Subgroups — #18011387

Order 4736.

348 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558585	C₂	1	2	yes	maximal, complemented
2	#14558585	C₂	1	2	yes	maximal, complemented
3	#14558584	C₂	1	2	yes	maximal, complemented
4	#14558583	C₂	1	2	yes	maximal, complemented
5	#14558539	C₂	1	2	yes	maximal
6	#14558536	C₂	1	1	yes	maximal, characteristic
7	#14558497	C₂	1	2	yes	maximal
8	C₂×#11924591	C₂	1	1	yes	maximal, characteristic
9	#14558085	C₂	1	1	yes	maximal, characteristic
10	C₂×#10596957	C₂²	1	2	yes	—
11	C₂²×#64438	C₂²	1	1	yes	characteristic
12	C₂²×#64438	C₂²	1	1	yes	characteristic
13	#11924603	C₂²	1	2	yes	—
14	#11924603	C₂²	1	2	yes	—
15	#11924603	C₂²	1	2	yes	—
16	C₄⋊#64438	C₂²	1	1	yes	characteristic
17	C₄⋊#64438	C₂²	1	1	yes	characteristic
18	C₄⋊#64438	C₂²	1	2	yes	—
19	C₄⋊#64438	C₂²	1	2	yes	—
20	#11924591	C₂²	1	2	yes	—
21	#11924591	C₂²	1	2	yes	—
22	#11924591	C₂²	1	2	yes	—
23	#11924591	C₂²	1	2	yes	—
24	C₄×#64438	C₂²	1	2	yes	—
25	C₄×#64438	C₂²	1	2	yes	—
26	#11924535	C₄	1	1	yes	characteristic
27	#11924534	C₄	1	1	yes	characteristic
28	#11924526	C₄	1	2	yes	—
29	#11924515	C₄	1	1	yes	characteristic
30	#11924513	C₄	1	2	yes	—
31	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
32	#11924501	C₂²	1	2	yes	—
33	#11924501	C₂²	1	1	yes	characteristic
34	#11924492	C₂²	1	2	yes	—
35	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₃₇	C₄	1	1	yes	characteristic
37	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
43	#10596957	—	2	4	no	—
44	C₂×#64438	—	2	2	no	—
45	C₂×#64438	Q₈	1	4	yes	complemented
46	C₂×#64438	C₂³	1	2	yes	—
47	C₂×#64438	C₂³	1	2	yes	—
48	C₂×#64438	C₂³	1	2	yes	—
49	C₂×#64438	C₂³	1	2	yes	—
50	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
53	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
56	#10596936	C₂×C₄	1	2	yes	—
57	#10596936	C₂×C₄	1	2	yes	—
58	#10596936	C₂×C₄	1	2	yes	—
59	#10596936	C₂×C₄	1	2	yes	—
60	#10596936	C₂×C₄	1	2	yes	—
61	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
64	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₃₇	C₂³	1	2	yes	—
66	C₄×Dic₃₇	C₂³	1	2	yes	—
67	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	Q₈×C₃₇	—	4	1	no	—
69	C₂×C₄×C₃₇	—	4	1	no	—
70	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
74	C₂×Dic₃₇	—	4	1	no	—
75	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
81	C₂×Dic₃₇	#41	1	2	yes	—
82	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
83	Dic₇₄	—	4	2	no	—
84	Dic₇₄	—	4	1	no	—
85	#64438	—	2	2	no	—
86	#64438	—	2	2	no	—
87	#64438	—	2	2	no	—
88	#64438	C₂×Q₈	1	4	yes	—
89	C₂²×C₃₇	—	2	1	no	—
90	C₂²×C₃₇	—	2	1	no	—
91	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
92	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₃₇	—	2	1	no	—
94	C₄×C₃₇	—	2	2	no	—
95	C₄×C₃₇	—	2	1	no	—
96	C₄×C₃₇	—	2	2	no	—
97	Dic₃₇	—	2	1	no	—
98	Dic₃₇	—	2	2	no	—
99	Dic₃₇	—	2	2	no	—
100	Dic₃₇	—	2	1	no	—
101	Dic₃₇	C₄×Q₈	1	2	yes	—
102	Dic₃₇	C₂²×Q₈	1	1	yes	characteristic
103	Dic₃₇	C₂×#41	1	1	yes	characteristic
104	#2961	—	37	1	no	maximal, Sylow
105	C₂×C₃₇	—	2	2	no	—
106	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂×C₃₇	#568	1	1	yes	characteristic
108	C₂×C₃₇	#568	1	1	yes	characteristic
109	#533	—	37	1	no	—
110	#446	—	37	2	no	—
111	#446	—	37	2	no	—
112	#446	—	37	2	no	—
113	#446	—	37	2	no	—
114	#424	—	37	2	no	—
115	#423	—	37	1	no	—
116	#405	—	37	2	no	—
117	C₂×#97	—	37	1	no	—
118	C₃₇	#2961	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	37	2	no	—
120	C₂²×C₈	—	37	1	no	—
121	C₂²×C₈	—	37	1	no	—
122	C₄⋊Q₈	—	37	1	no	—
123	C₄⋊Q₈	—	37	1	no	—
124	#125	—	37	2	no	—
125	C₂²⋊Q₈	—	37	1	no	—
126	C₂²⋊Q₈	—	37	2	no	—
127	C₂²⋊Q₈	#3866	1	1	yes	characteristic
128	C₄×Q₈	—	37	1	no	—
129	C₄×Q₈	—	37	1	no	—
130	C₄×Q₈	—	37	2	no	—
131	#117	—	37	2	no	—
132	C₂×C₄²	—	37	1	no	—
133	C₄⋊C₈	—	37	1	no	—
134	C₄⋊C₈	—	37	1	no	—
135	C₄⋊C₈	—	37	2	no	—
136	C₄⋊C₈	—	37	2	no	—
137	C₄⋊C₈	—	37	2	no	—
138	C₄⋊C₈	—	37	2	no	—
139	C₄⋊C₈	—	37	2	no	—
140	#97	—	37	2	no	—
141	#97	—	37	2	no	—
142	#97	—	37	2	no	—
143	#97	—	37	2	no	—
144	C₄×C₈	—	37	2	no	—
145	C₄×C₈	—	37	2	no	—
146	C₂×Q₈	—	37	2	no	—
147	C₂×Q₈	—	37	1	no	—
148	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
149	C₂²×C₄	—	37	1	no	—
150	C₂²×C₄	—	37	1	no	—
151	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
152	M₁₆	—	74	4	no	—
153	C₂×C₈	—	37	4	no	—
154	C₂×C₈	—	37	2	no	—
155	C₂×C₈	—	37	2	no	—
156	C₂×C₈	—	37	2	no	—
157	C₂×C₈	—	37	2	no	—
158	C₂×C₈	—	74	2	no	—
159	C₄⋊C₄	—	37	2	no	—
160	C₄⋊C₄	—	37	2	no	—
161	C₄⋊C₄	—	37	2	no	—
162	C₄⋊C₄	—	37	2	no	—
163	C₄⋊C₄	—	37	2	no	—
164	C₄⋊C₄	—	37	2	no	—
165	C₄⋊C₄	—	37	1	no	—
166	C₄⋊C₄	C₂×#3866	1	2	yes	—
167	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
168	C₂²⋊C₄	—	37	2	no	—
169	C₂²⋊C₄	C₂×#3866	1	2	yes	—
170	C₄²	—	37	1	no	—
171	C₄²	—	37	1	no	—
172	C₄²	—	37	2	no	—
173	C₄²	—	37	2	no	—
174	C₄²	—	37	1	no	—
175	C₄²	—	37	1	no	—
176	C₂³	C₂²×#3866	1	1	yes	characteristic
177	Q₈	—	4	1	no	—
178	Q₈	—	148	1	no	—
179	Q₈	—	148	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	37	2	no	—
182	C₂×C₄	—	37	2	no	—
183	C₂×C₄	—	37	1	no	—
184	C₂×C₄	—	37	1	no	—
185	C₂×C₄	—	37	1	no	—
186	C₂×C₄	—	37	1	no	—
187	C₂×C₄	—	37	2	no	—
188	C₂×C₄	—	37	2	no	—
189	C₂×C₄	—	148	1	no	—
190	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
191	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
192	C₂×C₄	C₂²×#3866	1	2	yes	—
193	C₂×C₄	C₂²×#3866	1	2	yes	—
194	C₈	—	37	4	no	—
195	C₈	—	74	2	no	—
196	C₈	—	74	2	no	—
197	C₈	—	74	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	Q₈×#3866	1	2	yes	—
201	C₂²	C₂³×#3866	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	37	1	no	—
207	C₄	—	37	1	no	—
208	C₄	—	37	2	no	—
209	C₄	—	74	2	no	—
210	C₄	—	74	1	no	—
211	C₄	—	74	1	no	—
212	C₄	—	74	2	no	—
213	C₂	—	2	2	no	—
214	C₂	C₂×Q₈×#3866	1	1	yes	characteristic
215	C₂	#14558611	1	1	yes	characteristic
216	C₂	#14558612	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₂²⋊Q₈×C₃₇	C₄
prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₂²⋊Q₈	#3866
hypocenter	C₃₇	#2961
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558585	C₂	semidirect
3. #14558584	C₂	semidirect
4. #14558583	C₂	semidirect
45. C₂×#64438	Q₈	semidirect
118. C₃₇	#2961	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
#14558583	2
#14558539	2
#14558536	2
#14558497	2
C₂×#11924591	2
#14558085	2
#2961	37