Subgroups — C₂×#19136717 #25251157

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583315	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583171	C₂	1	2	yes	maximal
3	#19136717	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582729	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582715	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063356	—	2	2	no	—
8	C₂×#13063331	C₂²	1	1	yes	characteristic
9	#14583315	—	2	2	no	—
10	#14583315	C₂²	1	4	yes	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063264	—	2	1	no	—
13	C₂×C₁₆×D₄₃	—	2	1	no	—
14	#14583171	—	2	4	no	—
15	#14583171	C₂²	1	8	yes	—
16	C₂×#13063261	C₂²	1	2	yes	—
17	C₂×#13063244	—	2	2	no	—
18	#14582729	C₂²	1	4	yes	complemented
19	#14582715	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₄₃	—	2	1	no	—
22	C₂×Q₃₂×C₄₃	—	2	1	no	—
23	C₂×Q₃₂×C₄₃	—	2	2	no	—
24	C₂×Q₃₂×C₄₃	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#10621898	D₄	1	1	yes	characteristic
27	#13063356	—	2	8	no	—
28	#13063356	—	4	4	no	—
29	C₂×#10621885	—	4	2	no	—
30	C₂×#10621884	—	4	2	no	—
31	#13063331	—	2	1	no	—
32	#13063331	—	2	1	no	—
33	#13063331	D₄	1	2	yes	—
34	#13063331	C₂³	1	2	yes	—
35	C₂×#10621877	—	2	2	no	—
36	C₂×#10621877	D₄	1	1	yes	characteristic
37	C₂×#10621872	—	2	1	no	—
38	C₂×C₈×D₄₃	—	2	1	no	—
39	#13063265	—	2	1	no	—
40	#13063265	C₂³	1	2	yes	—
41	#13063264	—	2	4	no	—
42	C₂×C₄₃⋊C₁₆	—	2	1	no	—
43	C₁₆×D₄₃	—	2	4	no	—
44	#13063261	—	2	2	no	—
45	#13063261	C₂³	1	4	yes	—
46	#13063244	—	2	8	no	—
47	C₂²×Q₈×C₄₃	—	2	2	no	—
48	C₂×Q₁₆×C₄₃	—	2	2	no	—
49	C₂×Q₁₆×C₄₃	—	2	2	no	—
50	C₂×Q₁₆×C₄₃	—	2	2	no	—
51	C₂×Q₁₆×C₄₃	—	2	2	no	—
52	C₂×Q₁₆×C₄₃	—	2	2	no	—
53	C₂×Q₁₆×C₄₃	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	Q₃₂×C₄₃	—	2	4	no	—
56	Q₃₂×C₄₃	—	2	4	no	—
57	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
61	C₂×Q₈×C₄₃	—	2	4	no	—
62	C₂×Q₈×C₄₃	—	4	2	no	—
63	C₂×Q₈×C₄₃	—	4	2	no	—
64	C₂×Q₈×C₄₃	—	4	2	no	—
65	C₂×Q₈×C₄₃	—	4	2	no	—
66	C₂×Q₈×C₄₃	—	4	2	no	—
67	C₂²×C₄×C₄₃	—	4	2	no	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₃	—	2	1	no	—
70	#10621898	—	2	1	no	—
71	#10621898	—	2	1	no	—
72	#10621898	D₈	1	2	yes	—
73	#10621898	C₂×D₄	1	2	yes	—
74	C₂×D₁₇₂	D₈	1	1	yes	characteristic
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×Dic₈₆	D₈	1	1	yes	characteristic
77	Q₁₆×C₄₃	—	2	4	no	—
78	Q₁₆×C₄₃	—	2	4	no	—
79	Q₁₆×C₄₃	—	4	2	no	—
80	Q₁₆×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
87	#10621885	—	4	8	no	—
88	#10621884	—	4	8	no	—
89	#10621877	—	2	1	no	—
90	#10621877	—	2	4	no	—
91	#10621877	—	4	2	no	—
92	#10621877	C₂×D₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	4	2	no	—
95	#10621872	—	2	2	no	—
96	#10621872	—	2	2	no	—
97	C₈×D₄₃	—	2	2	no	—
98	C₈×D₄₃	—	2	2	no	—
99	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
101	C₄₃⋊C₁₆	—	2	2	no	—
102	C₂³×C₄₃	#135	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	1	no	—
104	Q₈×C₄₃	—	4	4	no	—
105	Q₈×C₄₃	—	4	4	no	—
106	Q₈×C₄₃	—	8	2	no	—
107	Q₈×C₄₃	—	8	2	no	—
108	C₂×C₄×C₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	8	2	no	—
110	C₂×C₄×C₄₃	—	8	2	no	—
111	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₄₃	—	2	2	no	—
118	C₂²⋊D₄₃	—	2	2	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	D₁₇₂	—	2	1	no	—
121	D₁₇₂	C₂×D₈	1	2	yes	—
122	C₄×D₄₃	—	2	2	no	—
123	C₄×D₄₃	—	2	2	no	—
124	Dic₈₆	—	2	1	no	—
125	Dic₈₆	C₂×D₈	1	2	yes	—
126	C₈×C₄₃	#121	1	1	yes	characteristic
127	C₈×C₄₃	#121	1	1	yes	characteristic
128	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
130	C₄₃⋊C₈	—	2	2	no	—
131	C₄₃⋊C₈	—	4	4	no	—
132	C₂×#2222	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	#466	1	1	yes	characteristic
135	C₂²×C₄₃	#466	1	1	yes	characteristic
136	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
137	C₂×D₄₃	—	2	2	no	—
138	C₂×D₄₃	—	4	2	no	—
139	C₄×C₄₃	—	8	4	no	—
140	C₄×C₄₃	#466	1	1	yes	characteristic
141	C₄×C₄₃	#466	1	1	yes	characteristic
142	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
143	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
144	Dic₄₃	—	2	2	no	—
145	C₂²×Q₃₂	D₄₃	1	1	yes	characteristic, complemented
146	C₂×#504	—	43	1	no	—
147	C₂×#473	—	43	2	no	—
148	#2222	—	43	8	no	—
149	C₂×#368	—	43	1	no	—
150	C₂×#362	—	43	2	no	—
151	C₂×C₄₃	—	2	1	no	—
152	C₂×C₄₃	—	2	1	no	—
153	C₂×C₄₃	#2222	1	2	yes	complemented
154	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
155	D₄₃	—	4	2	no	—
156	C₂×#137	—	86	2	no	—
157	C₂²×Q₁₆	C₂×D₄₃	1	2	yes	—
158	C₂×#131	—	43	1	no	—
159	C₂×Q₃₂	—	2	1	no	—
160	C₂×Q₃₂	—	2	1	no	—
161	C₂×Q₃₂	—	2	2	no	—
162	C₂×Q₃₂	C₂×D₄₃	1	4	yes	complemented
163	#504	—	43	4	no	—
164	#504	—	86	2	no	—
165	C₂×M₃₂	—	43	1	no	—
166	C₂×M₃₂	—	86	1	no	—
167	C₂²×C₁₆	—	86	1	no	—
168	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
169	#473	—	43	8	no	—
170	#473	—	86	4	no	—
171	C₂×#108	—	43	2	no	—
172	C₂×#101	—	86	2	no	—
173	#368	—	43	4	no	—
174	#362	—	43	8	no	—
175	C₄₃	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	43	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#137	—	86	8	no	—
179	#137	—	172	4	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	172	2	no	—
186	C₂×Q₁₆	C₂²×D₄₃	1	4	yes	—
187	C₂×SD₁₆	—	172	2	no	—
188	#131	—	43	2	no	—
189	#131	—	43	2	no	—
190	#131	—	86	1	no	—
191	#131	—	86	1	no	—
192	C₂×M₁₆	—	43	1	no	—
193	C₂×M₁₆	—	86	1	no	—
194	C₂×M₁₆	—	86	2	no	—
195	C₂²×C₈	—	86	1	no	—
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	Q₃₂	—	2	4	no	—
198	Q₃₂	—	2	4	no	—
199	M₃₂	—	43	2	no	—
200	M₃₂	—	86	1	no	—
201	M₃₂	—	86	4	no	—
202	C₂×C₁₆	—	86	1	no	—
203	C₂×C₁₆	—	86	4	no	—
204	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
207	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
208	#108	—	43	4	no	—
209	#108	—	86	2	no	—
210	#101	—	86	8	no	—
211	#41	—	43	2	no	—
212	#41	—	43	2	no	—
213	#41	—	86	1	no	—
214	#41	—	86	1	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	43	1	no	—
222	C₂×D₄	—	43	1	no	—
223	C₂×D₄	—	86	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	86	1	no	—
226	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	2	no	—
230	Q₁₆	—	4	2	no	—
231	Q₁₆	—	172	8	no	—
232	SD₁₆	—	172	8	no	—
233	M₁₆	—	43	2	no	—
234	M₁₆	—	86	1	no	—
235	M₁₆	—	86	2	no	—
236	M₁₆	—	86	2	no	—
237	M₁₆	—	86	4	no	—
238	M₁₆	—	172	2	no	—
239	C₂×C₈	—	86	1	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	—	172	2	no	—
243	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
244	C₂×C₈	#10621900	1	1	yes	characteristic
245	C₂×C₈	#10621900	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
249	C₁₆	—	86	2	no	—
250	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
251	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
252	C₂³	—	86	1	no	—
253	C₂³	#13063344	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	8	2	no	—
257	Q₈	—	8	2	no	—
258	Q₈	—	43	2	no	—
259	Q₈	—	86	1	no	—
260	D₄	—	43	2	no	—
261	D₄	—	86	2	no	—
262	D₄	—	86	2	no	—
263	D₄	—	86	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	86	1	no	—
268	C₂×C₄	—	86	2	no	—
269	C₂×C₄	—	86	2	no	—
270	C₂×C₄	#13063296	1	1	yes	characteristic
271	C₂×C₄	#13063296	1	1	yes	characteristic
272	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
273	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
274	C₂×C₄	#13063344	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
276	C₈	—	86	2	no	—
277	C₈	—	172	4	no	—
278	C₈	#13063296	1	1	yes	characteristic
279	C₈	#13063296	1	1	yes	characteristic
280	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
281	C₈	C₂×#10621900	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	86	2	no	—
284	C₂²	—	172	2	no	—
285	C₂²	#14583132	1	1	yes	characteristic
286	C₂²	#14583132	1	1	yes	characteristic
287	C₂²	C₂×#13063344	1	1	yes	characteristic
288	C₄	—	8	4	no	—
289	C₄	—	86	2	no	—
290	C₄	#14583132	1	1	yes	characteristic
291	C₄	#14583132	1	1	yes	characteristic
292	C₄	C₂×#13063296	1	1	yes	characteristic
293	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	172	2	no	—
297	C₂	#19136717	1	2	yes	direct product factor
298	C₂	C₂×#14583132	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂²×Q₃₂×C₄₃	C₂
socle	C₂²×C₄₃	#466
prime radical	C₂×#10621898	D₄
hypercenter	C₂²×Q₃₂	D₄₃
hypocenter	C₄₃	C₂×#2222
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #19136717	C₂	direct (N × H)
4. C₂×#14582729	C₂	semidirect
5. C₂×#14582715	C₂	semidirect
6. C₂²×Q₃₂×C₄₃	C₂	semidirect
18. #14582729	C₂²	semidirect
19. #14582715	C₂²	semidirect
24. C₂×Q₃₂×C₄₃	C₂²	semidirect
145. C₂²×Q₃₂	D₄₃	semidirect
153. C₂×C₄₃	#2222	semidirect
162. C₂×Q₃₂	C₂×D₄₃	semidirect
175. C₄₃	C₂×#2222	semidirect
297. C₂	#19136717	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×#14583315	2
C₂×#14583171	2
#19136717	2
C₂×#14582729	2
C₂×#14582715	2
C₂²×Q₃₂×C₄₃	2
C₂×#2222	43

Subgroups — C2×#19136717 #25251157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136717 #25251157