Subgroups — C₂×#22611704 #33238996

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556527	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988625	C₂	1	1	yes	maximal, characteristic
3	#22611704	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988410	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	—	2	1	no	—
7	C₂×#14556550	—	2	1	no	—
8	C₂×#14556527	—	2	1	no	—
9	C₂×#14556527	—	2	2	no	—
10	C₂×#14556527	C₂²	1	2	yes	—
11	C₂×#14556527	C₂²	1	4	yes	—
12	#17988625	—	2	2	no	—
13	#17988625	C₂²	1	4	yes	—
14	C₂×#14556525	—	2	1	no	—
15	C₂×#14556523	C₂²	1	2	yes	—
16	#17988410	C₂²	1	16	yes	complemented
17	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₇₃	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₇₃	C₄	1	4	yes	complemented
20	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
21	C₂×M₃₂×C₇₃	C₂²	1	4	yes	—
22	C₂²×C₁₆×C₇₃	C₄	1	2	yes	complemented
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	—	2	4	no	—
25	C₂×#11924012	—	4	1	no	—
26	C₂×#11924012	—	4	2	no	—
27	C₂²×#10596831	—	4	1	no	—
28	#14556550	—	2	4	no	—
29	#14556550	—	4	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
32	C₂×C₄×Dic₇₃	—	4	1	no	—
33	#14556527	—	2	2	no	—
34	#14556527	—	2	4	no	—
35	#14556527	C₂³	1	4	yes	—
36	#14556525	—	2	4	no	—
37	#14556523	—	2	2	no	—
38	#14556523	C₂³	1	4	yes	—
39	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	D₄	1	2	yes	—
45	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
47	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
50	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
51	M₃₂×C₇₃	C₂×C₄	1	8	yes	complemented
52	M₃₂×C₇₃	C₂³	1	4	yes	—
53	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
54	C₂×C₁₆×C₇₃	C₂×C₄	1	8	yes	complemented
55	C₂²×Dic₇₃	—	4	2	no	—
56	#11924012	—	2	4	no	—
57	#11924012	—	4	2	no	—
58	#11924012	—	4	4	no	—
59	C₂×#10596831	—	4	4	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₄⋊Dic₇₃	—	2	4	no	—
62	C₄⋊Dic₇₃	—	4	2	no	—
63	C₂²⋊Dic₇₃	—	4	4	no	—
64	C₄×Dic₇₃	—	4	4	no	—
65	C₂⁴×C₇₃	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
94	C₁₆×C₇₃	C₂²×C₄	1	8	yes	—
95	C₂³×C₇₃	—	4	2	no	—
96	C₂³×C₇₃	#106	1	1	yes	characteristic
97	C₂³×C₇₃	#106	1	1	yes	characteristic
98	C₂³×C₇₃	#106	1	2	yes	—
99	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂³×C₇₃	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₇₃	#106	1	2	yes	—
102	C₂×C₄×C₇₃	#106	1	2	yes	—
103	C₂×C₄×C₇₃	#106	1	2	yes	—
104	C₂×C₄×C₇₃	#106	1	2	yes	—
105	C₂×C₄×C₇₃	#106	1	2	yes	—
106	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
113	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₇₃	—	4	4	no	—
120	C₂×Dic₇₃	—	8	4	no	—
121	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
127	#10596831	—	4	4	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	4	2	no	—
132	C₂²×C₇₃	—	4	2	no	—
133	C₂²×C₇₃	C₂×#106	1	2	yes	—
134	C₂²×C₇₃	C₂×#106	1	2	yes	—
135	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
137	C₂²×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
139	C₄×C₇₃	C₂×#106	1	2	yes	—
140	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
141	C₄×C₇₃	C₂×#106	1	2	yes	—
142	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
144	Dic₇₃	—	8	4	no	—
145	C₂×#2155	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	1	no	—
147	C₂×C₇₃	—	2	1	no	—
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	#2155	1	2	yes	complemented
151	C₂×C₇₃	C₂²×#106	1	1	yes	characteristic
152	C₂²×M₃₂	D₇₃	1	1	yes	characteristic
153	C₂²×#108	—	73	1	no	—
154	C₂×#427	—	73	1	no	—
155	#2155	—	73	8	no	—
156	C₂×#365	—	73	4	no	—
157	C₇₃	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	146	1	no	—
159	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
160	C₂×#117	—	146	1	no	—
161	C₂×M₃₂	Dic₇₃	1	2	yes	complemented
162	C₂×M₃₂	Dic₇₃	1	4	yes	complemented
163	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
164	C₂×M₃₂	C₂×D₇₃	1	4	yes	—
165	C₂²×C₁₆	Dic₇₃	1	2	yes	complemented
166	C₂×#108	—	73	2	no	—
167	C₂×#108	—	73	4	no	—
168	C₂×#108	—	146	1	no	—
169	C₂×#108	—	146	2	no	—
170	#427	—	73	4	no	—
171	#427	—	146	2	no	—
172	C₂×#107	—	146	1	no	—
173	C₂×#106	—	73	2	no	—
174	#365	—	73	16	no	—
175	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
176	C₂×M₁₆	—	292	1	no	—
177	C₂×M₁₆	—	292	2	no	—
178	C₂×M₁₆	—	146	2	no	—
179	C₂×M₁₆	—	146	4	no	—
180	C₂²×C₈	—	292	1	no	—
181	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
182	C₂²×C₈	Dic₁₄₆	1	2	yes	—
183	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
184	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
185	C₂²×C₈	D₂₉₂	1	2	yes	—
186	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
189	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
191	#117	—	292	2	no	—
192	#117	—	146	4	no	—
193	C₂×C₄⋊C₄	—	146	2	no	—
194	C₂×C₂²⋊C₄	—	292	1	no	—
195	C₂×C₄²	—	292	1	no	—
196	M₃₂	C₂×Dic₇₃	1	4	yes	—
197	M₃₂	C₂×Dic₇₃	1	8	yes	complemented
198	M₃₂	C₂²×D₇₃	1	4	yes	—
199	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
200	C₂×C₁₆	C₂×Dic₇₃	1	8	yes	complemented
201	#108	—	73	4	no	—
202	#108	—	146	2	no	—
203	#108	—	146	4	no	—
204	#107	—	146	4	no	—
205	#106	—	73	4	no	—
206	#106	—	146	2	no	—
207	C₂⁴	#11924017	1	1	yes	characteristic
208	C₂²×C₄	—	292	2	no	—
209	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
212	C₂²×C₄	#11924017	1	2	yes	—
213	C₂²×C₄	#11924017	1	1	yes	characteristic
214	C₂²×C₄	#11924017	1	1	yes	characteristic
215	C₂²×C₄	#11924017	1	2	yes	—
216	C₂²×C₄	#11924017	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
219	M₁₆	—	292	2	no	—
220	M₁₆	—	292	4	no	—
221	M₁₆	—	146	4	no	—
222	C₂×C₈	—	292	2	no	—
223	C₂×C₈	—	292	4	no	—
224	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
225	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
226	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
227	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
228	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
229	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
231	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
233	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
235	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
236	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
237	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
238	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
239	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
240	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
241	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
242	C₄⋊C₄	—	292	2	no	—
243	C₄⋊C₄	—	146	4	no	—
244	C₂²⋊C₄	—	292	4	no	—
245	C₄²	—	292	4	no	—
246	C₁₆	C₂²×Dic₇₃	1	8	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#14556523	1	1	yes	characteristic
249	C₂³	#14556523	1	1	yes	characteristic
250	C₂³	#14556523	1	2	yes	—
251	C₂³	C₂×#11924017	1	1	yes	characteristic
252	C₂³	C₂×#11924017	1	2	yes	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	584	4	no	—
255	C₂×C₄	#14556523	1	2	yes	—
256	C₂×C₄	#14556523	1	2	yes	—
257	C₂×C₄	#14556523	1	1	yes	characteristic
258	C₂×C₄	#14556523	1	2	yes	—
259	C₂×C₄	#14556523	1	2	yes	—
260	C₂×C₄	#14556523	1	2	yes	—
261	C₂×C₄	#14556523	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
266	C₂×C₄	C₂×#11924017	1	2	yes	—
267	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
269	C₂×C₄	C₂×#11924017	1	2	yes	—
270	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
271	C₂×C₄	C₂×#11924017	1	2	yes	—
272	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
273	C₈	—	292	4	no	—
274	C₈	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
275	C₈	C₂×C₄⋊Dic₇₃	1	2	yes	—
276	C₈	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
277	C₈	C₂×C₄⋊Dic₇₃	1	2	yes	—
278	C₈	C₂³×Dic₇₃	1	1	yes	characteristic
279	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#14556523	1	2	yes	—
286	C₂²	C₂×#14556523	1	2	yes	—
287	C₂²	C₂×#14556523	1	1	yes	characteristic
288	C₂²	C₂×#14556523	1	1	yes	characteristic
289	C₂²	C₂²×#11924017	1	1	yes	characteristic
290	C₄	—	584	4	no	—
291	C₄	C₂×#14556523	1	2	yes	—
292	C₄	C₂×#14556523	1	1	yes	characteristic
293	C₄	C₂×#14556523	1	1	yes	characteristic
294	C₄	C₂×#14556523	1	2	yes	—
295	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
296	C₄	C₂²×#11924017	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#22611704	1	2	yes	direct product factor
302	C₂	C₂²×#14556523	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556523
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂²×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂⁴×C₇₃	SD₁₆
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂²×M₃₂	D₇₃
hypocenter	C₇₃	C₂×#2155
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22611704	C₂	direct (N × H)
4. C₂×#17988410	C₂	semidirect
16. #17988410	C₂²	semidirect
18. C₂×M₃₂×C₇₃	C₄	semidirect
19. C₂×M₃₂×C₇₃	C₄	semidirect
22. C₂²×C₁₆×C₇₃	C₄	semidirect
51. M₃₂×C₇₃	C₂×C₄	semidirect
54. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
150. C₂×C₇₃	#2155	semidirect
157. C₇₃	C₂×#2155	semidirect
161. C₂×M₃₂	Dic₇₃	semidirect
162. C₂×M₃₂	Dic₇₃	semidirect
165. C₂²×C₁₆	Dic₇₃	semidirect
197. M₃₂	C₂×Dic₇₃	semidirect
200. C₂×C₁₆	C₂×Dic₇₃	semidirect
301. C₂	#22611704	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×#14556527	2
C₂×#17988625	2
#22611704	2
C₂×#17988410	2
C₂²×M₃₂×C₇₃	2
C₂×#2155	73

Subgroups — C2×#22611704 #33238996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611704 #33238996