Subgroups — C₂×#14557303 #17994337

Order 4736.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10596934	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10596934	C₂	1	1	yes	maximal, characteristic
3	#14557303	C₂	1	4	yes	maximal, direct product factor
4	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596934	—	2	1	no	—
6	C₂×#10596934	—	2	1	no	—
7	C₂×#10596934	—	2	1	no	—
8	C₂×#10596934	—	2	1	no	—
9	C₂×#10596934	—	2	1	no	—
10	C₂×#10596934	—	2	1	no	—
11	C₂×#10596934	—	2	1	no	—
12	C₂×#10596934	C₄	1	2	yes	complemented
13	C₂×#10596934	C₄	1	2	yes	complemented
14	C₂×#10596934	C₄	1	2	yes	complemented
15	C₂×#10596934	C₂²	1	2	yes	—
16	C₂×#10596934	C₂²	1	2	yes	—
17	C₂²×#64436	—	2	1	no	—
18	C₂²×#64436	C₄	1	2	yes	complemented
19	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
20	#117×C₃₇	—	2	1	no	—
21	#117×C₃₇	—	2	1	no	—
22	#117×C₃₇	C₄	1	2	yes	—
23	#117×C₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₃₇	C₄	1	2	yes	—
26	C₂×C₄²×C₃₇	—	2	1	no	—
27	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₇	—	2	2	no	—
29	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂²⋊C₄×C₃₇	—	2	2	no	—
46	C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
47	C₄²×C₃₇	—	2	2	no	—
48	C₄²×C₃₇	—	2	2	no	—
49	#10596934	—	2	2	no	—
50	#10596934	—	2	2	no	—
51	#10596934	—	2	2	no	—
52	#10596934	—	2	2	no	—
53	#10596934	—	2	2	no	—
54	#10596934	—	2	2	no	—
55	#10596934	—	2	2	no	—
56	#10596934	C₂×C₄	1	4	yes	complemented
57	C₂×#64436	—	2	2	no	—
58	C₂×#64436	—	2	2	no	—
59	C₂×#64436	—	2	2	no	—
60	C₂×#64436	—	2	2	no	—
61	C₂×#64436	—	2	2	no	—
62	C₂×#64436	C₂×C₄	1	4	yes	complemented
63	C₂³×C₃₇	—	4	2	no	—
64	C₂³×C₃₇	C₄²	1	1	yes	characteristic
65	C₂³×C₃₇	C₄²	1	1	yes	characteristic
66	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₇	—	2	4	no	—
72	C₂×C₄×C₃₇	—	4	4	no	—
73	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
74	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
75	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
101	#64436	—	2	4	no	—
102	#64436	—	2	4	no	—
103	C₂²×C₃₇	—	2	2	no	—
104	C₂²×C₃₇	—	2	2	no	—
105	C₂²×C₃₇	—	2	2	no	—
106	C₂²×C₃₇	—	4	1	no	—
107	C₂²×C₃₇	—	4	1	no	—
108	C₂²×C₃₇	—	4	1	no	—
109	C₂²×C₃₇	—	4	1	no	—
110	C₂²×C₃₇	#95	1	1	yes	characteristic
111	C₂²×C₃₇	#95	1	1	yes	characteristic
112	C₂²×C₃₇	#95	1	1	yes	characteristic
113	C₂²×C₃₇	#95	1	1	yes	characteristic
114	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₃₇	—	4	4	no	—
118	C₄×C₃₇	#95	1	1	yes	characteristic
119	C₄×C₃₇	#95	1	1	yes	characteristic
120	C₄×C₃₇	#95	1	1	yes	characteristic
121	C₄×C₃₇	#95	1	1	yes	characteristic
122	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×#344	—	37	1	no	maximal, Sylow
127	C₂×C₃₇	—	2	1	no	—
128	C₂×C₃₇	—	2	1	no	—
129	C₂×C₃₇	—	2	1	no	—
130	C₂×C₃₇	—	2	1	no	—
131	C₂×C₃₇	—	2	1	no	—
132	C₂×C₃₇	—	2	1	no	—
133	C₂×C₃₇	#344	1	2	yes	complemented
134	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
135	C₂²×M₁₆	—	37	1	no	—
136	C₂²×M₁₆	—	37	1	no	—
137	C₂×#117	D₃₇	1	1	yes	characteristic
138	#344	—	37	4	no	—
139	C₃₇	C₂×#344	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
141	C₂×M₁₆	—	37	2	no	—
142	C₂×M₁₆	—	37	2	no	—
143	C₂×M₁₆	—	37	2	no	—
144	C₂×M₁₆	—	37	2	no	—
145	C₂×M₁₆	—	37	2	no	—
146	C₂×M₁₆	—	74	1	no	—
147	C₂×M₁₆	—	74	1	no	—
148	C₂×M₁₆	—	74	1	no	—
149	C₂×M₁₆	—	74	1	no	—
150	C₂×M₁₆	—	74	1	no	—
151	C₂×M₁₆	—	74	1	no	—
152	C₂×M₁₆	—	74	1	no	—
153	C₂²×C₈	—	37	2	no	—
154	C₂²×C₈	—	74	1	no	—
155	#117	—	2	1	no	—
156	#117	—	2	1	no	—
157	#117	Dic₃₇	1	2	yes	—
158	#117	C₂×D₃₇	1	2	yes	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₂²⋊C₄	Dic₃₇	1	2	yes	—
161	C₂×C₄²	—	2	1	no	—
162	C₂⁴	C₂×Dic₃₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	Dic₇₄	1	1	yes	characteristic
165	C₂²×C₄	Dic₇₄	1	1	yes	characteristic
166	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
170	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
171	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₃₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
178	M₁₆	—	37	4	no	—
179	M₁₆	—	74	2	no	—
180	M₁₆	—	74	2	no	—
181	M₁₆	—	74	2	no	—
182	M₁₆	—	74	2	no	—
183	M₁₆	—	74	2	no	—
184	M₁₆	—	74	2	no	—
185	M₁₆	—	74	2	no	—
186	C₂×C₈	—	37	4	no	—
187	C₂×C₈	—	74	2	no	—
188	C₂×C₈	—	74	2	no	—
189	C₂×C₈	—	74	2	no	—
190	C₂×C₈	—	74	2	no	—
191	C₂×C₈	—	74	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	C₂×Dic₃₇	1	4	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄×Dic₃₇	1	1	yes	characteristic
200	C₂³	C₄×Dic₃₇	1	1	yes	characteristic
201	C₂³	C₄⋊Dic₃₇	1	1	yes	characteristic
202	C₂³	C₄⋊Dic₃₇	1	1	yes	characteristic
203	C₂³	C₂²⋊Dic₃₇	1	1	yes	characteristic
204	C₂³	C₂²⋊Dic₃₇	1	1	yes	characteristic
205	C₂³	C₂²×Dic₃₇	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₄×Dic₃₇	1	1	yes	characteristic
209	C₂×C₄	C₄×Dic₃₇	1	1	yes	characteristic
210	C₂×C₄	#10596936	1	1	yes	characteristic
211	C₂×C₄	#10596936	1	1	yes	characteristic
212	C₂×C₄	#10596936	1	1	yes	characteristic
213	C₂×C₄	#10596936	1	1	yes	characteristic
214	C₂×C₄	#10596936	1	1	yes	characteristic
215	C₂×C₄	#10596936	1	1	yes	characteristic
216	C₂×C₄	#10596936	1	1	yes	characteristic
217	C₂×C₄	#10596936	1	1	yes	characteristic
218	C₂×C₄	C₄⋊Dic₃₇	1	1	yes	characteristic
219	C₂×C₄	C₄⋊Dic₃₇	1	1	yes	characteristic
220	C₂×C₄	#10596938	1	1	yes	characteristic
221	C₂×C₄	#10596938	1	1	yes	characteristic
222	C₂×C₄	#10596938	1	1	yes	characteristic
223	C₂×C₄	#10596938	1	1	yes	characteristic
224	C₂×C₄	#10596938	1	1	yes	characteristic
225	C₂×C₄	#10596938	1	1	yes	characteristic
226	C₂×C₄	#10596938	1	1	yes	characteristic
227	C₂×C₄	#10596938	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊Dic₃₇	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
236	C₈	—	74	4	no	—
237	C₈	—	74	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#11924443	1	1	yes	characteristic
246	C₂²	#11924443	1	1	yes	characteristic
247	C₂²	#11924443	1	1	yes	characteristic
248	C₂²	#11924443	1	1	yes	characteristic
249	C₂²	C₂×C₄×Dic₃₇	1	1	yes	characteristic
250	C₂²	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
251	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
252	C₄	—	4	4	no	—
253	C₄	#11924443	1	1	yes	characteristic
254	C₄	#11924443	1	1	yes	characteristic
255	C₄	#11924443	1	1	yes	characteristic
256	C₄	#11924443	1	1	yes	characteristic
257	C₄	C₂×#10596938	1	1	yes	characteristic
258	C₄	C₂×#10596938	1	1	yes	characteristic
259	C₄	C₂×#10596936	1	1	yes	characteristic
260	C₄	C₂×#10596936	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#14557303	1	2	yes	direct product factor
268	C₂	C₂×#11924443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10596938
commutator	C₂²×C₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#117×C₃₇	C₂
socle	C₂²×C₃₇	#95
prime radical	C₂⁴×C₃₇	C₂×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	C₂×#344
LCS	G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14557303	C₂	direct (N × H)
12. C₂×#10596934	C₄	semidirect
13. C₂×#10596934	C₄	semidirect
14. C₂×#10596934	C₄	semidirect
18. C₂²×#64436	C₄	semidirect
56. #10596934	C₂×C₄	semidirect
62. C₂×#64436	C₂×C₄	semidirect
133. C₂×C₃₇	#344	semidirect
139. C₃₇	C₂×#344	semidirect
267. C₂	#14557303	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10596934	2
#14557303	2
C₂×#117×C₃₇	2
C₂×#344	37

Subgroups — C2×#14557303 #17994337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557303 #17994337