Subgroups — C₂²×#17988666 #33949831

Order 18688.

3262 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924011	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988666	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14556503	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₉₂	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₇₃	C₂²	1	2	yes	—
9	C₂²×#11924011	—	2	16	no	—
10	C₂²×#11924011	C₂²	1	24	yes	complemented
11	C₂³×#10596831	C₂²	1	2	yes	—
12	C₂×C₄×D₂₉₂	C₄	1	12	yes	—
13	C₂×C₄×D₂₉₂	C₂²	1	12	yes	—
14	C₂²×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
15	C₂²×#11924005	C₄	1	2	yes	—
16	#17988666	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊#10596831	C₂²	1	12	yes	complemented
18	C₂×#14556503	C₂²	1	24	yes	complemented
19	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₇₃	C₂×C₄	1	2	yes	—
24	C₂²×D₂₉₂	—	2	8	no	—
25	C₂²×D₂₉₂	C₂×C₄	1	12	yes	—
26	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×D₇₃	—	2	8	no	—
28	C₂²×C₄×D₇₃	—	2	8	no	—
29	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
30	C₂²×C₄×D₇₃	C₂³	1	12	yes	—
31	C₂×#11924011	—	2	96	no	—
32	C₂×#11924011	C₂³	1	32	yes	complemented
33	C₂²×#10596831	—	2	8	no	—
34	C₂²×#10596831	C₂³	1	12	yes	—
35	C₄×D₂₉₂	C₂×C₄	1	48	yes	—
36	C₄×D₂₉₂	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₇₃	C₂×C₄	1	12	yes	—
38	C₂×#11924005	C₂×C₄	1	24	yes	—
39	C₄⋊#10596831	C₂³	1	16	yes	complemented
40	#14556503	C₂³	1	32	yes	complemented
41	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	—	2	4	no	—
45	C₂²×C₈×C₇₃	D₄	1	8	yes	complemented
46	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
47	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
48	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
49	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
50	C₄×C₈×C₇₃	C₂³	1	16	yes	complemented
51	C₂³×D₇₃	—	2	8	no	—
52	C₂³×D₇₃	—	2	8	no	—
53	C₂³×D₇₃	—	2	8	no	—
54	C₂³×D₇₃	C₂²×C₄	1	12	yes	—
55	C₂×D₂₉₂	—	2	48	no	—
56	C₂×D₂₉₂	C₂²×C₄	1	16	yes	—
57	C₂²×Dic₇₃	—	2	8	no	—
58	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
59	C₂×C₄×D₇₃	—	2	48	no	—
60	C₂×C₄×D₇₃	—	2	24	no	—
61	C₂×C₄×D₇₃	—	2	24	no	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	24	yes	—
63	C₂×C₄×D₇₃	C₂⁴	1	8	yes	—
64	#11924011	—	2	64	no	—
65	C₂×#10596831	—	2	24	no	—
66	C₂×#10596831	C₂⁴	1	8	yes	—
67	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
68	#11924005	C₂²×C₄	1	32	yes	—
69	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	—	2	4	no	—
71	C₂²×C₄×C₇₃	M₁₆	1	8	yes	complemented
72	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₇₃	#41	1	4	yes	—
77	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
78	C₂×C₈×C₇₃	—	2	12	no	—
79	C₂×C₈×C₇₃	C₂×D₄	1	24	yes	complemented
80	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
81	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
82	C₄²×C₇₃	C₂²×C₄	1	12	yes	—
83	C₄²×C₇₃	C₂⁴	1	4	yes	—
84	C₂³×C₇₃	C₄×D₄	1	4	yes	—
85	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
86	C₂³×C₇₃	#131	1	4	yes	—
87	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
88	C₂²×D₇₃	—	2	24	no	—
89	C₂²×D₇₃	—	2	24	no	—
90	C₂²×D₇₃	—	2	24	no	—
91	C₂²×D₇₃	—	4	32	no	—
92	C₂²×D₇₃	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₇₃	—	2	12	no	—
94	C₂×C₄×C₇₃	C₄×D₄	1	12	yes	—
95	C₂×C₄×C₇₃	C₂×M₁₆	1	24	yes	complemented
96	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
98	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
101	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₇₃	—	2	24	no	—
103	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
104	D₂₉₂	—	2	32	no	—
105	C₄×D₇₃	—	2	48	no	—
106	C₄×D₇₃	—	2	8	no	—
107	C₄×D₇₃	—	2	8	no	—
108	C₈×C₇₃	—	2	4	no	—
109	C₈×C₇₃	C₂²×D₄	1	8	yes	complemented
110	#10596831	—	2	8	no	—
111	C₂²×C₇₃	#436	1	16	yes	complemented
112	C₂²×C₇₃	C₂×C₄×D₄	1	6	yes	—
113	C₂²×C₇₃	C₂²×M₁₆	1	6	yes	—
114	C₂²×C₇₃	C₂×#131	1	6	yes	—
115	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
116	C₂×D₇₃	—	2	8	no	—
117	C₂×D₇₃	—	2	8	no	—
118	C₂×D₇₃	—	2	8	no	—
119	C₂×D₇₃	—	4	48	no	—
120	C₄×C₇₃	—	2	4	no	—
121	C₄×C₇₃	C₂×C₄×D₄	1	6	yes	—
122	C₄×C₇₃	C₂²×M₁₆	1	8	yes	complemented
123	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
125	Dic₇₃	—	2	8	no	—
126	C₂²×#436	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	C₂×#436	1	12	yes	complemented
128	C₂×C₇₃	C₂²×C₄×D₄	1	1	yes	characteristic
129	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
130	C₂×C₇₃	C₂²×#131	1	1	yes	characteristic
131	D₇₃	—	4	8	no	—
132	C₂³×M₁₆	—	73	2	no	—
133	C₂²×C₄×D₄	—	73	1	no	—
134	C₂×#436	—	73	24	no	—
135	C₂²×C₄⋊C₈	—	73	1	no	—
136	C₂²×C₂²⋊C₈	—	73	2	no	—
137	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic, complemented
138	C₇₃	C₂²×#436	1	1	yes	characteristic, complemented, Sylow
139	C₂³×D₄	—	73	1	no	—
140	C₂⁴×C₄	—	73	2	no	—
141	C₂²×M₁₆	—	73	24	no	—
142	C₂²×M₁₆	—	146	16	no	—
143	C₂³×C₈	—	73	2	no	—
144	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
145	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
146	C₂×C₄×D₄	—	73	12	no	—
147	C₂×C₄×D₄	—	73	12	no	—
148	C₂²×C₄⋊C₄	—	73	1	no	—
149	C₂²×C₂²⋊C₄	—	73	2	no	—
150	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
151	#436	—	73	64	no	—
152	C₂×C₄⋊C₈	—	73	12	no	—
153	C₂×C₂²⋊C₈	—	73	24	no	—
154	C₂×C₄×C₈	C₂×D₇₃	1	12	yes	complemented
155	C₂⁵	—	73	2	no	—
156	C₂²×D₄	—	73	12	no	—
157	C₂²×D₄	—	146	8	no	—
158	C₂³×C₄	—	73	2	no	—
159	C₂³×C₄	—	73	12	no	—
160	C₂³×C₄	—	73	12	no	—
161	C₂³×C₄	—	146	8	no	—
162	C₂³×C₄	—	146	8	no	—
163	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
166	C₂×M₁₆	—	73	32	no	—
167	C₂×M₁₆	—	146	96	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	73	12	no	—
170	C₂²×C₈	—	146	8	no	—
171	C₂²×C₈	D₂₉₂	1	8	yes	complemented
172	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
173	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
174	C₄×D₄	—	73	48	no	—
175	C₄×D₄	—	73	16	no	—
176	C₂×C₄⋊C₄	—	73	12	no	—
177	C₂×C₂²⋊C₄	—	73	24	no	—
178	C₂×C₄²	C₄×D₇₃	1	6	yes	—
179	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
180	C₄⋊C₈	—	73	16	no	—
181	C₂²⋊C₈	—	73	32	no	—
182	C₄×C₈	C₂²×D₇₃	1	16	yes	complemented
183	C₂⁴	—	73	12	no	—
184	C₂⁴	—	146	8	no	—
185	C₂⁴	—	146	8	no	—
186	C₂⁴	—	146	8	no	—
187	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
188	C₂×D₄	—	73	16	no	—
189	C₂×D₄	—	146	48	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	73	12	no	—
192	C₂²×C₄	—	73	24	no	—
193	C₂²×C₄	—	73	8	no	—
194	C₂²×C₄	—	146	8	no	—
195	C₂²×C₄	—	146	48	no	—
196	C₂²×C₄	—	146	24	no	—
197	C₂²×C₄	—	146	24	no	—
198	C₂²×C₄	#11924011	1	8	yes	complemented
199	C₂²×C₄	C₂×C₄×D₇₃	1	6	yes	—
200	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
201	C₂²×C₄	C₂×C₄×D₇₃	1	6	yes	—
202	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
203	C₂²×C₄	#11924028	1	4	yes	—
204	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
205	M₁₆	—	146	64	no	—
206	C₂×C₈	—	2	12	no	—
207	C₂×C₈	—	73	8	no	—
208	C₂×C₈	—	146	24	no	—
209	C₂×C₈	C₂×D₂₉₂	1	24	yes	complemented
210	C₂×C₈	C₂³×D₇₃	1	4	yes	—
211	C₂×C₈	C₂³×D₇₃	1	4	yes	—
212	C₄⋊C₄	—	73	16	no	—
213	C₂²⋊C₄	—	73	32	no	—
214	C₄²	C₂×C₄×D₇₃	1	12	yes	—
215	C₄²	C₂³×D₇₃	1	4	yes	—
216	C₂³	—	292	32	no	—
217	C₂³	—	73	8	no	—
218	C₂³	—	146	24	no	—
219	C₂³	—	146	24	no	—
220	C₂³	—	146	24	no	—
221	C₂³	C₄×D₂₉₂	1	4	yes	—
222	C₂³	C₂×#11924011	1	4	yes	—
223	C₂³	#14556597	1	4	yes	—
224	C₂³	C₂²×C₄×D₇₃	1	3	yes	—
225	D₄	—	146	32	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	73	8	no	—
228	C₂×C₄	—	146	24	no	—
229	C₂×C₄	—	146	8	no	—
230	C₂×C₄	—	146	8	no	—
231	C₂×C₄	—	146	48	no	—
232	C₂×C₄	C₄×D₂₉₂	1	12	yes	—
233	C₂×C₄	C₂×#11924011	1	24	yes	complemented
234	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
235	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
236	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
237	C₂×C₄	C₂²×D₂₉₂	1	6	yes	—
238	C₂×C₄	C₂×#11924028	1	6	yes	—
239	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	146	8	no	—
242	C₈	C₂²×D₂₉₂	1	8	yes	complemented
243	C₂²	—	292	48	no	—
244	C₂²	—	146	8	no	—
245	C₂²	—	146	8	no	—
246	C₂²	—	146	8	no	—
247	C₂²	#17988666	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₂₉₂	1	6	yes	—
249	C₂²	C₂²×#11924011	1	6	yes	—
250	C₂²	C₂×#14556597	1	6	yes	—
251	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	146	8	no	—
254	C₄	C₂×C₄×D₂₉₂	1	6	yes	—
255	C₄	C₂²×#11924011	1	8	yes	complemented
256	C₄	C₂³×D₂₉₂	1	1	yes	characteristic
257	C₄	C₂²×#11924028	1	1	yes	characteristic
258	C₂	—	292	8	no	—
259	C₂	C₂×#17988666	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₂₉₂	1	1	yes	characteristic
261	C₂	C₂³×#11924011	1	1	yes	characteristic
262	C₂	C₂²×#14556597	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂³×D₂₉₂	C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	C₂²×#436
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11924011	C₂	semidirect
3. C₂×#17988666	C₂	direct (N × H)
4. C₂²×C₄⋊#10596831	C₂	semidirect
5. C₂²×#14556503	C₂	semidirect
6. C₂²×C₄×C₈×C₇₃	C₂	semidirect
10. C₂²×#11924011	C₂²	semidirect
16. #17988666	C₂²	direct (N × H)
17. C₂×C₄⋊#10596831	C₂²	semidirect
18. C₂×#14556503	C₂²	semidirect
22. C₂×C₄×C₈×C₇₃	C₂²	semidirect
32. C₂×#11924011	C₂³	semidirect
39. C₄⋊#10596831	C₂³	semidirect
40. #14556503	C₂³	semidirect
45. C₂²×C₈×C₇₃	D₄	semidirect
50. C₄×C₈×C₇₃	C₂³	semidirect
71. C₂²×C₄×C₇₃	M₁₆	semidirect
79. C₂×C₈×C₇₃	C₂×D₄	semidirect
95. C₂×C₄×C₇₃	C₂×M₁₆	semidirect
109. C₈×C₇₃	C₂²×D₄	semidirect
111. C₂²×C₇₃	#436	semidirect
122. C₄×C₇₃	C₂²×M₁₆	semidirect
127. C₂×C₇₃	C₂×#436	semidirect
137. C₂²×C₄×C₈	D₇₃	semidirect
138. C₇₃	C₂²×#436	semidirect
154. C₂×C₄×C₈	C₂×D₇₃	semidirect
171. C₂²×C₈	D₂₉₂	semidirect
182. C₄×C₈	C₂²×D₇₃	semidirect
198. C₂²×C₄	#11924011	semidirect
209. C₂×C₈	C₂×D₂₉₂	semidirect
233. C₂×C₄	C₂×#11924011	semidirect
242. C₈	C₂²×D₂₉₂	semidirect
247. C₂²	#17988666	direct (N × H)
255. C₄	C₂²×#11924011	semidirect
259. C₂	C₂×#17988666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924011	2
C₂²×C₄×D₂₉₂	2
C₂×#17988666	2
C₂²×C₄⋊#10596831	2
C₂²×#14556503	2
C₂²×C₄×C₈×C₇₃	2
C₂²×#436	73

Subgroups — C22×#17988666 #33949831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988666 #33949831