Subgroups — C₂²×#17988845 #33953111

Order 18688.

2878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556570	C₂	1	2	yes	maximal
3	C₂×#17988845	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556523	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14556516	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₆	C₂²	1	2	yes	—
8	C₂²×Dic₂₉₂	—	2	8	no	—
9	C₂²×Dic₂₉₂	C₂²	1	12	yes	complemented
10	C₂×#14556570	—	2	12	no	—
11	C₂×#14556570	C₂²	1	24	yes	—
12	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₂²×#11924007	—	2	2	no	—
14	C₂²×C₂²⋊Dic₇₃	—	2	2	no	—
15	#17988845	C₂²	1	64	yes	direct product factor
16	C₂×#14556523	C₂²	1	12	yes	complemented
17	C₂×#14556516	C₂²	1	24	yes	complemented
18	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₇₃	—	2	8	no	—
20	C₂²×M₁₆×C₇₃	C₂²	1	12	yes	—
21	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₄₆	—	2	6	no	—
24	C₂²×Dic₁₄₆	—	2	8	no	—
25	C₂²×Dic₁₄₆	—	2	8	no	—
26	C₂²×Dic₁₄₆	C₂³	1	12	yes	—
27	C₂³×Dic₇₃	—	2	2	no	—
28	C₂³×Dic₇₃	—	2	2	no	—
29	C₂×Dic₂₉₂	—	2	48	no	—
30	C₂×Dic₂₉₂	C₂³	1	16	yes	complemented
31	#14556570	—	2	48	no	—
32	#14556570	C₂³	1	32	yes	—
33	C₂×C₄⋊Dic₇₃	—	2	6	no	—
34	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
35	C₂×#11924007	—	2	24	no	—
36	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
37	#14556523	C₂³	1	16	yes	complemented
38	#14556516	C₂³	1	32	yes	complemented
39	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	—	2	4	no	—
41	C₂³×C₄×C₇₃	—	2	4	no	—
42	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₃	D₄	1	6	yes	—
44	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
45	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₇₃	—	2	48	no	—
47	C₂×M₁₆×C₇₃	C₂³	1	16	yes	—
48	C₂²×C₈×C₇₃	—	2	4	no	—
49	C₂²×C₈×C₇₃	D₄	1	8	yes	—
50	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
52	C₂×Dic₁₄₆	—	2	24	no	—
53	C₂×Dic₁₄₆	—	2	24	no	—
54	C₂×Dic₁₄₆	—	2	12	no	—
55	C₂×Dic₁₄₆	—	4	24	no	—
56	C₂×Dic₁₄₆	C₂⁴	1	8	yes	—
57	C₂²×Dic₇₃	—	2	12	no	—
58	C₂²×Dic₇₃	—	2	12	no	—
59	C₂²×Dic₇₃	—	4	8	no	—
60	C₂²×Dic₇₃	—	4	8	no	—
61	Dic₂₉₂	—	2	32	no	—
62	C₄⋊Dic₇₃	—	2	12	no	—
63	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
64	#11924007	—	2	32	no	—
65	C₂²⋊Dic₇₃	—	2	32	no	—
66	C₂⁴×C₇₃	—	2	4	no	—
67	C₂⁴×C₇₃	—	2	8	no	—
68	C₂⁴×C₇₃	C₂×D₄	1	6	yes	—
69	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	—	2	4	no	—
71	C₂²×C₄×C₇₃	—	2	24	no	—
72	C₂²×C₄×C₇₃	—	2	12	no	—
73	C₂²×C₄×C₇₃	—	2	12	no	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₇₃	#41	1	4	yes	—
79	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
81	M₁₆×C₇₃	—	2	32	no	—
82	C₂×C₈×C₇₃	—	2	12	no	—
83	C₂×C₈×C₇₃	C₂×D₄	1	24	yes	—
84	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
85	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
86	C₂³×C₇₃	—	2	12	no	—
87	C₂³×C₇₃	—	2	24	no	—
88	C₂³×C₇₃	—	4	16	no	—
89	C₂³×C₇₃	#121	1	4	yes	—
90	C₂³×C₇₃	#137	1	8	yes	—
91	C₂³×C₇₃	C₂²×D₄	1	4	yes	—
92	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₇₃	—	2	12	no	—
94	C₂×C₄×C₇₃	—	2	24	no	—
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	#121	1	12	yes	—
98	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
100	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
102	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₇₃	—	2	8	no	—
104	C₂×Dic₇₃	—	2	8	no	—
105	C₂×Dic₇₃	—	4	24	no	—
106	C₂×Dic₇₃	—	4	24	no	—
107	Dic₁₄₆	—	2	8	no	—
108	Dic₁₄₆	—	2	8	no	—
109	Dic₁₄₆	—	4	24	no	—
110	C₈×C₇₃	—	2	4	no	—
111	C₈×C₇₃	C₂²×D₄	1	8	yes	—
112	C₂²×C₇₃	—	2	4	no	—
113	C₂²×C₇₃	—	2	8	no	—
114	C₂²×C₇₃	—	4	24	no	—
115	C₂²×C₇₃	#470	1	16	yes	complemented
116	C₂²×C₇₃	C₂×#121	1	6	yes	—
117	C₂²×C₇₃	C₂×#137	1	12	yes	—
118	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₇₃	—	2	4	no	—
120	C₄×C₇₃	C₂×#121	1	6	yes	—
121	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
122	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
123	Dic₇₃	—	4	8	no	—
124	Dic₇₃	—	4	8	no	—
125	C₂²×#470	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	4	4	no	—
127	C₂×C₇₃	C₂×#470	1	12	yes	complemented
128	C₂×C₇₃	C₂²×#121	1	1	yes	characteristic
129	C₂×C₇₃	C₂²×#137	1	2	yes	—
130	C₂³×Q₁₆	—	73	1	no	—
131	C₂³×M₁₆	D₇₃	1	1	yes	characteristic
132	C₂²×C₂²⋊Q₈	—	73	2	no	—
133	C₂×#470	—	73	24	no	—
134	C₂²×#106	—	73	1	no	—
135	C₂²×#103	—	73	2	no	—
136	C₇₃	C₂²×#470	1	1	yes	characteristic, complemented, Sylow
137	C₂³×Q₈	—	73	2	no	—
138	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
139	C₂²×Q₁₆	—	73	12	no	—
140	C₂²×Q₁₆	—	146	8	no	—
141	C₂²×M₁₆	—	2	8	no	—
142	C₂²×M₁₆	C₂×D₇₃	1	12	yes	—
143	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
144	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
145	C₂×C₂²⋊Q₈	—	73	24	no	—
146	C₂×C₂²⋊Q₈	—	146	12	no	—
147	C₂²×C₄⋊C₄	—	73	2	no	—
148	C₂²×C₄⋊C₄	—	146	2	no	—
149	C₂²×C₂²⋊C₄	—	146	2	no	—
150	#470	—	73	64	no	—
151	C₂×#106	—	73	12	no	—
152	C₂×#103	—	73	24	no	—
153	C₂⁵	D₂₉₂	1	1	yes	characteristic
154	C₂²×Q₈	—	73	12	no	—
155	C₂²×Q₈	—	146	8	no	—
156	C₂²×Q₈	—	146	8	no	—
157	C₂²×Q₈	—	146	6	no	—
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	—	146	2	no	—
161	C₂³×C₄	—	146	2	no	—
162	C₂³×C₄	D₂₉₂	1	6	yes	—
163	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
165	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
166	C₂×Q₁₆	—	73	16	no	—
167	C₂×Q₁₆	—	146	48	no	—
168	C₂×M₁₆	—	2	48	no	—
169	C₂×M₁₆	C₂²×D₇₃	1	16	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂²⋊D₇₃	1	8	yes	—
172	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
173	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
174	C₂²⋊Q₈	—	73	32	no	—
175	C₂²⋊Q₈	—	146	48	no	—
176	C₂×C₄⋊C₄	—	73	12	no	—
177	C₂×C₄⋊C₄	—	146	6	no	—
178	C₂×C₄⋊C₄	—	146	24	no	—
179	C₂×C₂²⋊C₄	—	146	24	no	—
180	#106	—	73	16	no	—
181	#103	—	73	32	no	—
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	8	no	—
184	C₂⁴	C₂×D₂₉₂	1	6	yes	—
185	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
186	C₂×Q₈	—	292	24	no	—
187	C₂×Q₈	—	73	8	no	—
188	C₂×Q₈	—	146	24	no	—
189	C₂×Q₈	—	146	24	no	—
190	C₂×Q₈	—	146	12	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	12	no	—
194	C₂²×C₄	—	2	12	no	—
195	C₂²×C₄	—	292	8	no	—
196	C₂²×C₄	—	292	8	no	—
197	C₂²×C₄	—	146	12	no	—
198	C₂²×C₄	—	146	12	no	—
199	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
200	C₂²×C₄	C₂×D₂₉₂	1	6	yes	—
201	C₂²×C₄	C₂×D₂₉₂	1	12	yes	—
202	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
203	C₂²×C₄	#11924028	1	4	yes	—
204	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
205	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
206	Q₁₆	—	146	32	no	—
207	M₁₆	—	2	32	no	—
208	C₂×C₈	—	2	12	no	—
209	C₂×C₈	C₂×C₂²⋊D₇₃	1	24	yes	—
210	C₂×C₈	C₂³×D₇₃	1	4	yes	—
211	C₂×C₈	C₂³×D₇₃	1	4	yes	—
212	C₄⋊C₄	—	73	8	no	—
213	C₄⋊C₄	—	146	32	no	—
214	C₄⋊C₄	—	146	12	no	—
215	C₂²⋊C₄	—	146	32	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	12	no	—
218	C₂³	—	4	16	no	—
219	C₂³	#14556565	1	4	yes	—
220	C₂³	#14556624	1	8	yes	—
221	C₂³	C₂²×D₂₉₂	1	4	yes	—
222	C₂³	C₂²×D₂₉₂	1	3	yes	—
223	Q₈	—	292	24	no	—
224	Q₈	—	146	8	no	—
225	Q₈	—	146	8	no	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	292	24	no	—
231	C₂×C₄	—	292	24	no	—
232	C₂×C₄	—	146	8	no	—
233	C₂×C₄	—	146	8	no	—
234	C₂×C₄	#14556565	1	12	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₇₃	1	6	yes	—
236	C₂×C₄	C₂²×D₂₉₂	1	4	yes	—
237	C₂×C₄	C₂²×D₂₉₂	1	3	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	C₂²×C₂²⋊D₇₃	1	8	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	24	no	—
245	C₂²	#17988845	1	16	yes	direct product factor
246	C₂²	C₂×#14556565	1	6	yes	—
247	C₂²	C₂×#14556624	1	12	yes	—
248	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	292	8	no	—
251	C₄	—	292	8	no	—
252	C₄	C₂×#14556565	1	6	yes	—
253	C₄	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
254	C₄	C₂²×#11924028	1	1	yes	characteristic
255	C₂	—	4	4	no	—
256	C₂	C₂×#17988845	1	12	yes	direct product factor
257	C₂	C₂²×#14556565	1	1	yes	characteristic
258	C₂	C₂²×#14556624	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₉₂
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂³×M₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
prime radical	C₂⁵×C₇₃	D₄
hypercenter	C₂³×M₁₆	D₇₃
hypocenter	C₇₃	C₂²×#470
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₉₂	C₂	semidirect
3. C₂×#17988845	C₂	direct (N × H)
4. C₂²×#14556523	C₂	semidirect
5. C₂²×#14556516	C₂	semidirect
9. C₂²×Dic₂₉₂	C₂²	semidirect
15. #17988845	C₂²	direct (N × H)
16. C₂×#14556523	C₂²	semidirect
17. C₂×#14556516	C₂²	semidirect
30. C₂×Dic₂₉₂	C₂³	semidirect
37. #14556523	C₂³	semidirect
38. #14556516	C₂³	semidirect
115. C₂²×C₇₃	#470	semidirect
127. C₂×C₇₃	C₂×#470	semidirect
136. C₇₃	C₂²×#470	semidirect
245. C₂²	#17988845	direct (N × H)
256. C₂	C₂×#17988845	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×Dic₂₉₂	2
C₂²×#14556570	2
C₂×#17988845	2
C₂²×#14556523	2
C₂²×#14556516	2
C₂³×M₁₆×C₇₃	2
C₂²×#470	73

Subgroups — C22×#17988845 #33953111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988845 #33953111