Subgroups — #49167482

Order 37504.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22627631	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#34036636	C₂	1	1	yes	maximal, characteristic
4	#34036203	C₂	1	4	yes	maximal, complemented
5	#34036192	C₂	1	4	yes	maximal, complemented
6	#34036155	C₂	1	1	yes	maximal, characteristic, complemented
7	#34036092	C₂	1	1	yes	maximal, characteristic, complemented
8	#34036087	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₂₉₃	C₂²	1	1	yes	characteristic
11	#22627631	—	2	4	no	—
12	C₂×#17989786	C₂²	1	1	yes	characteristic
13	C₂×#17989786	C₂²	1	2	yes	complemented
14	C₂²×#14556873	C₂²	1	1	yes	characteristic
15	C₂²×#14556873	C₂²	1	2	yes	complemented
16	C₂×C₈×D₂₉₃	—	2	4	no	—
17	C₂×C₈×D₂₉₃	C₂²	1	2	yes	—
18	C₂×C₈×D₂₉₃	C₂²	1	2	yes	—
19	#22627605	C₄	1	2	yes	—
20	#22627600	C₄	1	2	yes	—
21	#22627595	C₄	1	2	yes	—
22	#22627592	C₄	1	2	yes	—
23	C₄×C₂²⋊D₂₉₃	C₂²	1	4	yes	—
24	#22627583	C₂²	1	1	yes	characteristic
25	#22627553	C₂²	1	4	yes	complemented
26	#22627536	C₂²	1	4	yes	—
27	#22627534	C₂²	1	2	yes	complemented
28	C₈×Dic₂₉₃	C₂²	1	2	yes	complemented
29	C₂×#41×C₂₉₃	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₂₉₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
32	C₂²⋊C₈×C₂₉₃	C₂²	1	4	yes	complemented
33	C₂³×D₂₉₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₂₉₃	C₂×C₄	1	4	yes	—
35	C₂²×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₉₃	—	2	2	no	—
37	C₂×C₄×D₂₉₃	—	2	2	no	—
38	C₂×C₄×D₂₉₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₉₃	C₂³	1	2	yes	—
40	#17989786	—	2	4	no	—
41	#17989786	—	2	2	no	—
42	C₂×#14556873	—	2	4	no	—
43	C₂×#14556873	D₄	1	4	yes	complemented
44	C₂×#14556873	C₂³	1	2	yes	—
45	C₂×#14556873	C₂³	1	1	yes	characteristic
46	C₂×#14556873	C₂³	1	1	yes	characteristic
47	C₈×D₂₉₃	—	2	4	no	—
48	C₈×D₂₉₃	—	2	4	no	—
49	C₄⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
50	#17989781	C₂×C₄	1	4	yes	—
51	C₂²⋊Dic₂₉₃	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
53	#17989779	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₉₃	C₂³	1	2	yes	—
55	#41×C₂₉₃	—	2	2	no	—
56	#41×C₂₉₃	—	2	2	no	—
57	C₂×Q₈×C₂₉₃	C₂×C₄	1	1	yes	characteristic
58	C₂×D₄×C₂₉₃	C₂×C₄	1	2	yes	—
59	C₂×D₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
62	M₁₆×C₂₉₃	—	2	2	no	—
63	C₂×C₈×C₂₉₃	—	2	2	no	—
64	C₂×C₈×C₂₉₃	C₂³	1	2	yes	—
65	C₂×C₈×C₂₉₃	C₂³	1	2	yes	—
66	C₂²×D₂₉₃	—	2	2	no	—
67	C₂²×D₂₉₃	—	2	4	no	—
68	C₂²×D₂₉₃	C₂²×C₄	1	2	yes	—
69	C₂²⋊D₂₉₃	—	2	8	no	—
70	C₂×Dic₂₉₃	—	2	2	no	—
71	C₂×Dic₂₉₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₂₉₃	C₂²×C₄	1	2	yes	—
73	C₄×D₂₉₃	—	2	4	no	—
74	C₄×D₂₉₃	—	2	2	no	—
75	C₄×D₂₉₃	—	2	2	no	—
76	#14556873	—	2	2	no	—
77	#14556873	C₂×D₄	1	2	yes	—
78	#14556873	C₂×D₄	1	2	yes	—
79	C₂³×C₂₉₃	C₂²×C₄	1	2	yes	—
80	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₂₉₃	—	2	2	no	—
82	D₄×C₂₉₃	—	2	4	no	—
83	D₄×C₂₉₃	—	2	2	no	—
84	C₂×C₄×C₂₉₃	—	2	2	no	—
85	C₂×C₄×C₂₉₃	—	2	2	no	—
86	C₂×C₄×C₂₉₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₉₃	#41	1	2	yes	—
90	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₂₉₃	—	2	2	no	—
92	C₈×C₂₉₃	—	2	2	no	—
93	C₂²×C₂₉₃	—	2	2	no	—
94	C₂²×C₂₉₃	—	2	2	no	—
95	C₂²×C₂₉₃	—	2	4	no	—
96	C₂²×C₂₉₃	C₄×D₄	1	2	yes	—
97	C₂²×C₂₉₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₉₃	—	2	8	no	—
99	C₂×D₂₉₃	—	2	2	no	—
100	C₂×D₂₉₃	—	2	2	no	—
101	C₂×D₂₉₃	#131	1	4	yes	complemented
102	C₄×C₂₉₃	—	2	2	no	—
103	C₄×C₂₉₃	C₄×D₄	1	2	yes	—
104	C₄×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₉₃	C₂×#41	1	1	yes	characteristic
106	Dic₂₉₃	—	2	2	no	—
107	Dic₂₉₃	—	2	4	no	—
108	C₂×C₂₉₃	—	2	2	no	—
109	C₂×C₂₉₃	—	4	2	no	—
110	C₂×C₂₉₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₂₉₃	C₂×#131	1	2	yes	—
112	D₂₉₃	—	2	4	no	—
113	C₂₉₃	#2929	1	1	yes	characteristic, complemented, Sylow
114	#2929	—	293	1	no	maximal, Sylow
115	C₂×#131	—	293	1	no	—
116	C₂³×C₈	—	293	1	no	—
117	#525	—	293	1	no	—
118	#435	—	293	4	no	—
119	C₈×D₄	—	293	4	no	—
120	#424	—	293	1	no	—
121	#408	D₂₉₃	1	1	yes	characteristic, complemented
122	#407	—	293	1	no	—
123	#405	—	293	1	no	—
124	C₂×#41	C₂×D₂₉₃	1	1	yes	characteristic
125	C₂³×C₄	—	293	1	no	—
126	#131	—	586	4	no	—
127	C₂×M₁₆	—	293	1	no	—
128	C₂×M₁₆	—	293	2	no	—
129	C₂×M₁₆	C₂×D₂₉₃	1	1	yes	characteristic
130	C₂²×C₈	—	293	1	no	—
131	C₂²×C₈	—	293	2	no	—
132	C₂²×C₈	—	293	2	no	—
133	C₂²×C₈	—	293	2	no	—
134	C₂²×C₈	—	586	4	no	—
135	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
136	#123	—	293	2	no	—
137	C₂²⋊Q₈	—	293	2	no	—
138	#121	—	293	2	no	—
139	#120	—	293	2	no	—
140	C₄×D₄	—	293	4	no	—
141	#117	—	293	1	no	—
142	C₄⋊C₈	—	293	4	no	—
143	C₂²⋊C₈	—	293	4	no	—
144	C₂²⋊C₈	C₂×D₂₉₃	1	4	yes	complemented
145	#97	—	293	2	no	—
146	C₄×C₈	—	293	2	no	—
147	C₂⁴	—	293	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₂₉₃	1	1	yes	characteristic
151	C₂×D₄	—	293	4	no	—
152	C₂×D₄	C₄×D₂₉₃	1	2	yes	—
153	C₂×D₄	C₄×D₂₉₃	1	1	yes	characteristic
154	C₂²×C₄	—	293	1	no	—
155	C₂²×C₄	—	293	2	no	—
156	C₂²×C₄	—	293	2	no	—
157	C₂²×C₄	—	586	2	no	—
158	C₂²×C₄	—	586	2	no	—
159	C₂²×C₄	C₂²×D₂₉₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	586	4	no	—
163	M₁₆	—	586	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	293	4	no	—
166	C₂×C₈	—	293	1	no	—
167	C₂×C₈	—	293	1	no	—
168	C₂×C₈	—	293	2	no	—
169	C₂×C₈	—	586	4	no	—
170	C₂×C₈	—	586	4	no	—
171	C₂×C₈	—	586	4	no	—
172	C₂×C₈	C₂²×D₂₉₃	1	2	yes	—
173	C₂×C₈	C₂²×D₂₉₃	1	2	yes	—
174	C₄⋊C₄	—	293	2	no	—
175	C₄⋊C₄	—	293	4	no	—
176	C₂²⋊C₄	—	293	2	no	—
177	C₂²⋊C₄	—	293	4	no	—
178	C₂²⋊C₄	—	293	4	no	—
179	C₄²	—	293	2	no	—
180	C₂³	—	293	2	no	—
181	C₂³	—	586	4	no	—
182	C₂³	—	586	2	no	—
183	C₂³	C₂×C₄×D₂₉₃	1	2	yes	—
184	C₂³	C₂×C₄×D₂₉₃	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	586	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	293	4	no	—
192	C₂×C₄	—	293	2	no	—
193	C₂×C₄	—	586	2	no	—
194	C₂×C₄	—	586	4	no	—
195	C₂×C₄	—	586	2	no	—
196	C₂×C₄	—	586	2	no	—
197	C₂×C₄	C₂×C₄×D₂₉₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
199	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
200	C₂×C₄	#17989801	1	2	yes	—
201	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	293	2	no	—
205	C₈	—	293	2	no	—
206	C₈	—	586	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	293	4	no	—
211	C₂²	—	586	8	no	—
212	C₂²	—	586	2	no	—
213	C₂²	—	586	2	no	—
214	C₂²	#22627584	1	2	yes	—
215	C₂²	C₂²×C₄×D₂₉₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	586	2	no	—
218	C₄	—	586	4	no	—
219	C₄	#22627584	1	2	yes	—
220	C₄	C₂×D₄×D₂₉₃	1	1	yes	characteristic
221	C₄	C₂×#17989801	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	586	4	no	—
225	C₂	C₂×#22627584	1	1	yes	characteristic
226	C₂	#131×D₂₉₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉₃
commutator, socle	C₂²×C₂₉₃	C₂³×C₄
Fitting	#408×C₂₉₃	C₂
prime radical	#34036636	C₂
cosocle	C₂×C₄×C₂₉₃	C₂⁴
hypercenter	#408	D₂₉₃
hypocenter	C₂₉₃	#2929
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#22627631	C₂	semidirect
2. C₂²×C₈×D₂₉₃	C₂	semidirect
4. #34036203	C₂	semidirect
5. #34036192	C₂	semidirect
6. #34036155	C₂	semidirect
7. #34036092	C₂	semidirect
8. #34036087	C₂	semidirect
9. #408×C₂₉₃	C₂	semidirect
13. C₂×#17989786	C₂²	semidirect
15. C₂²×#14556873	C₂²	semidirect
25. #22627553	C₂²	semidirect
27. #22627534	C₂²	semidirect
28. C₈×Dic₂₉₃	C₂²	semidirect
32. C₂²⋊C₈×C₂₉₃	C₂²	semidirect
43. C₂×#14556873	D₄	semidirect
101. C₂×D₂₉₃	#131	semidirect
113. C₂₉₃	#2929	semidirect
121. #408	D₂₉₃	semidirect
144. C₂²⋊C₈	C₂×D₂₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22627631	2
C₂²×C₈×D₂₉₃	2
#34036636	2
#34036203	2
#34036192	2
#34036155	2
#34036092	2
#34036087	2
#408×C₂₉₃	2
#2929	293