Subgroups — C₂×#22608696 #33055000

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988644	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
3	#22608696	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988339	C₂	1	2	yes	maximal, complemented
5	C₂×#17988337	C₂	1	2	yes	maximal, complemented
6	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	C₄	1	2	yes	—
8	#17988644	—	2	2	no	—
9	#17988644	C₂²	1	4	yes	—
10	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#10596831	—	2	1	no	—
13	C₄×#11924012	C₄	1	4	yes	—
14	C₄×#11924012	C₂²	1	4	yes	—
15	C₂×#14556502	—	2	1	no	—
16	C₂×#14556502	C₄	1	1	yes	characteristic
17	C₂×#14556502	C₄	1	1	yes	characteristic
18	C₂×C₄×#10596831	C₂²	1	2	yes	—
19	#17988339	C₂²	1	8	yes	complemented
20	#17988337	C₂²	1	8	yes	complemented
21	C₂²×M₁₆×C₇₃	—	2	1	no	—
22	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	#423×C₇₃	—	2	2	no	—
24	#423×C₇₃	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
26	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
27	C₂×#11924012	—	2	8	no	—
28	C₂×#11924012	C₂×C₄	1	8	yes	—
29	C₂²×#10596831	—	2	1	no	—
30	C₂²×#10596831	—	2	1	no	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	2	yes	—
33	C₄⋊#10596831	—	2	1	no	—
34	C₄⋊#10596831	—	2	1	no	—
35	C₄⋊#10596831	C₂³	1	2	yes	—
36	C₄⋊#10596831	C₂³	1	2	yes	—
37	C₂²⋊#10596831	—	2	4	no	—
38	#14556502	—	2	4	no	—
39	#14556502	C₂×C₄	1	4	yes	—
40	#14556502	C₂×C₄	1	4	yes	—
41	C₄×#10596831	C₂×C₄	1	4	yes	—
42	C₄×#10596831	C₂³	1	4	yes	—
43	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₇₃	Q₈	1	1	yes	characteristic
46	C₂×M₁₆×C₇₃	—	2	4	no	—
47	C₂×M₁₆×C₇₃	—	4	4	no	—
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₇₃	D₄	1	2	yes	—
52	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
57	C₄⋊C₈×C₇₃	—	2	4	no	—
58	C₄⋊C₈×C₇₃	—	2	2	no	—
59	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
60	#11924012	—	2	16	no	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	—	2	2	no	—
64	C₂×#10596831	—	4	2	no	—
65	C₂×#10596831	—	4	2	no	—
66	C₂×#10596831	M₁₆	1	8	yes	complemented
67	C₂×#10596831	C₂²×C₄	1	4	yes	—
68	C₂×#10596831	C₂²×C₄	1	4	yes	—
69	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	—	2	4	no	—
71	C₂²×C₄×C₇₃	—	2	2	no	—
72	C₂²×C₄×C₇₃	—	2	2	no	—
73	C₂²×C₄×C₇₃	—	4	2	no	—
74	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
86	M₁₆×C₇₃	—	4	8	no	—
87	C₂×C₈×C₇₃	—	2	4	no	—
88	C₂×C₈×C₇₃	—	4	4	no	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₇₃	C₂×D₄	1	2	yes	—
95	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
97	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
99	C₂³×C₇₃	—	2	2	no	—
100	C₂³×C₇₃	—	2	2	no	—
101	C₂³×C₇₃	—	2	2	no	—
102	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	4	no	—
112	C₂×C₄×C₇₃	—	2	4	no	—
113	C₂×C₄×C₇₃	—	4	2	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₃	#137	1	2	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₇₃	—	4	4	no	—
128	#10596831	—	2	4	no	—
129	#10596831	—	4	2	no	—
130	#10596831	—	4	2	no	—
131	#10596831	C₂×M₁₆	1	8	yes	complemented
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	4	4	no	—
136	C₂²×C₇₃	#423	1	2	yes	—
137	C₂²×C₇₃	#428	1	2	yes	—
138	C₂²×C₇₃	#430	1	2	yes	—
139	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	—	2	2	no	—
143	C₄×C₇₃	—	2	1	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	#423	1	2	yes	—
146	C₄×C₇₃	#428	1	2	yes	—
147	C₄×C₇₃	C₂²×M₁₆	1	2	yes	—
148	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
150	C₂×#1569	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	4	2	no	—
152	C₂×C₇₃	#1569	1	4	yes	complemented
153	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
154	C₂×C₇₃	C₂×#428	1	1	yes	characteristic
155	C₂×C₇₃	C₂×#430	1	1	yes	characteristic
156	C₂×#432	—	73	1	no	—
157	C₂×#423	D₇₃	1	1	yes	characteristic
158	C₂×C₄×M₁₆	—	73	1	no	—
159	#1569	—	73	8	no	—
160	C₂×#335	—	73	2	no	—
161	C₂×#334	—	73	2	no	—
162	C₇₃	C₂×#1569	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	2	1	no	—
164	C₂²×M₁₆	—	73	2	no	—
165	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
166	#432	—	73	4	no	—
167	#432	—	146	2	no	—
168	#423	—	2	2	no	—
169	#423	C₂×D₇₃	1	4	yes	—
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	—	73	1	no	—
172	C₂×C₄⋊C₈	—	73	1	no	—
173	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
174	C₂×C₂²⋊C₈	—	146	1	no	—
175	C₄×M₁₆	—	73	4	no	—
176	C₄×M₁₆	—	73	4	no	—
177	C₂×#97	—	73	1	no	—
178	C₂×#97	—	73	1	no	—
179	C₂×#97	—	146	1	no	—
180	C₂×C₄×C₈	—	73	2	no	—
181	#335	—	73	8	no	—
182	#334	—	73	8	no	—
183	C₂³×C₄	Dic₁₄₆	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	4	4	no	—
188	C₂×M₁₆	—	73	8	no	—
189	C₂×M₁₆	—	146	8	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	73	2	no	—
193	C₂²×C₈	—	146	1	no	—
194	C₂²×C₈	—	146	1	no	—
195	C₂×C₄²	Dic₁₄₆	1	1	yes	characteristic
196	C₂×C₄²	Dic₁₄₆	1	2	yes	—
197	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₇₃	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	73	2	no	—
206	C₄⋊C₈	—	73	2	no	—
207	C₄⋊C₈	—	146	1	no	—
208	C₄⋊C₈	—	146	1	no	—
209	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
210	C₂²⋊C₈	—	146	4	no	—
211	#97	—	73	4	no	—
212	#97	—	73	4	no	—
213	#97	—	146	4	no	—
214	C₄×C₈	—	73	4	no	—
215	C₄×C₈	—	73	4	no	—
216	C₂⁴	#11924007	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	#11924007	1	1	yes	characteristic
222	C₂²×C₄	#11924007	1	2	yes	—
223	C₂²×C₄	#11924007	1	1	yes	characteristic
224	C₂²×C₄	#11924007	1	2	yes	—
225	C₂²×C₄	#11924007	1	1	yes	characteristic
226	C₂²×C₄	#11924007	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
233	M₁₆	—	4	8	no	—
234	M₁₆	—	146	16	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	292	2	no	—
238	C₂×C₈	—	292	2	no	—
239	C₂×C₈	—	73	8	no	—
240	C₂×C₈	—	73	4	no	—
241	C₂×C₈	—	73	4	no	—
242	C₂×C₈	—	146	2	no	—
243	C₂×C₈	—	146	2	no	—
244	C₂×C₈	—	146	4	no	—
245	C₄²	#11924007	1	2	yes	—
246	C₄²	#11924007	1	2	yes	—
247	C₄²	#11924007	1	2	yes	—
248	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₇₃	1	2	yes	—
250	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
251	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
252	C₄²	C₂×Dic₁₄₆	1	2	yes	—
253	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
254	C₄²	C₂³×D₇₃	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×#11924007	1	2	yes	—
259	C₂³	C₂×#11924007	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
272	C₂×C₄	C₂×#11924007	1	2	yes	—
273	C₂×C₄	C₂×#11924007	1	2	yes	—
274	C₂×C₄	C₂×#11924007	1	2	yes	—
275	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
276	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
277	C₂×C₄	M₁₆×D₇₃	1	4	yes	—
278	C₂×C₄	#14556617	1	2	yes	—
279	C₂×C₄	#14556622	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	292	2	no	—
285	C₈	—	292	2	no	—
286	C₈	—	73	8	no	—
287	C₈	—	146	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#17988610	1	2	yes	—
293	C₂²	#17988627	1	2	yes	—
294	C₂²	#17988635	1	2	yes	—
295	C₂²	C₂²×#11924007	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	#17988610	1	2	yes	—
302	C₄	#17988627	1	2	yes	—
303	C₄	C₂×M₁₆×D₇₃	1	2	yes	—
304	C₄	C₂×#14556617	1	1	yes	characteristic
305	C₄	C₂×#14556622	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	#22608696	1	4	yes	direct product factor
308	C₂	C₂×#17988610	1	1	yes	characteristic
309	C₂	C₂×#17988627	1	1	yes	characteristic
310	C₂	C₂×#17988635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	C₂×#1569
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22608696	C₂	direct (N × H)
4. C₂×#17988339	C₂	semidirect
5. C₂×#17988337	C₂	semidirect
19. #17988339	C₂²	semidirect
20. #17988337	C₂²	semidirect
66. C₂×#10596831	M₁₆	semidirect
131. #10596831	C₂×M₁₆	semidirect
152. C₂×C₇₃	#1569	semidirect
162. C₇₃	C₂×#1569	semidirect
307. C₂	#22608696	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×#17988644	2
C₂×C₄×#11924012	2
#22608696	2
C₂×#17988339	2
C₂×#17988337	2
C₂×#423×C₇₃	2
C₂×#1569	73

Subgroups — C2×#22608696 #33055000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608696 #33055000