Subgroups — C₂×#22626139 #34029787

Order 18688.

854 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989550	C₂	1	1	yes	maximal, characteristic
3	#22626139	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17989474	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#64408	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₇₃⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×#14556670	C₄	1	1	yes	characteristic
9	C₂×#14556666	C₄	1	2	yes	complemented
10	#17989643	C₄	1	4	yes	complemented
11	#17989643	C₂²	1	4	yes	—
12	C₂×#14556665	C₄	1	2	yes	complemented
13	C₂×#14556664	C₂²	1	2	yes	—
14	#17989550	C₄	1	4	yes	—
15	#17989550	C₂²	1	4	yes	—
16	C₂×#14556645	C₄	1	2	yes	—
17	C₂×C₈×#64408	C₂²	1	2	yes	—
18	#17989474	C₂²	1	8	yes	complemented
19	C₁₆×#64408	C₂²	1	8	yes	complemented
20	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
21	C₂²×#11924003	C₄	1	1	yes	characteristic, complemented
22	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
24	C₂×C₁₆×D₇₃	C₄	1	4	yes	complemented
25	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₇₃	C₂²	1	4	yes	—
28	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
29	C₂×#11924038	C₈	1	2	yes	—
30	D₇₃⋊M₁₆	C₈	1	4	yes	—
31	D₇₃⋊M₁₆	C₂×C₄	1	4	yes	—
32	C₂×#11924037	C₈	1	2	yes	—
33	C₂×#11924036	C₂×C₄	1	2	yes	—
34	#14556670	C₈	1	4	yes	—
35	#14556670	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊#64408	C₈	1	2	yes	—
37	C₂×C₂²⋊#64408	C₈	1	2	yes	—
38	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
39	#14556666	C₂×C₄	1	8	yes	complemented
40	#14556665	C₂×C₄	1	8	yes	complemented
41	#14556664	C₂×C₄	1	4	yes	—
42	#14556664	C₂³	1	4	yes	—
43	C₂×#11924031	C₂×C₄	1	4	yes	—
44	#14556645	C₂×C₄	1	8	yes	—
45	C₈×#64408	C₂×C₄	1	4	yes	—
46	C₈×#64408	C₂³	1	4	yes	—
47	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×D₇₃	C₂³	1	2	yes	—
57	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
58	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
59	C₂×#11924003	C₂×C₄	1	4	yes	complemented
60	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
61	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
62	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
63	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
64	C₁₆×D₇₃	C₂³	1	2	yes	—
65	C₁₆×D₇₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
70	C₂²×#64408	C₂×C₈	1	4	yes	—
71	#11924038	C₂×C₈	1	8	yes	—
72	#11924037	C₂×C₈	1	8	yes	—
73	#11924036	C₂×C₈	1	4	yes	—
74	#11924036	C₂²×C₄	1	4	yes	—
75	C₂×#10596833	C₂×C₈	1	4	yes	—
76	C₄⋊#64408	C₂×C₈	1	8	yes	—
77	C₂²⋊#64408	C₂×C₈	1	8	yes	—
78	C₄×#64408	C₂×C₈	1	4	yes	—
79	C₄×#64408	C₂²×C₄	1	4	yes	—
80	#11924031	C₂²×C₄	1	8	yes	—
81	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₄²	1	2	yes	—
84	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
91	C₂×#10596831	C₄²	1	1	yes	characteristic
92	C₂×#10596831	C₄²	1	2	yes	—
93	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
94	C₂×#10596831	C₂²×C₄	1	2	yes	—
95	C₈×D₇₃	C₄²	1	2	yes	—
96	C₈×D₇₃	C₄²	1	2	yes	—
97	C₈×D₇₃	C₄²	1	2	yes	—
98	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₈×D₇₃	C₂²×C₄	1	2	yes	—
100	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₇₃	C₂²×C₄	1	2	yes	—
102	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₇₃	C₂²×C₄	1	2	yes	—
104	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
105	#11924003	C₂²×C₄	1	2	yes	—
106	#11924003	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₄²	1	2	yes	—
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
112	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
113	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
115	C₂²×D₇₃	—	2	4	no	—
116	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
117	C₂²×D₇₃	C₄×C₈	1	2	yes	—
118	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
119	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×#64408	—	2	8	no	—
121	C₂×#64408	C₂²×C₈	1	8	yes	—
122	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
123	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
127	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
128	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
130	C₄×D₇₃	C₄×C₈	1	2	yes	—
131	C₄×D₇₃	C₄×C₈	1	2	yes	—
132	C₄×D₇₃	C₄×C₈	1	2	yes	—
133	C₄×D₇₃	C₂×C₄²	1	2	yes	—
134	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
135	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
136	C₄×D₇₃	C₂²×C₈	1	2	yes	—
137	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₇₃	C₂²×C₈	1	2	yes	—
139	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
140	#10596833	C₂²×C₈	1	8	yes	—
141	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
142	C₈×C₇₃	C₂×C₄²	1	2	yes	—
143	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
144	#10596831	C₂×C₄²	1	1	yes	characteristic
145	#10596831	C₂×C₄²	1	2	yes	—
146	#10596831	C₂³×C₄	1	1	yes	characteristic
147	C₂²×C₇₃	—	2	2	no	—
148	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
149	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
150	C₂×D₇₃	—	2	2	no	—
151	C₂×D₇₃	—	2	4	no	—
152	C₂×D₇₃	—	2	4	no	—
153	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
154	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂×D₇₃	#504	1	4	yes	—
156	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
157	#64408	—	2	8	no	—
158	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
159	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
160	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
161	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
162	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
163	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
164	C₂×#2108	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	2	2	no	—
166	C₂×C₇₃	#2108	1	2	yes	complemented
167	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
168	D₇₃	—	2	2	no	—
169	D₇₃	#2108	1	2	yes	complemented
170	D₇₃	C₂×#504	1	2	yes	—
171	C₂²×M₃₂	—	73	1	no	—
172	C₂³×C₁₆	—	73	1	no	—
173	C₂×#405	—	73	1	no	—
174	#2108	—	73	8	no	—
175	C₂×#346	—	73	2	no	—
176	C₂×C₄×C₁₆	—	73	2	no	—
177	C₇₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	73	1	no	—
179	C₂³×C₈	—	73	1	no	—
180	C₂×#117	—	73	1	no	—
181	C₂×M₃₂	—	73	2	no	—
182	C₂×M₃₂	—	73	4	no	—
183	C₂×M₃₂	—	73	4	no	—
184	C₂×M₃₂	—	73	2	no	—
185	C₂²×C₁₆	—	73	2	no	—
186	C₂²×C₁₆	—	73	1	no	—
187	C₂²×C₁₆	—	73	1	no	—
188	C₂²×C₁₆	—	73	1	no	—
189	C₂²×C₁₆	—	73	1	no	—
190	C₂²×C₁₆	—	73	1	no	—
191	C₂²×C₁₆	—	73	4	no	—
192	C₂²×C₁₆	—	73	4	no	—
193	C₂²×C₁₆	#64408	1	1	yes	characteristic, complemented
194	#405	—	73	4	no	—
195	#405	—	73	4	no	—
196	C₂×#97	—	73	2	no	—
197	C₂×C₄×C₈	—	73	2	no	—
198	#346	—	73	8	no	—
199	C₄×C₁₆	—	73	8	no	—
200	C₂³×C₄	—	73	1	no	—
201	C₂×M₁₆	—	73	2	no	—
202	C₂×M₁₆	—	73	4	no	—
203	C₂×M₁₆	—	73	4	no	—
204	C₂×M₁₆	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	73	1	no	—
212	C₂²×C₈	—	73	2	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	2	no	—
215	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
216	#117	—	73	4	no	—
217	#117	—	73	4	no	—
218	C₂×C₄⋊C₄	—	73	2	no	—
219	C₂×C₂²⋊C₄	—	73	2	no	—
220	C₂×C₄²	—	73	2	no	—
221	M₃₂	—	73	8	no	—
222	M₃₂	—	73	8	no	—
223	C₂×C₁₆	—	73	4	no	—
224	C₂×C₁₆	—	73	4	no	—
225	C₂×C₁₆	—	73	4	no	—
226	C₂×C₁₆	—	73	1	no	—
227	C₂×C₁₆	—	73	1	no	—
228	C₂×C₁₆	—	73	4	no	—
229	C₂×C₁₆	—	73	2	no	—
230	C₂×C₁₆	—	73	2	no	—
231	C₂×C₁₆	—	73	2	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	—	73	4	no	—
234	C₂×C₁₆	—	73	4	no	—
235	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
237	C₂×C₁₆	C₂×#64408	1	4	yes	complemented
238	#97	—	73	8	no	—
239	C₄×C₈	—	73	4	no	—
240	C₄×C₈	—	73	4	no	—
241	C₂⁴	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	2	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	2	no	—
247	C₂²×C₄	—	73	2	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	2	no	—
251	C₂²×C₄	—	73	4	no	—
252	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
253	M₁₆	—	73	8	no	—
254	M₁₆	—	73	8	no	—
255	C₂×C₈	—	73	4	no	—
256	C₂×C₈	—	73	4	no	—
257	C₂×C₈	—	73	4	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	2	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	2	no	—
272	C₂×C₈	C₄×#64408	1	1	yes	characteristic
273	C₂×C₈	C₄×#64408	1	2	yes	—
274	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
275	C₂×C₈	C₂²×#64408	1	2	yes	—
276	C₄⋊C₄	—	73	8	no	—
277	C₂²⋊C₄	—	73	8	no	—
278	C₄²	—	73	4	no	—
279	C₄²	—	73	4	no	—
280	C₁₆	—	73	8	no	—
281	C₁₆	—	73	2	no	—
282	C₁₆	—	73	2	no	—
283	C₁₆	C₂²×#64408	1	2	yes	—
284	C₁₆	C₂²×#64408	1	2	yes	—
285	C₂³	—	73	1	no	—
286	C₂³	—	73	2	no	—
287	C₂³	—	73	2	no	—
288	C₂³	—	73	1	no	—
289	C₂³	—	146	4	no	—
290	C₂³	C₈×#64408	1	1	yes	characteristic
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	1	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	2	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	2	no	—
305	C₂×C₄	—	73	8	no	—
306	C₂×C₄	—	146	8	no	—
307	C₂×C₄	C₈×#64408	1	2	yes	—
308	C₂×C₄	C₈×#64408	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
310	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
311	C₈	—	73	8	no	—
312	C₈	—	73	1	no	—
313	C₈	—	73	1	no	—
314	C₈	—	73	2	no	—
315	C₈	C₂×C₄×#64408	1	1	yes	characteristic
316	C₈	C₂×C₄×#64408	1	2	yes	—
317	C₈	C₂³×#64408	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	73	4	no	—
320	C₂²	—	73	1	no	—
321	C₂²	—	73	1	no	—
322	C₂²	—	73	2	no	—
323	C₂²	—	146	4	no	—
324	C₂²	—	146	4	no	—
325	C₂²	—	146	2	no	—
326	C₂²	C₂×C₈×#64408	1	2	yes	—
327	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
328	C₄	—	73	1	no	—
329	C₄	—	73	1	no	—
330	C₄	—	73	2	no	—
331	C₄	—	146	8	no	—
332	C₄	C₂×C₈×#64408	1	1	yes	characteristic
333	C₄	C₂×C₈×#64408	1	2	yes	—
334	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	73	2	no	—
337	C₂	—	73	2	no	—
338	C₂	—	146	2	no	—
339	C₂	#22626139	1	2	yes	direct product factor
340	C₂	C₂²×C₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₈	C₂³×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	C₂×#2108
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22626139	C₂	direct (N × H)
4. C₂×#17989474	C₂	semidirect
5. C₂×C₁₆×#64408	C₂	semidirect
9. C₂×#14556666	C₄	semidirect
10. #17989643	C₄	semidirect
12. C₂×#14556665	C₄	semidirect
18. #17989474	C₂²	semidirect
19. C₁₆×#64408	C₂²	semidirect
21. C₂²×#11924003	C₄	semidirect
22. C₂×C₁₆×D₇₃	C₄	semidirect
23. C₂×C₁₆×D₇₃	C₄	semidirect
24. C₂×C₁₆×D₇₃	C₄	semidirect
28. C₂²×C₁₆×C₇₃	C₄	semidirect
39. #14556666	C₂×C₄	semidirect
40. #14556665	C₂×C₄	semidirect
59. C₂×#11924003	C₂×C₄	semidirect
62. C₁₆×D₇₃	C₂×C₄	semidirect
63. C₁₆×D₇₃	C₂×C₄	semidirect
69. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
166. C₂×C₇₃	#2108	semidirect
169. D₇₃	#2108	semidirect
177. C₇₃	C₂×#2108	semidirect
193. C₂²×C₁₆	#64408	semidirect
237. C₂×C₁₆	C₂×#64408	semidirect
339. C₂	#22626139	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×#17989643	2
C₂×#17989550	2
#22626139	2
C₂×#17989474	2
C₂×C₁₆×#64408	2
C₂²×C₁₆×D₇₃	2
C₂×#2108	73

Subgroups — C2×#22626139 #34029787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626139 #34029787