Subgroups — C₂×#22624986 #34012153

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556665	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22624986	C₂	1	4	yes	maximal, direct product factor
4	C₂×#17988545	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	—	2	1	no	—
6	C₂×#14556665	—	2	2	no	—
7	C₂×#14556665	C₄	1	4	yes	complemented
8	C₂×#14556665	C₂²	1	2	yes	—
9	C₂²×#11924031	—	2	1	no	—
10	C₂²×#11924031	—	2	1	no	—
11	C₂²×#11924031	—	2	2	no	—
12	C₂²×#11924031	—	2	1	no	—
13	C₂²×#11924031	C₄	1	4	yes	complemented
14	C₂²×#11924031	C₂²	1	2	yes	—
15	C₂³×#10596831	C₂²	1	1	yes	characteristic
16	C₂×#14556550	C₄	1	1	yes	characteristic
17	#17988545	—	2	1	no	—
18	#17988545	—	2	1	no	—
19	#17988545	C₄	1	2	yes	—
20	#17988545	C₂²	1	2	yes	—
21	C₂×#14556501	—	2	1	no	—
22	C₂×C₈×Dic₇₃	—	2	1	no	—
23	C₂²×M₁₆×C₇₃	C₄	1	1	yes	characteristic
24	#14556665	—	2	4	no	—
25	#14556665	—	2	2	no	—
26	#14556665	C₂×C₄	1	4	yes	complemented
27	C₂×#11924031	—	2	4	no	—
28	C₂×#11924031	—	2	4	no	—
29	C₂×#11924031	—	2	4	no	—
30	C₂×#11924031	—	2	2	no	—
31	C₂×#11924031	—	2	2	no	—
32	C₂×#11924031	—	2	2	no	—
33	C₂×#11924031	C₂×C₄	1	4	yes	complemented
34	C₂²×#10596831	C₂×C₄	1	2	yes	—
35	C₂²×#10596831	C₂×C₄	1	2	yes	—
36	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₂²×#10596831	D₄	1	1	yes	characteristic
38	C₂²×#10596831	D₄	1	1	yes	characteristic
39	C₂²×#10596831	D₄	1	1	yes	characteristic
40	C₂²×#10596831	D₄	1	1	yes	characteristic
41	C₂²×#10596831	D₄	1	2	yes	—
42	C₂²×#10596831	Q₈	1	2	yes	—
43	C₂²×#10596831	C₂³	1	1	yes	characteristic
44	#14556550	—	2	1	no	—
45	#14556550	—	2	1	no	—
46	#14556550	C₈	1	2	yes	—
47	#14556550	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₇₃	C₈	1	2	yes	—
49	C₂×C₂²⋊Dic₇₃	—	2	1	no	—
50	C₂×C₄×Dic₇₃	—	2	1	no	—
51	#14556501	—	2	2	no	—
52	#14556501	—	2	2	no	—
53	C₈×Dic₇₃	—	2	2	no	—
54	C₈×Dic₇₃	—	2	2	no	—
55	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₃	—	2	1	no	—
57	C₂×M₁₆×C₇₃	—	2	2	no	—
58	C₂×M₁₆×C₇₃	C₈	1	4	yes	—
59	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₇₃	—	2	1	no	—
61	#11924031	—	2	4	no	—
62	#11924031	—	2	4	no	—
63	C₂²×Dic₇₃	—	2	2	no	—
64	C₂×#10596831	C₄²	1	2	yes	—
65	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
68	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
69	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
70	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
71	C₂×#10596831	C₄⋊C₄	1	2	yes	—
72	C₂×#10596831	C₄⋊C₄	1	2	yes	—
73	C₂×#10596831	C₄⋊C₄	1	2	yes	—
74	C₂×#10596831	C₄⋊C₄	1	2	yes	—
75	C₂×#10596831	C₄⋊C₄	1	2	yes	—
76	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
77	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
78	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
79	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
80	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
81	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
84	C₂²⋊Dic₇₃	—	2	2	no	—
85	C₂²⋊Dic₇₃	—	2	2	no	—
86	C₄×Dic₇₃	—	2	2	no	—
87	C₄×Dic₇₃	—	2	2	no	—
88	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
97	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	M₁₆×C₇₃	—	2	4	no	—
100	M₁₆×C₇₃	—	2	2	no	—
101	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
102	C₂×C₈×C₇₃	—	2	4	no	—
103	C₂×C₈×C₇₃	—	2	2	no	—
104	C₂³×C₇₃	—	2	4	no	—
105	C₂³×C₇₃	#97	1	2	yes	—
106	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
109	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#95	1	2	yes	—
111	C₂×C₄×C₇₃	#95	1	2	yes	—
112	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	#97	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×Dic₇₃	—	2	4	no	—
129	C₂×Dic₇₃	—	4	4	no	—
130	C₈×C₇₃	—	2	4	no	—
131	#10596831	#95	1	2	yes	—
132	#10596831	#95	1	2	yes	—
133	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
136	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	4	no	—
140	C₂²×C₇₃	—	2	4	no	—
141	C₂²×C₇₃	#336	1	2	yes	—
142	C₂²×C₇₃	#336	1	2	yes	—
143	C₂²×C₇₃	#350	1	4	yes	—
144	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₇₃	#336	1	2	yes	—
148	C₄×C₇₃	#336	1	2	yes	—
149	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
150	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
151	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
153	Dic₇₃	—	4	4	no	—
154	C₂×#1379	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	—	2	1	no	—
157	C₂×C₇₃	—	2	1	no	—
158	C₂×C₇₃	#1379	1	4	yes	complemented
159	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#350	1	2	yes	—
161	C₂²×M₃₂	—	73	1	no	—
162	C₂³×C₁₆	—	73	1	no	—
163	C₂×#405	—	73	1	no	—
164	#1379	—	73	4	no	—
165	C₇₃	C₂×#1379	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	#64408	1	1	yes	characteristic
167	C₂³×C₈	—	73	1	no	—
168	C₂×#117	—	73	1	no	—
169	C₂×M₃₂	—	73	4	no	—
170	C₂×M₃₂	—	73	2	no	—
171	C₂×M₃₂	—	146	1	no	—
172	C₂×M₃₂	—	146	2	no	—
173	C₂²×C₁₆	—	73	4	no	—
174	C₂²×C₁₆	—	73	2	no	—
175	C₂²×C₁₆	—	146	1	no	—
176	C₂²×C₁₆	—	146	1	no	—
177	C₂²×C₁₆	—	146	2	no	—
178	C₂²×C₁₆	—	146	1	no	—
179	#405	—	73	2	no	—
180	#405	—	73	2	no	—
181	#405	—	146	1	no	—
182	#405	—	146	1	no	—
183	C₂×#97	—	146	1	no	—
184	C₂×C₄×C₈	—	146	1	no	—
185	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	#10596833	1	4	yes	—
189	C₂×M₁₆	C₂×#64408	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	73	2	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	73	1	no	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	1	no	—
200	C₂²×C₈	—	73	1	no	—
201	#117	—	73	2	no	—
202	#117	—	73	2	no	—
203	#117	—	146	1	no	—
204	#117	—	146	1	no	—
205	C₂×C₄⋊C₄	—	73	2	no	—
206	C₂×C₂²⋊C₄	—	146	1	no	—
207	C₂×C₄²	—	146	1	no	—
208	M₃₂	—	73	4	no	—
209	M₃₂	—	146	4	no	—
210	M₃₂	—	146	2	no	—
211	C₂×C₁₆	—	73	4	no	—
212	C₂×C₁₆	—	146	4	no	—
213	C₂×C₁₆	—	146	4	no	—
214	C₂×C₁₆	—	146	2	no	—
215	C₂×C₁₆	—	146	2	no	—
216	C₂×C₁₆	—	146	4	no	—
217	C₂×C₁₆	—	146	2	no	—
218	#97	—	146	2	no	—
219	#97	—	146	2	no	—
220	C₄×C₈	—	146	2	no	—
221	C₄×C₈	—	146	2	no	—
222	C₂⁴	#11924038	1	1	yes	characteristic
223	C₂²×C₄	—	146	2	no	—
224	C₂²×C₄	C₄×#64408	1	2	yes	—
225	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊#64408	1	2	yes	—
228	C₂²×C₄	C₂×#10596833	1	2	yes	—
229	C₂²×C₄	#11924036	1	1	yes	characteristic
230	C₂²×C₄	#11924036	1	1	yes	characteristic
231	C₂²×C₄	#11924037	1	2	yes	—
232	C₂²×C₄	#11924038	1	1	yes	characteristic
233	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	C₂×#10596833	1	4	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	2	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	73	2	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	2	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	73	1	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	1	no	—
254	C₂×C₈	—	73	2	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	1	no	—
257	C₄⋊C₄	—	73	4	no	—
258	C₄⋊C₄	—	146	2	no	—
259	C₂²⋊C₄	—	146	2	no	—
260	C₂²⋊C₄	—	146	2	no	—
261	C₄²	—	146	2	no	—
262	C₄²	—	146	2	no	—
263	C₁₆	—	146	4	no	—
264	C₁₆	—	146	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	#14556646	1	2	yes	—
267	C₂³	#14556648	1	1	yes	characteristic
268	C₂³	#14556648	1	1	yes	characteristic
269	C₂³	C₄⋊#10596833	1	2	yes	—
270	C₂³	C₂×#11924038	1	1	yes	characteristic
271	C₂×C₄	—	292	4	no	—
272	C₂×C₄	—	146	4	no	—
273	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
274	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
275	C₂×C₄	C₄×#10596833	1	2	yes	—
276	C₂×C₄	C₄×#10596833	1	2	yes	—
277	C₂×C₄	#14556646	1	2	yes	—
278	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
279	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
280	C₂×C₄	#14556648	1	1	yes	characteristic
281	C₂×C₄	#14556648	1	1	yes	characteristic
282	C₂×C₄	C₄⋊#10596833	1	2	yes	—
283	C₂×C₄	#14556658	1	2	yes	—
284	C₂×C₄	#14556658	1	2	yes	—
285	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
288	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
289	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
290	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	73	1	no	—
293	C₈	—	73	1	no	—
294	C₈	—	73	2	no	—
295	C₈	—	73	2	no	—
296	C₈	—	73	1	no	—
297	C₈	—	73	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#17989453	1	2	yes	—
303	C₂²	#17989453	1	2	yes	—
304	C₂²	#17989486	1	4	yes	—
305	C₂²	C₂×#14556646	1	1	yes	characteristic
306	C₂²	C₂×#14556648	1	1	yes	characteristic
307	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
308	C₄	—	292	4	no	—
309	C₄	#17989453	1	2	yes	—
310	C₄	#17989453	1	2	yes	—
311	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
312	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
313	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
314	C₄	C₂×#14556658	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	#22624986	1	4	yes	direct product factor
319	C₂	C₂×#17989453	1	1	yes	characteristic
320	C₂	C₂×#17989486	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂²×M₁₆×C₇₃	C₄
socle	C₂³×C₇₃	C₂²⋊C₈
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂²×M₁₆	#64408
hypocenter	C₇₃	C₂×#1379
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22624986	C₂	direct (N × H)
7. C₂×#14556665	C₄	semidirect
13. C₂²×#11924031	C₄	semidirect
26. #14556665	C₂×C₄	semidirect
33. C₂×#11924031	C₂×C₄	semidirect
158. C₂×C₇₃	#1379	semidirect
165. C₇₃	C₂×#1379	semidirect
318. C₂	#22624986	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×#14556665	2
C₂³×#11924031	2
#22624986	2
C₂×#17988545	2
C₂×#1379	73

Subgroups — C2×#22624986 #34012153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624986 #34012153