Subgroups — C₂×#22618865 #33950349

Order 18688.

1982 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989144	C₂	1	1	yes	maximal, characteristic
2	#22618865	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988717	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Dic₁₄₆	C₂	1	4	yes	maximal, complemented
5	C₂×#17988613	C₂	1	2	yes	maximal
6	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988542	C₂	1	2	yes	maximal
8	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11924012	C₂²	1	2	yes	—
10	#17989144	C₄	1	8	yes	—
11	#17989144	C₂²	1	8	yes	—
12	C₂×#14556591	C₄	1	2	yes	—
13	C₂×#14556583	C₄	1	2	yes	—
14	C₂×#14556570	C₄	1	4	yes	—
15	C₂×C₄×Dic₁₄₆	C₂²	1	4	yes	—
16	C₂×#14556550	C₂²	1	2	yes	—
17	#17988717	C₂²	1	32	yes	complemented
18	C₈×Dic₁₄₆	C₂²	1	32	yes	complemented
19	#17988613	C₂²	1	16	yes	—
20	#17988611	C₂²	1	8	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
22	C₂×#14556520	C₂²	1	8	yes	—
23	#17988542	C₂²	1	16	yes	—
24	C₂×#14556501	C₂²	1	4	yes	—
25	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
26	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	8	yes	—
32	C₂²×Dic₁₄₆	C₂×C₄	1	4	yes	—
33	C₂×#11924012	—	2	8	no	—
34	C₂×#11924012	C₂³	1	8	yes	—
35	C₂²×#10596831	C₂³	1	4	yes	—
36	#14556591	C₂×C₄	1	16	yes	—
37	#14556583	C₂×C₄	1	16	yes	—
38	#14556570	C₂×C₄	1	32	yes	—
39	C₄×Dic₁₄₆	C₂×C₄	1	16	yes	—
40	C₄×Dic₁₄₆	C₂³	1	16	yes	—
41	#14556550	C₂×C₄	1	8	yes	—
42	#14556550	C₂³	1	8	yes	—
43	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
45	C₂×#11924007	C₂×C₄	1	8	yes	—
46	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
48	C₄⋊#10596831	C₂³	1	16	yes	—
49	#14556520	C₂³	1	32	yes	—
50	#14556501	C₂³	1	16	yes	—
51	C₈×Dic₇₃	C₂³	1	16	yes	—
52	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	—	2	4	no	—
55	C₂²×C₈×C₇₃	Q₈	1	8	yes	complemented
56	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
57	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
65	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
66	C₄×C₈×C₇₃	C₂³	1	8	yes	—
67	C₄×C₈×C₇₃	C₂³	1	8	yes	—
68	C₂×Dic₁₄₆	—	2	16	no	—
69	C₂×Dic₁₄₆	C₂²×C₄	1	16	yes	—
70	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
71	#11924012	—	2	16	no	—
72	C₂×#10596831	—	2	8	no	—
73	C₂×#10596831	C₂⁴	1	8	yes	—
74	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
75	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
76	#11924007	C₂²×C₄	1	32	yes	—
77	C₂²⋊Dic₇₃	C₂²×C₄	1	16	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
79	C₄×Dic₇₃	C₂⁴	1	8	yes	—
80	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	—	2	8	no	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇₃	#41	1	4	yes	—
89	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
91	C₂×C₈×C₇₃	—	2	4	no	—
92	C₂×C₈×C₇₃	—	2	8	no	—
93	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
94	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₇₃	C₂×Q₈	1	16	yes	complemented
96	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
97	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
98	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
99	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
100	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₇₃	C₂⁴	1	4	yes	—
103	C₂³×C₇₃	—	2	4	no	—
104	C₂³×C₇₃	C₄×Q₈	1	4	yes	—
105	C₂³×C₇₃	C₂³×C₄	1	2	yes	—
106	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	16	no	—
109	C₂×C₄×C₇₃	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₇₃	#131	1	8	yes	—
112	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
116	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
120	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
121	C₂×Dic₇₃	—	2	16	no	—
122	C₂×Dic₇₃	C₂³×C₄	1	16	yes	—
123	Dic₁₄₆	—	2	32	no	—
124	C₈×C₇₃	—	2	4	no	—
125	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
126	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
127	#10596831	—	2	8	no	—
128	C₂²×C₇₃	—	2	8	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	C₂×C₄×Q₈	1	4	yes	—
131	C₂²×C₇₃	C₂×C₄×Q₈	1	2	yes	—
132	C₂²×C₇₃	C₂×#131	1	4	yes	—
133	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₇₃	—	2	4	no	—
135	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₇₃	C₂×C₄×Q₈	1	4	yes	—
137	C₄×C₇₃	C₂×#131	1	8	yes	—
138	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
139	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
140	Dic₇₃	—	2	16	no	—
141	C₂×#2960	—	73	1	no	maximal, Sylow
142	C₂×C₇₃	—	2	4	no	—
143	C₂×C₇₃	#2960	1	4	yes	complemented
144	C₂×C₇₃	C₂²×C₄×Q₈	1	1	yes	characteristic
145	C₂×C₇₃	C₂²×#131	1	2	yes	—
146	C₂×#533	—	73	1	no	—
147	#2960	—	73	16	no	—
148	C₂×#446	—	73	4	no	—
149	C₂×C₈×Q₈	—	73	4	no	—
150	C₂×#424	—	73	2	no	—
151	C₂×#423	—	73	1	no	—
152	C₂×#405	—	73	2	no	—
153	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic
154	C₇₃	C₂×#2960	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	73	2	no	—
156	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
158	#533	—	73	8	no	—
159	#533	—	73	8	no	—
160	C₂×C₄⋊Q₈	—	73	2	no	—
161	C₂×#125	—	73	2	no	—
162	C₂×C₂²⋊Q₈	—	73	4	no	—
163	C₂×C₄×Q₈	—	73	4	no	—
164	C₂×#117	—	73	2	no	—
165	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
166	#446	—	73	32	no	—
167	C₈×Q₈	—	73	32	no	—
168	#424	—	73	16	no	—
169	#423	—	73	8	no	—
170	C₂×C₄⋊C₈	—	73	8	no	—
171	C₂×C₄⋊C₈	—	73	4	no	—
172	#405	—	73	16	no	—
173	C₂×#97	—	73	4	no	—
174	C₂×C₄×C₈	—	73	4	no	—
175	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₇₃	1	8	yes	—
177	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
178	C₂²×Q₈	—	73	4	no	—
179	C₂³×C₄	C₄×D₇₃	1	2	yes	—
180	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
181	C₂×M₁₆	—	73	8	no	—
182	C₂×M₁₆	—	146	8	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	73	4	no	—
185	C₂²×C₈	Dic₁₄₆	1	8	yes	complemented
186	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
188	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
190	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	73	16	no	—
192	#125	—	73	16	no	—
193	C₂²⋊Q₈	—	73	32	no	—
194	C₄×Q₈	—	73	16	no	—
195	C₄×Q₈	—	73	16	no	—
196	#117	—	73	8	no	—
197	#117	—	73	8	no	—
198	C₂×C₄⋊C₄	—	73	4	no	—
199	C₂×C₄⋊C₄	—	73	8	no	—
200	C₂×C₄⋊C₄	—	73	4	no	—
201	C₂×C₂²⋊C₄	—	73	4	no	—
202	C₂×C₄²	—	73	4	no	—
203	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
205	C₂×C₄²	C₄×D₇₃	1	4	yes	—
206	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₇₃	1	4	yes	—
208	C₄⋊C₈	—	73	32	no	—
209	C₄⋊C₈	—	73	16	no	—
210	#97	—	73	16	no	—
211	C₄×C₈	—	73	16	no	—
212	C₄×C₈	C₂²×D₇₃	1	8	yes	—
213	C₄×C₈	C₂²×D₇₃	1	8	yes	—
214	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
215	C₂×Q₈	—	73	16	no	—
216	C₂×Q₈	—	146	16	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	73	8	no	—
219	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₇₃	1	8	yes	—
221	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
224	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
225	C₂²×C₄	#11924028	1	4	yes	—
226	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
228	M₁₆	—	146	16	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	73	8	no	—
232	C₂×C₈	—	146	8	no	—
233	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
234	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
235	C₂×C₈	C₂×Dic₁₄₆	1	16	yes	complemented
236	C₂×C₈	C₂³×D₇₃	1	4	yes	—
237	C₂×C₈	C₂³×D₇₃	1	2	yes	—
238	C₂×C₈	C₂³×D₇₃	1	2	yes	—
239	C₄⋊C₄	—	73	32	no	—
240	C₄⋊C₄	—	73	16	no	—
241	C₄⋊C₄	—	73	16	no	—
242	C₂²⋊C₄	—	73	16	no	—
243	C₄²	—	73	8	no	—
244	C₄²	—	73	8	no	—
245	C₄²	C₂×C₄×D₇₃	1	4	yes	—
246	C₄²	C₂×C₄×D₇₃	1	4	yes	—
247	C₄²	C₂×C₄×D₇₃	1	4	yes	—
248	C₄²	C₂³×D₇₃	1	4	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₄×Dic₁₄₆	1	4	yes	—
251	C₂³	C₂²×C₄×D₇₃	1	2	yes	—
252	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
253	Q₈	—	146	32	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	73	16	no	—
257	C₂×C₄	—	146	16	no	—
258	C₂×C₄	C₄×Dic₁₄₆	1	8	yes	—
259	C₂×C₄	C₄×Dic₁₄₆	1	4	yes	—
260	C₂×C₄	#14556597	1	8	yes	—
261	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
262	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
265	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
266	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
267	C₂×C₄	C₂×#11924028	1	2	yes	—
268	C₂×C₄	C₂×#11924028	1	4	yes	—
269	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	146	8	no	—
272	C₈	C₂²×Dic₁₄₆	1	4	yes	—
273	C₈	C₂²×Dic₁₄₆	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×C₄×Dic₁₄₆	1	4	yes	—
277	C₂²	C₂×C₄×Dic₁₄₆	1	2	yes	—
278	C₂²	C₂×#14556597	1	4	yes	—
279	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	146	16	no	—
282	C₄	C₂×C₄×Dic₁₄₆	1	2	yes	—
283	C₄	C₂×C₄×Dic₁₄₆	1	4	yes	—
284	C₄	C₂×#14556597	1	8	yes	—
285	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
286	C₄	C₂²×#11924028	1	1	yes	characteristic
287	C₂	—	2	4	no	—
288	C₂	#22618865	1	4	yes	direct product factor
289	C₂	C₂²×C₄×Dic₁₄₆	1	1	yes	characteristic
290	C₂	C₂²×#14556597	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	C₂×#2960
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22618865	C₂	direct (N × H)
3. C₂×#17988717	C₂	semidirect
4. C₂×C₈×Dic₁₄₆	C₂	semidirect
17. #17988717	C₂²	semidirect
18. C₈×Dic₁₄₆	C₂²	semidirect
55. C₂²×C₈×C₇₃	Q₈	semidirect
95. C₂×C₈×C₇₃	C₂×Q₈	semidirect
143. C₂×C₇₃	#2960	semidirect
154. C₇₃	C₂×#2960	semidirect
185. C₂²×C₈	Dic₁₄₆	semidirect
235. C₂×C₈	C₂×Dic₁₄₆	semidirect
288. C₂	#22618865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989144	2
#22618865	2
C₂×#17988717	2
C₂×C₈×Dic₁₄₆	2
C₂×#17988613	2
C₂×#17988611	2
C₂×#17988542	2
C₂²×C₄×C₈×C₇₃	2
C₂×#2960	73

Subgroups — C2×#22618865 #33950349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22618865 #33950349