Subgroups — C₂×#22612076 #33240591

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988991	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22612076	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988415	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988414	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988405	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988388	C₂	1	1	yes	maximal, characteristic
8	C₂×#357×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556590	—	2	1	no	—
10	C₂×D₄×Dic₇₃	—	2	1	no	—
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	#17988991	—	2	2	no	—
13	#17988991	C₂²	1	4	yes	complemented
14	D₈×Dic₇₃	—	2	2	no	—
15	D₈×Dic₇₃	C₂²	1	4	yes	—
16	C₂×#14556525	C₂²	1	1	yes	characteristic
17	C₂×#14556525	C₂²	1	1	yes	characteristic
18	C₂×#14556524	—	2	1	no	—
19	C₂×#14556512	—	2	1	no	—
20	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
21	#17988415	C₂²	1	4	yes	complemented
22	#17988414	C₂²	1	4	yes	complemented
23	#17988405	C₂²	1	4	yes	complemented
24	#17988388	C₂²	1	4	yes	—
25	C₂²×D₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
28	#357×C₇₃	C₂²	1	4	yes	—
29	C₂²×Dic₁₄₆	—	4	1	no	—
30	C₂³×Dic₇₃	—	4	1	no	—
31	C₂²×#10596831	D₄	1	1	yes	characteristic
32	#14556590	—	2	4	no	—
33	#14556590	—	4	2	no	—
34	D₄×Dic₇₃	—	2	4	no	—
35	D₄×Dic₇₃	—	4	2	no	—
36	C₂×C₄⋊Dic₇₃	—	2	1	no	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×#11924007	—	4	1	no	—
39	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
40	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×#11924003	—	2	2	no	—
42	C₂×#11924003	C₂³	1	2	yes	—
43	#14556525	—	2	1	no	—
44	#14556525	—	2	1	no	—
45	#14556525	C₂³	1	2	yes	—
46	#14556525	C₂³	1	2	yes	—
47	#14556524	—	2	4	no	—
48	#14556512	—	2	4	no	—
49	C₈×Dic₇₃	D₄	1	2	yes	—
50	C₈×Dic₇₃	C₂³	1	2	yes	—
51	C₂²×D₄×C₇₃	—	2	1	no	—
52	C₂×D₈×C₇₃	—	2	1	no	—
53	C₂×D₈×C₇₃	—	2	2	no	—
54	C₂×D₈×C₇₃	Q₈	1	4	yes	complemented
55	C₂×D₈×C₇₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
58	C₂×C₁₆×C₇₃	—	2	2	no	—
59	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
60	#107×C₇₃	—	2	1	no	—
61	#107×C₇₃	C₂³	1	2	yes	—
62	C₂×Dic₁₄₆	—	4	4	no	—
63	C₂×Dic₁₄₆	—	8	4	no	—
64	C₂²×Dic₇₃	—	4	1	no	—
65	C₂²×Dic₇₃	—	4	1	no	—
66	C₂²×Dic₇₃	—	4	4	no	—
67	C₂²×Dic₇₃	—	8	4	no	—
68	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
69	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	C₂×D₄	1	2	yes	—
72	C₄⋊Dic₇₃	—	2	2	no	—
73	C₄⋊Dic₇₃	—	2	2	no	—
74	C₄⋊Dic₇₃	—	4	1	no	—
75	C₄⋊Dic₇₃	—	4	1	no	—
76	#11924007	—	4	4	no	—
77	C₂²⋊Dic₇₃	—	4	4	no	—
78	C₄×Dic₇₃	D₈	1	2	yes	—
79	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
80	#11924003	—	2	2	no	—
81	C₂⁴×C₇₃	—	4	1	no	—
82	C₂×D₄×C₇₃	—	2	4	no	—
83	C₂×D₄×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	—	4	1	no	—
85	C₂×D₄×C₇₃	—	4	2	no	—
86	C₂²×C₄×C₇₃	—	4	1	no	—
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	D₈×C₇₃	—	2	4	no	—
89	D₈×C₇₃	—	2	2	no	—
90	D₈×C₇₃	C₂×Q₈	1	4	yes	complemented
91	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₇₃	#41	1	2	yes	—
94	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₇₃	—	2	2	no	—
96	C₄⋊C₄×C₇₃	—	4	1	no	—
97	C₁₆×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	—	4	2	no	—
99	C₂³×C₇₃	—	4	4	no	—
100	C₂³×C₇₃	—	8	2	no	—
101	C₂³×C₇₃	—	8	2	no	—
102	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
103	D₄×C₇₃	—	2	4	no	—
104	D₄×C₇₃	—	4	2	no	—
105	D₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	8	2	no	—
108	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#137	1	2	yes	—
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇₃	—	2	2	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	8	2	no	—
116	C₂×Dic₇₃	—	8	2	no	—
117	C₂×Dic₇₃	—	8	8	no	—
118	C₂×Dic₇₃	SD₃₂	1	4	yes	—
119	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
120	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
121	Dic₁₄₆	—	8	8	no	—
122	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
125	#10596831	—	2	2	no	—
126	C₂²×C₇₃	—	4	4	no	—
127	C₂²×C₇₃	—	8	2	no	—
128	C₂²×C₇₃	—	8	2	no	—
129	C₂²×C₇₃	—	8	8	no	—
130	C₂²×C₇₃	#474	1	2	yes	—
131	C₂²×C₇₃	C₂×SD₃₂	1	2	yes	—
132	C₂²×C₇₃	#509	1	2	yes	—
133	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₇₃	—	8	2	no	—
135	C₄×C₇₃	#474	1	2	yes	—
136	C₄×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
138	Dic₇₃	—	2	2	no	—
139	Dic₇₃	—	8	2	no	—
140	Dic₇₃	—	8	2	no	—
141	Dic₇₃	C₂×SD₃₂	1	4	yes	—
142	C₂×#2226	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	8	4	no	—
144	C₂×C₇₃	#2226	1	4	yes	complemented
145	C₂×C₇₃	C₂×#474	1	1	yes	characteristic
146	C₂×C₇₃	C₂²×SD₃₂	1	1	yes	characteristic
147	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
148	C₂×#500	—	73	1	no	—
149	C₂×C₄×D₈	—	73	1	no	—
150	#2226	—	73	8	no	—
151	C₂×#367	—	73	1	no	—
152	C₂×#367	—	73	1	no	—
153	C₂×C₄⋊C₁₆	—	73	1	no	—
154	C₂×#357	—	73	1	no	—
155	C₂×#357	D₇₃	1	1	yes	characteristic
156	C₇₃	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
157	C₂²×D₈	C₂×D₇₃	1	1	yes	characteristic
158	C₂×C₄⋊Q₈	—	146	1	no	—
159	C₂×C₄×D₄	—	146	1	no	—
160	C₂²×C₁₆	—	73	1	no	—
161	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
162	#500	—	73	4	no	—
163	#500	—	146	2	no	—
164	C₄×D₈	—	73	4	no	—
165	C₄×D₈	—	146	2	no	—
166	C₂×#107	—	73	1	no	—
167	C₂×#107	—	73	1	no	—
168	C₂×#107	—	146	1	no	—
169	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
170	C₂×#102	—	146	1	no	—
171	C₂×C₄×C₈	—	73	1	no	—
172	#367	—	73	4	no	—
173	#367	—	73	4	no	—
174	C₄⋊C₁₆	—	73	4	no	—
175	#357	—	73	4	no	—
176	#357	C₂×D₇₃	1	4	yes	—
177	C₂²×Q₈	—	292	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂³×C₄	—	292	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	Dic₁₄₆	1	4	yes	complemented
183	C₂×D₈	C₂²×D₇₃	1	2	yes	—
184	C₂²×C₈	—	73	1	no	—
185	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	292	2	no	—
187	C₄⋊Q₈	—	146	4	no	—
188	C₄×D₄	—	292	2	no	—
189	C₄×D₄	—	146	4	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	292	1	no	—
192	C₂×C₄⋊C₄	—	146	1	no	—
193	C₂×C₄⋊C₄	—	146	1	no	—
194	C₂×C₂²⋊C₄	—	292	1	no	—
195	C₂×C₄²	—	73	1	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	73	2	no	—
198	C₂×C₁₆	—	146	2	no	—
199	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
200	#107	—	2	1	no	—
201	#107	—	73	2	no	—
202	#107	—	73	2	no	—
203	#107	—	146	4	no	—
204	#107	—	146	1	no	—
205	#107	—	146	1	no	—
206	#107	C₂²×D₇₃	1	2	yes	—
207	#102	—	146	4	no	—
208	C₄×C₈	—	73	2	no	—
209	C₄×C₈	—	73	2	no	—
210	C₂⁴	—	4	1	no	—
211	C₂×Q₈	—	292	4	no	—
212	C₂×Q₈	—	584	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	292	1	no	—
219	C₂²×C₄	—	292	1	no	—
220	C₂²×C₄	—	292	4	no	—
221	C₂²×C₄	—	584	4	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	1	no	—
224	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
225	D₈	—	2	4	no	—
226	D₈	—	2	2	no	—
227	D₈	C₂×Dic₁₄₆	1	4	yes	complemented
228	C₂×C₈	—	73	2	no	—
229	C₂×C₈	—	146	2	no	—
230	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
231	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
232	C₂×C₈	#11924027	1	2	yes	—
233	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	292	4	no	—
237	C₄⋊C₄	—	292	1	no	—
238	C₄⋊C₄	—	292	1	no	—
239	C₄⋊C₄	—	146	2	no	—
240	C₄⋊C₄	—	146	2	no	—
241	C₂²⋊C₄	—	292	4	no	—
242	C₄²	—	73	2	no	—
243	C₄²	—	73	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	146	2	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	D₈×D₇₃	1	1	yes	characteristic
251	Q₈	—	584	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	292	2	no	—
259	C₂×C₄	—	584	2	no	—
260	C₂×C₄	—	584	2	no	—
261	C₂×C₄	—	584	8	no	—
262	C₂×C₄	—	73	4	no	—
263	C₂×C₄	—	73	2	no	—
264	C₂×C₄	—	73	2	no	—
265	C₂×C₄	—	146	2	no	—
266	C₂×C₄	#14556568	1	2	yes	—
267	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
268	C₂×C₄	#14556620	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
270	C₈	—	146	2	no	—
271	C₈	#14556568	1	2	yes	—
272	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
273	C₈	C₂×#11924027	1	1	yes	characteristic
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	8	no	—
278	C₂²	#17988865	1	2	yes	—
279	C₂²	SD₃₂×D₇₃	1	2	yes	—
280	C₂²	#17989021	1	2	yes	—
281	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	584	2	no	—
284	C₄	—	584	2	no	—
285	C₄	—	73	4	no	—
286	C₄	—	146	2	no	—
287	C₄	#17988865	1	2	yes	—
288	C₄	C₂×#14556568	1	1	yes	characteristic
289	C₄	C₂×#14556620	1	1	yes	characteristic
290	C₂	—	8	4	no	—
291	C₂	#22612076	1	4	yes	direct product factor
292	C₂	C₂×#17988865	1	1	yes	characteristic
293	C₂	C₂×SD₃₂×D₇₃	1	1	yes	characteristic
294	C₂	C₂×#17989021	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#357×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₈×C₇₃	C₂²
hypercenter	C₂×#357	D₇₃
hypocenter	C₇₃	C₂×#2226
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. C₂×#17988991	C₂	semidirect
3. #22612076	C₂	direct (N × H)
4. C₂×#17988415	C₂	semidirect
5. C₂×#17988414	C₂	semidirect
6. C₂×#17988405	C₂	semidirect
13. #17988991	C₂²	semidirect
21. #17988415	C₂²	semidirect
22. #17988414	C₂²	semidirect
23. #17988405	C₂²	semidirect
54. C₂×D₈×C₇₃	Q₈	semidirect
90. D₈×C₇₃	C₂×Q₈	semidirect
144. C₂×C₇₃	#2226	semidirect
156. C₇₃	C₂×#2226	semidirect
182. C₂×D₈	Dic₁₄₆	semidirect
227. D₈	C₂×Dic₁₄₆	semidirect
291. C₂	#22612076	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×#17988991	2
C₂×D₈×Dic₇₃	2
#22612076	2
C₂×#17988415	2
C₂×#17988414	2
C₂×#17988405	2
C₂×#17988388	2
C₂×#357×C₇₃	2
C₂×#2226	73

Subgroups — C2×#22612076 #33240591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612076 #33240591