes	—
78	C₄×D₃₇	#41	1	2	yes	—
79	Dic₇₄	—	2	4	no	—
80	Dic₇₄	M₁₆	1	4	yes	complemented
81	#64438	—	2	4	no	—
82	#64438	—	2	2	no	—
83	#64438	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×D₃₇	—	2	1	no	—
86	C₂×D₃₇	—	2	1	no	—
87	C₂×D₃₇	C₄×Q₈	1	2	yes	—
88	C₄×C₃₇	—	2	2	no	—
89	C₄×C₃₇	—	2	2	no	—
90	C₄×C₃₇	—	2	1	no	—
91	C₄×C₃₇	C₄×Q₈	1	2	yes	—
92	C₄×C₃₇	C₂×M₁₆	1	2	yes	—
93	Dic₃₇	—	2	1	no	—
94	Dic₃₇	—	2	1	no	—
95	Dic₃₇	—	2	2	no	—
96	Dic₃₇	C₂×M₁₆	1	4	yes	—
97	Dic₃₇	C₂²×Q₈	1	1	yes	characteristic
98	Dic₃₇	C₂×#41	1	1	yes	characteristic
99	Q₈×M₁₆	—	37	1	no	maximal, Sylow
100	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₃₇	#568	1	1	yes	characteristic
103	D₃₇	—	2	2	no	—
104	C₂×C₄×Q₈	—	37	1	no	—
105	#446	—	37	2	no	—
106	#446	—	37	4	no	—
107	C₈×Q₈	—	37	2	no	—
108	#423	—	37	1	no	—
109	#423	—	37	2	no	—
110	C₄×M₁₆	—	37	1	no	—
111	C₄×M₁₆	—	37	2	no	—
112	C₃₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	37	1	no	—
114	C₂×M₁₆	—	37	1	no	—
115	C₂×M₁₆	—	37	2	no	—
116	C₂×M₁₆	—	37	1	no	—
117	C₄×Q₈	—	37	1	no	—
118	C₄×Q₈	—	37	1	no	—
119	C₄×Q₈	—	37	2	no	—
120	C₄×Q₈	—	37	1	no	—
121	C₄×Q₈	—	37	2	no	—
122	C₄×Q₈	#3866	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	37	2	no	—
124	C₂×C₄⋊C₄	—	37	1	no	—
125	C₂×C₄²	—	37	1	no	—
126	C₂×C₄²	—	37	2	no	—
127	C₄⋊C₈	—	37	2	no	—
128	C₄⋊C₈	—	37	2	no	—
129	C₄⋊C₈	—	37	4	no	—
130	C₄⋊C₈	—	37	4	no	—
131	#97	—	37	4	no	—
132	#97	—	37	2	no	—
133	C₄×C₈	—	37	2	no	—
134	C₄×C₈	—	37	4	no	—
135	C₂×Q₈	—	37	1	no	—
136	C₂×Q₈	—	37	2	no	—
137	C₂×Q₈	—	74	4	no	—
138	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
139	C₂²×C₄	—	37	1	no	—
140	C₂²×C₄	—	37	2	no	—
141	C₂²×C₄	—	37	1	no	—
142	C₂²×C₄	—	37	1	no	—
143	C₂²×C₄	—	37	2	no	—
144	M₁₆	—	37	4	no	—
145	M₁₆	—	74	4	no	—
146	M₁₆	—	74	2	no	—
147	C₂×C₈	—	37	2	no	—
148	C₂×C₈	—	37	4	no	—
149	C₂×C₈	—	37	2	no	—
150	C₄⋊C₄	—	37	2	no	—
151	C₄⋊C₄	—	37	2	no	—
152	C₄⋊C₄	—	37	2	no	—
153	C₄⋊C₄	—	37	2	no	—
154	C₄⋊C₄	—	37	1	no	—
155	C₄⋊C₄	C₂×#3866	1	2	yes	—
156	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
157	C₄²	—	37	2	no	—
158	C₄²	—	37	1	no	—
159	C₄²	—	37	2	no	—
160	C₄²	—	37	1	no	—
161	C₄²	—	37	2	no	—
162	C₄²	—	37	1	no	—
163	C₄²	C₂×#3866	1	2	yes	—
164	C₄²	C₂×#3866	1	1	yes	characteristic
165	C₂³	—	37	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	37	4	no	—
168	Q₈	—	74	4	no	—
169	C₂×C₄	—	37	1	no	—
170	C₂×C₄	—	37	2	no	—
171	C₂×C₄	—	37	2	no	—
172	C₂×C₄	—	37	2	no	—
173	C₂×C₄	—	37	1	no	—
174	C₂×C₄	—	37	2	no	—
175	C₂×C₄	—	37	1	no	—
176	C₂×C₄	—	74	4	no	—
177	C₂×C₄	—	74	2	no	—
178	C₂×C₄	—	148	2	no	—
179	C₂×C₄	—	148	1	no	—
180	C₂×C₄	C₂²×#3866	1	2	yes	—
181	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
182	C₂×C₄	C₂²×#3866	1	2	yes	—
183	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
184	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
185	C₈	—	37	4	no	—
186	C₈	—	74	4	no	—
187	C₈	—	74	2	no	—
188	C₂²	—	37	2	no	—
189	C₂²	—	74	1	no	—
190	C₂²	—	74	1	no	—
191	C₂²	C₂³×#3866	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	37	1	no	—
196	C₄	—	37	1	no	—
197	C₄	—	37	4	no	—
198	C₄	—	74	1	no	—
199	C₄	—	74	1	no	—
200	C₄	—	74	2	no	—
201	C₄	Q₈×#3866	1	2	yes	—
202	C₄	D₃₇⋊M₁₆	1	2	yes	—
203	C₂	—	74	2	no	—
204	C₂	C₂×Q₈×#3866	1	1	yes	characteristic
205	C₂	C₂×D₃₇⋊M₁₆	1	1	yes	characteristic
206	C₂	#14558612	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×Q₈	#3866
hypocenter	C₃₇	Q₈×M₁₆
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558584	C₂	semidirect
3. #14558582	C₂	semidirect
45. #10596953	Q₈	semidirect
80. Dic₇₄	M₁₆	semidirect
83. #64438	C₂×Q₈	semidirect
112. C₃₇	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#14558585	2
#14558584	2
#14558582	2
#14558538	2
#14558537	2
C₄×#10596953	2
C₄×Q₈×D₃₇	2
Q₈×M₁₆	37