Subgroups — C₂×#22626273 #34030077

Order 18688.

698 conjugacy classes of subgroups.

The subgroup lattice has 282 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	2	yes	maximal
2	#22626273	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989510	C₂	1	2	yes	maximal, complemented
4	C₂×#17989509	C₂	1	2	yes	maximal, complemented
5	C₂×#17988544	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	—	2	2	no	—
7	#17989643	—	2	4	no	—
8	#17989643	C₂²	1	8	yes	—
9	C₂×#14556665	C₂²	1	4	yes	—
10	C₂×#14556664	—	2	2	no	—
11	#17989510	C₂²	1	8	yes	complemented
12	#17989509	C₂²	1	8	yes	complemented
13	C₂²×#11924011	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂×#14556548	C₄	1	1	yes	characteristic
16	#17988544	C₄	1	4	yes	—
17	#17988544	C₂²	1	4	yes	—
18	C₂×#14556501	C₄	1	1	yes	characteristic
19	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
22	#14556666	—	2	8	no	—
23	#14556665	—	2	4	no	—
24	#14556665	C₂³	1	8	yes	—
25	#14556664	—	2	8	no	—
26	C₂×#11924031	—	2	4	no	—
27	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924011	C₈	1	4	yes	—
29	C₂×#11924011	C₈	1	4	yes	—
30	C₂×#11924011	C₂×C₄	1	4	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂³	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₃	D₄	1	2	yes	—
35	C₂×C₈×D₇₃	D₄	1	2	yes	—
36	C₂×C₈×D₇₃	Q₈	1	2	yes	—
37	C₂×C₈×D₇₃	Q₈	1	2	yes	—
38	C₂×C₈×D₇₃	C₂³	1	2	yes	—
39	#14556548	—	2	2	no	—
40	#14556548	C₂×C₄	1	4	yes	—
41	C₂×#11924007	—	2	1	no	—
42	C₂×#11924005	—	2	1	no	—
43	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	#14556501	C₂×C₄	1	4	yes	—
45	C₄×#10596831	C₂×C₄	1	2	yes	—
46	C₄×#10596831	C₂³	1	2	yes	—
47	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	#97×C₇₃	C₂×C₄	1	4	yes	—
53	#11924031	—	2	8	no	—
54	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
55	C₂²×Dic₇₃	—	2	1	no	—
56	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
63	#11924011	C₂×C₈	1	16	yes	—
64	C₂×#10596831	C₂×C₈	1	2	yes	—
65	C₂×#10596831	C₂×C₈	1	2	yes	—
66	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
67	C₂×#10596831	C₂²×C₄	1	2	yes	—
68	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
69	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
70	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
71	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
72	C₂×#10596831	C₂⁴	1	1	yes	characteristic
73	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
74	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
75	C₈×D₇₃	C₂×D₄	1	4	yes	—
76	C₈×D₇₃	C₂×Q₈	1	4	yes	—
77	#11924007	—	2	4	no	—
78	#11924005	—	2	4	no	—
79	C₄×Dic₇₃	—	2	1	no	—
80	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	—	2	1	no	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₇₃	—	2	1	no	—
90	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂²×D₇₃	—	4	2	no	—
93	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
94	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
95	C₂²×D₇₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	4	2	no	—
98	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₇₃	—	2	2	no	—
105	C₂×Dic₇₃	—	4	2	no	—
106	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
107	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
108	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
109	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
110	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
111	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₄×D₇₃	C₂²×C₈	1	4	yes	—
113	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₈×C₇₃	C₂²×C₈	1	4	yes	—
115	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
116	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
117	#10596831	C₂²×C₈	1	4	yes	—
118	#10596831	C₂²×D₄	1	1	yes	characteristic
119	#10596831	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₇₃	—	2	2	no	—
121	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×D₇₃	—	2	4	no	—
125	C₂×D₇₃	—	4	2	no	—
126	C₂×D₇₃	—	4	2	no	—
127	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
128	C₄×C₇₃	—	4	2	no	—
129	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
133	Dic₇₃	—	4	2	no	—
134	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
135	C₂×#2153	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	—	2	1	no	—
137	C₂×C₇₃	—	2	1	no	—
138	C₂×C₇₃	#2153	1	2	yes	complemented
139	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	D₇₃	—	2	4	no	—
141	C₂²×M₃₂	—	73	2	no	—
142	C₂×#405	—	73	1	no	—
143	#2153	—	73	8	no	—
144	C₂×#364	—	73	2	no	—
145	C₂×#364	—	73	2	no	—
146	C₇₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
147	C₂²×M₁₆	—	73	1	no	—
148	C₂³×C₈	—	73	1	no	—
149	C₂×#117	—	73	1	no	—
150	C₂×M₃₂	—	73	4	no	—
151	C₂×M₃₂	—	73	8	no	—
152	C₂×M₃₂	—	146	4	no	—
153	C₂×M₃₂	—	146	2	no	—
154	C₂²×C₁₆	—	146	2	no	—
155	#405	—	73	4	no	—
156	#405	—	73	4	no	—
157	C₂×#97	—	73	1	no	—
158	C₂×#97	#64408	1	1	yes	characteristic
159	C₂×C₄×C₈	—	73	1	no	—
160	C₂×C₄×C₈	—	73	1	no	—
161	#364	—	73	8	no	—
162	#364	—	73	8	no	—
163	C₂³×C₄	—	73	1	no	—
164	C₂×M₁₆	—	73	4	no	—
165	C₂×M₁₆	—	73	4	no	—
166	C₂×M₁₆	—	73	4	no	—
167	C₂²×C₈	—	73	1	no	—
168	C₂²×C₈	—	73	1	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	—	73	2	no	—
171	C₂²×C₈	—	73	2	no	—
172	C₂²×C₈	—	73	2	no	—
173	C₂²×C₈	—	73	2	no	—
174	C₂²×C₈	—	73	2	no	—
175	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
176	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
177	#117	—	73	4	no	—
178	#117	—	146	2	no	—
179	C₂×C₄⋊C₄	—	146	1	no	—
180	C₂×C₂²⋊C₄	—	146	1	no	—
181	C₂×C₄²	—	73	1	no	—
182	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
183	M₃₂	—	73	8	no	—
184	M₃₂	—	146	4	no	—
185	M₃₂	—	146	8	no	—
186	C₂×C₁₆	—	146	4	no	—
187	C₂×C₁₆	—	146	8	no	—
188	#97	—	73	4	no	—
189	#97	C₂×#64408	1	4	yes	—
190	C₄×C₈	—	73	2	no	—
191	C₄×C₈	—	73	2	no	—
192	C₄×C₈	—	73	2	no	—
193	C₄×C₈	—	73	2	no	—
194	C₂⁴	—	73	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	2	no	—
198	C₂²×C₄	—	73	2	no	—
199	C₂²×C₄	—	73	2	no	—
200	C₂²×C₄	—	73	2	no	—
201	C₂²×C₄	—	73	2	no	—
202	C₂²×C₄	—	73	2	no	—
203	C₂²×C₄	—	146	1	no	—
204	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
205	M₁₆	—	73	16	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	73	1	no	—
209	C₂×C₈	—	73	2	no	—
210	C₂×C₈	—	73	1	no	—
211	C₂×C₈	—	73	1	no	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	1	no	—
214	C₂×C₈	—	73	1	no	—
215	C₂×C₈	—	73	4	no	—
216	C₂×C₈	—	73	4	no	—
217	C₂×C₈	—	73	4	no	—
218	C₂×C₈	—	73	4	no	—
219	C₂×C₈	C₄⋊#64408	1	2	yes	—
220	C₂×C₈	C₄⋊#64408	1	2	yes	—
221	C₂×C₈	#11924036	1	2	yes	—
222	C₂×C₈	#11924036	1	2	yes	—
223	C₂×C₈	C₂²×#64408	1	2	yes	—
224	C₂×C₈	C₂²×#64408	1	2	yes	—
225	C₄⋊C₄	—	146	4	no	—
226	C₂²⋊C₄	—	146	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	73	2	no	—
229	C₄²	—	146	1	no	—
230	C₄²	C₂²×#64408	1	2	yes	—
231	C₁₆	—	146	8	no	—
232	C₂³	—	292	2	no	—
233	C₂³	—	73	2	no	—
234	C₂³	—	73	2	no	—
235	C₂³	—	73	2	no	—
236	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	292	2	no	—
240	C₂×C₄	—	73	2	no	—
241	C₂×C₄	—	73	2	no	—
242	C₂×C₄	—	73	2	no	—
243	C₂×C₄	—	73	4	no	—
244	C₂×C₄	—	73	4	no	—
245	C₂×C₄	—	73	4	no	—
246	C₂×C₄	—	73	4	no	—
247	C₂×C₄	—	146	2	no	—
248	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
250	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
251	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
252	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
253	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
254	C₈	—	73	1	no	—
255	C₈	—	73	1	no	—
256	C₈	—	73	1	no	—
257	C₈	—	73	1	no	—
258	C₈	—	73	4	no	—
259	C₈	C₂×C₄⋊#64408	1	4	yes	—
260	C₈	C₂×#11924036	1	4	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	292	2	no	—
263	C₂²	—	292	2	no	—
264	C₂²	—	73	4	no	—
265	C₂²	—	146	4	no	—
266	C₂²	#17989588	1	1	yes	characteristic
267	C₂²	#17989588	1	1	yes	characteristic
268	C₂²	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	292	2	no	—
271	C₄	—	73	4	no	—
272	C₄	#17989588	1	1	yes	characteristic
273	C₄	#17989588	1	1	yes	characteristic
274	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
275	C₄	C₂²×#11924036	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	146	4	no	—
279	C₂	#22626273	1	2	yes	direct product factor
280	C₂	C₂×#17989588	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×#97×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	C₂×#2153
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22626273	C₂	direct (N × H)
3. C₂×#17989510	C₂	semidirect
4. C₂×#17989509	C₂	semidirect
11. #17989510	C₂²	semidirect
12. #17989509	C₂²	semidirect
138. C₂×C₇₃	#2153	semidirect
146. C₇₃	C₂×#2153	semidirect
279. C₂	#22626273	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
#22626273	2
C₂×#17989510	2
C₂×#17989509	2
C₂×#17988544	2
C₂×#2153	73

Subgroups — C2×#22626273 #34030077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626273 #34030077