Subgroups — C₂×#19147797 #26003194

Order 11008.

11534 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10621902	C₂	1	1	yes	maximal, characteristic
2	#19147797	C₂	1	32	yes	maximal, direct product factor
3	C₂×#14583381	C₂	1	8	yes	maximal, complemented
4	C₂×#14583377	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14583373	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×#10621902	C₂	1	8	yes	maximal, complemented
7	C₂×#117×D₄₃	C₂	1	6	yes	maximal, complemented
8	C₂×C₄₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621902	C₄	1	32	yes	complemented
10	C₂²×#10621902	C₂²	1	4	yes	—
11	C₂²×#10621902	C₂²	1	4	yes	—
12	C₂²×#10621902	C₂²	1	16	yes	—
13	C₂³×D₁₇₂	C₂²	1	3	yes	—
14	C₂³×C₄×D₄₃	C₂²	1	3	yes	—
15	#14583381	C₂²	1	128	yes	complemented
16	#14583377	C₂²	1	16	yes	complemented
17	#14583373	C₂²	1	96	yes	complemented
18	C₄×#10621902	C₂²	1	128	yes	complemented
19	C₂×Q₈×Dic₄₃	C₂²	1	8	yes	complemented
20	C₂×C₄×D₁₇₂	C₂²	1	24	yes	complemented
21	C₂×#13063287	C₂²	1	24	yes	complemented
22	C₂×#13063284	C₂²	1	6	yes	—
23	C₂×#13063283	C₂²	1	12	yes	complemented
24	C₂×#13063282	C₂²	1	12	yes	complemented
25	C₂×#13063281	C₂²	1	4	yes	—
26	#117×D₄₃	C₂²	1	96	yes	complemented
27	C₂×C₄⋊C₄×D₄₃	C₂²	1	12	yes	complemented
28	C₂×C₂²⋊C₄×D₄₃	C₂²	1	12	yes	—
29	C₂×C₄²×D₄₃	C₂²	1	12	yes	complemented
30	C₂³×Q₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#519	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₄₃	C₂²	1	8	yes	complemented
33	C₂×#117×C₄₃	C₂²	1	6	yes	—
34	C₂⁴×D₄₃	C₂³	1	3	yes	—
35	C₂×#10621902	C₂×C₄	1	64	yes	complemented
36	C₂×#10621902	C₂×C₄	1	64	yes	complemented
37	C₂×#10621902	C₂×C₄	1	256	yes	complemented
38	C₂×#10621902	C₂³	1	32	yes	—
39	C₂×#10621902	C₂³	1	32	yes	—
40	C₂²×D₁₇₂	C₂×C₄	1	48	yes	complemented
41	C₂²×D₁₇₂	C₂³	1	12	yes	—
42	C₂²×D₁₇₂	C₂³	1	24	yes	—
43	C₂³×Dic₄₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×D₄₃	C₂×C₄	1	48	yes	complemented
45	C₂²×C₄×D₄₃	C₂³	1	24	yes	—
46	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
47	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
48	C₂²×C₄×D₄₃	C₂³	1	24	yes	—
49	Q₈×Dic₄₃	C₂³	1	64	yes	complemented
50	C₄×D₁₇₂	C₂³	1	192	yes	complemented
51	#13063287	C₂³	1	192	yes	complemented
52	#13063284	C₂³	1	48	yes	—
53	#13063283	C₂³	1	96	yes	complemented
54	#13063282	C₂³	1	96	yes	complemented
55	#13063281	C₂³	1	32	yes	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
57	C₂×Dic₄₃⋊C₄	C₂³	1	8	yes	—
58	C₄⋊C₄×D₄₃	C₂³	1	96	yes	complemented
59	C₂×C₂²⋊Dic₄₃	C₂³	1	4	yes	—
60	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	24	yes	—
61	C₂²⋊C₄×D₄₃	C₂³	1	96	yes	—
62	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
63	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
64	C₄²×D₄₃	C₂³	1	96	yes	complemented
65	C₂²×Q₈×C₄₃	C₂×C₄	1	16	yes	complemented
66	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
67	C₂²×Q₈×C₄₃	C₂³	1	8	yes	—
68	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
69	C₄×Q₈×C₄₃	C₂³	1	64	yes	complemented
70	#117×C₄₃	C₂³	1	48	yes	—
71	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
72	C₂×C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
73	C₂×C₄²×C₄₃	C₂³	1	12	yes	—
74	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂³×D₄₃	—	4	12	no	—
76	C₂³×D₄₃	C₂²×C₄	1	24	yes	—
77	C₂³×D₄₃	C₂⁴	1	6	yes	—
78	C₂³×D₄₃	C₂⁴	1	12	yes	—
79	C₂×Q₈×C₄₃	C₂²×C₄	1	32	yes	—
80	C₂×Q₈×C₄₃	C₂²×C₄	1	64	yes	complemented
81	C₂×Q₈×C₄₃	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
83	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
86	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
87	C₂²×Dic₄₃	C₂⁴	1	8	yes	—
88	C₂²×Dic₄₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₃	C₂⁴	1	4	yes	—
91	#10621902	C₂²×C₄	1	256	yes	complemented
92	#10621902	C₂²×C₄	1	256	yes	complemented
93	C₂×D₁₇₂	C₂²×C₄	1	96	yes	—
94	C₂×D₁₇₂	C₂²×C₄	1	192	yes	complemented
95	C₂×D₁₇₂	C₂⁴	1	48	yes	—
96	C₂×C₄×D₄₃	—	4	48	no	—
97	C₂×C₄×D₄₃	C₂²×C₄	1	48	yes	—
98	C₂×C₄×D₄₃	C₂²×C₄	1	48	yes	—
99	C₂×C₄×D₄₃	C₂²×C₄	1	192	yes	complemented
100	C₂×C₄×D₄₃	C₂⁴	1	96	yes	—
101	C₂×C₄×D₄₃	C₂⁴	1	24	yes	—
102	C₂×C₄×D₄₃	C₂⁴	1	24	yes	—
103	C₄⋊C₄×C₄₃	C₂⁴	1	48	yes	—
104	C₂²⋊C₄×C₄₃	C₂⁴	1	16	yes	—
105	C₄²×C₄₃	C₂⁴	1	48	yes	—
106	C₂²⋊Dic₄₃	C₂⁴	1	16	yes	—
107	(C₂×D₄₃)⋊C₄	C₂⁴	1	96	yes	—
108	C₄⋊Dic₄₃	C₂⁴	1	48	yes	—
109	Dic₄₃⋊C₄	C₂⁴	1	32	yes	—
110	C₄×Dic₄₃	C₂⁴	1	32	yes	—
111	C₄×Dic₄₃	C₂⁴	1	48	yes	—
112	C₂³×C₄₃	—	2	4	no	—
113	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
114	C₂³×C₄₃	C₂⁵	1	2	yes	—
115	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	24	no	—
117	C₂²×D₄₃	—	4	48	no	—
118	C₂²×D₄₃	—	4	24	no	—
119	C₂²×D₄₃	C₂³×C₄	1	24	yes	—
120	C₂²×D₄₃	C₂³×C₄	1	48	yes	—
121	C₂²×D₄₃	C₂⁵	1	12	yes	—
122	Q₈×C₄₃	C₂³×C₄	1	64	yes	—
123	C₂×C₄×C₄₃	—	2	16	no	—
124	C₂×C₄×C₄₃	C₂³×C₄	1	24	yes	—
125	C₂×C₄×C₄₃	C₂³×C₄	1	48	yes	—
126	C₂×C₄×C₄₃	C₂⁵	1	16	yes	—
127	C₂×C₄×C₄₃	C₂⁵	1	12	yes	—
128	C₂×Dic₄₃	—	2	16	no	—
129	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
130	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
131	C₂×Dic₄₃	C₂³×C₄	1	16	yes	—
132	C₂×Dic₄₃	C₂⁵	1	16	yes	—
133	C₂×Dic₄₃	C₂⁵	1	2	yes	—
134	C₂×Dic₄₃	C₂⁵	1	2	yes	—
135	D₁₇₂	C₂³×C₄	1	192	yes	—
136	C₄×D₄₃	—	4	96	no	—
137	C₄×D₄₃	C₂³×C₄	1	96	yes	—
138	C₄×D₄₃	C₂³×C₄	1	96	yes	—
139	C₂×#3428	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	2	8	no	—
142	C₂²×C₄₃	C₂⁴×C₄	1	4	yes	—
143	C₂²×C₄₃	C₂⁴×C₄	1	2	yes	—
144	C₂²×C₄₃	#585	1	4	yes	—
145	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
146	C₂×D₄₃	—	2	24	no	—
147	C₂×D₄₃	—	2	48	no	—
148	C₂×D₄₃	—	4	48	no	—
149	C₂×D₄₃	C₂⁴×C₄	1	24	yes	—
150	C₄×C₄₃	—	2	16	no	—
151	C₄×C₄₃	C₂⁴×C₄	1	24	yes	—
152	Dic₄₃	—	2	16	no	—
153	Dic₄₃	C₂⁴×C₄	1	4	yes	—
154	Dic₄₃	C₂⁴×C₄	1	4	yes	—
155	C₂³×#41	—	43	1	no	—
156	#3428	—	43	32	no	—
157	C₂×#520	—	43	8	no	—
158	C₂×#519	—	43	1	no	—
159	C₂×#519	D₄₃	1	1	yes	characteristic, complemented
160	C₂×#518	—	43	6	no	—
161	C₂×C₄×#41	—	43	8	no	—
162	C₂²×#117	—	43	6	no	—
163	C₂×C₄₃	—	2	4	no	—
164	C₂×C₄₃	#3428	1	4	yes	complemented
165	C₂×C₄₃	C₂⁵×C₄	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#585	1	2	yes	—
167	D₄₃	—	2	24	no	—
168	C₂²×#41	—	43	4	no	—
169	C₂²×#41	—	43	4	no	—
170	C₂²×#41	—	43	16	no	—
171	C₂²×#41	—	43	32	no	—
172	C₂³×Q₈	C₂×D₄₃	1	1	yes	characteristic
173	C₂³×D₄	—	43	3	no	—
174	C₂⁴×C₄	—	43	3	no	—
175	#520	—	43	128	no	—
176	#519	—	43	16	no	—
177	#519	C₂×D₄₃	1	16	yes	complemented
178	#518	—	43	96	no	—
179	C₄×#41	—	43	128	no	—
180	C₂×C₄×Q₈	—	43	8	no	—
181	C₂×C₄×Q₈	C₂×D₄₃	1	8	yes	complemented
182	C₂×C₄×D₄	—	43	24	no	—
183	C₂×C₄×D₄	—	43	24	no	—
184	C₂×#117	—	43	6	no	—
185	C₂×#117	—	43	96	no	—
186	C₂×#117	—	43	12	no	—
187	C₂×#117	—	43	4	no	—
188	C₂×#117	—	43	12	no	—
189	C₂×#117	C₂×D₄₃	1	6	yes	—
190	C₂²×C₄⋊C₄	—	43	12	no	—
191	C₂²×C₂²⋊C₄	—	43	12	no	—
192	C₂²×C₄²	—	43	12	no	—
193	C₄₃	C₂×#3428	1	1	yes	characteristic, complemented, Sylow
194	C₂⁵	—	43	3	no	—
195	C₂×#41	—	43	64	no	—
196	C₂×#41	—	43	64	no	—
197	C₂×#41	—	43	256	no	—
198	C₂×#41	—	43	32	no	—
199	C₂×#41	—	43	32	no	—
200	C₂²×Q₈	C₄×D₄₃	1	16	yes	complemented
201	C₂²×Q₈	C₂²×D₄₃	1	4	yes	—
202	C₂²×Q₈	C₂²×D₄₃	1	8	yes	—
203	C₂²×D₄	—	43	48	no	—
204	C₂²×D₄	—	43	12	no	—
205	C₂²×D₄	—	43	24	no	—
206	C₂³×C₄	—	43	1	no	—
207	C₂³×C₄	—	43	48	no	—
208	C₂³×C₄	—	43	6	no	—
209	C₂³×C₄	—	43	24	no	—
210	C₂³×C₄	—	43	6	no	—
211	C₂³×C₄	—	43	24	no	—
212	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
213	C₄×Q₈	—	43	64	no	—
214	C₄×Q₈	C₂²×D₄₃	1	64	yes	complemented
215	C₄×D₄	—	43	192	no	—
216	C₄×D₄	—	43	192	no	—
217	#117	—	43	48	no	—
218	#117	—	43	96	no	—
219	#117	—	43	32	no	—
220	#117	—	43	96	no	—
221	#117	C₂²×D₄₃	1	48	yes	—
222	C₂×C₄⋊C₄	—	43	8	no	—
223	C₂×C₄⋊C₄	—	43	12	no	—
224	C₂×C₄⋊C₄	—	43	96	no	—
225	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
226	C₂×C₂²⋊C₄	—	43	4	no	—
227	C₂×C₂²⋊C₄	—	43	24	no	—
228	C₂×C₂²⋊C₄	—	43	96	no	—
229	C₂×C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
230	C₂×C₄²	—	43	8	no	—
231	C₂×C₄²	—	43	12	no	—
232	C₂×C₄²	—	43	96	no	—
233	C₂×C₄²	C₂²×D₄₃	1	12	yes	—
234	C₂⁴	—	43	6	no	—
235	C₂⁴	—	43	24	no	—
236	C₂⁴	—	43	12	no	—
237	C₂⁴	—	172	12	no	—
238	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
239	#41	—	43	256	no	—
240	#41	—	43	256	no	—
241	C₂×Q₈	C₂×C₄×D₄₃	1	32	yes	—
242	C₂×Q₈	C₂×C₄×D₄₃	1	64	yes	complemented
243	C₂×Q₈	C₂³×D₄₃	1	16	yes	—
244	C₂×D₄	—	43	96	no	—
245	C₂×D₄	—	43	192	no	—
246	C₂×D₄	—	43	48	no	—
247	C₂²×C₄	—	43	8	no	—
248	C₂²×C₄	—	43	4	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	43	48	no	—
252	C₂²×C₄	—	43	48	no	—
253	C₂²×C₄	—	43	192	no	—
254	C₂²×C₄	—	43	24	no	—
255	C₂²×C₄	—	43	24	no	—
256	C₂²×C₄	—	43	8	no	—
257	C₂²×C₄	—	43	96	no	—
258	C₂²×C₄	—	172	48	no	—
259	C₂²×C₄	C₂×C₄×D₄₃	1	24	yes	—
260	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
261	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
262	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
263	C₄⋊C₄	—	43	32	no	—
264	C₄⋊C₄	—	43	48	no	—
265	C₄⋊C₄	C₂³×D₄₃	1	48	yes	—
266	C₂²⋊C₄	—	43	16	no	—
267	C₂²⋊C₄	—	43	96	no	—
268	C₂²⋊C₄	C₂³×D₄₃	1	16	yes	—
269	C₄²	—	43	32	no	—
270	C₄²	—	43	48	no	—
271	C₄²	C₂³×D₄₃	1	48	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	43	24	no	—
274	C₂³	—	43	12	no	—
275	C₂³	—	43	48	no	—
276	C₂³	—	86	24	no	—
277	C₂³	—	172	48	no	—
278	C₂³	—	172	24	no	—
279	C₂³	C₂²×C₄×D₄₃	1	4	yes	—
280	C₂³	C₂⁴×D₄₃	1	2	yes	—
281	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
282	Q₈	C₂²×C₄×D₄₃	1	64	yes	—
283	D₄	—	43	192	no	—
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	43	16	no	—
286	C₂×C₄	—	43	4	no	—
287	C₂×C₄	—	43	4	no	—
288	C₂×C₄	—	43	2	no	—
289	C₂×C₄	—	43	2	no	—
290	C₂×C₄	—	43	96	no	—
291	C₂×C₄	—	43	96	no	—
292	C₂×C₄	—	43	16	no	—
293	C₂×C₄	—	86	16	no	—
294	C₂×C₄	—	172	96	no	—
295	C₂×C₄	C₂²×C₄×D₄₃	1	24	yes	—
296	C₂×C₄	C₂²×C₄×D₄₃	1	48	yes	—
297	C₂×C₄	C₂⁴×D₄₃	1	16	yes	—
298	C₂×C₄	C₂⁴×D₄₃	1	12	yes	—
299	C₂²	—	2	8	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	43	24	no	—
302	C₂²	—	86	48	no	—
303	C₂²	—	86	24	no	—
304	C₂²	—	172	48	no	—
305	C₂²	C₂³×C₄×D₄₃	1	4	yes	—
306	C₂²	C₂³×C₄×D₄₃	1	2	yes	—
307	C₂²	#14583736	1	4	yes	—
308	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
309	C₄	—	2	16	no	—
310	C₄	—	43	4	no	—
311	C₄	—	43	4	no	—
312	C₄	—	86	16	no	—
313	C₄	C₂³×C₄×D₄₃	1	24	yes	—
314	C₂	—	2	4	no	—
315	C₂	—	86	24	no	—
316	C₂	#19147797	1	4	yes	direct product factor
317	C₂	C₂⁴×C₄×D₄₃	1	1	yes	characteristic
318	C₂	C₂×#14583736	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₄₃
commutator	C₂×C₄₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂×C₄₃×#519	C₂
socle	C₂³×C₄₃	C₂⁵
prime radical	C₂³×#10621902	C₂
cosocle	C₂²×C₄₃	C₂⁶
hypercenter	C₂×#519	D₄₃
hypocenter	C₄₃	C₂×#3428
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
2. #19147797	C₂	direct (N × H)
3. C₂×#14583381	C₂	semidirect
4. C₂×#14583377	C₂	semidirect
5. C₂×#14583373	C₂	semidirect
6. C₂×C₄×#10621902	C₂	semidirect
7. C₂×#117×D₄₃	C₂	semidirect
8. C₂×C₄₃×#519	C₂	semidirect
9. C₂²×#10621902	C₄	semidirect
15. #14583381	C₂²	semidirect
16. #14583377	C₂²	semidirect
17. #14583373	C₂²	semidirect
18. C₄×#10621902	C₂²	semidirect
19. C₂×Q₈×Dic₄₃	C₂²	semidirect
20. C₂×C₄×D₁₇₂	C₂²	semidirect
21. C₂×#13063287	C₂²	semidirect
23. C₂×#13063283	C₂²	semidirect
24. C₂×#13063282	C₂²	semidirect
26. #117×D₄₃	C₂²	semidirect
27. C₂×C₄⋊C₄×D₄₃	C₂²	semidirect
29. C₂×C₄²×D₄₃	C₂²	semidirect
31. C₄₃×#519	C₂²	semidirect
32. C₂×C₄×Q₈×C₄₃	C₂²	semidirect
35. C₂×#10621902	C₂×C₄	semidirect
36. C₂×#10621902	C₂×C₄	semidirect
37. C₂×#10621902	C₂×C₄	semidirect
40. C₂²×D₁₇₂	C₂×C₄	semidirect
44. C₂²×C₄×D₄₃	C₂×C₄	semidirect
49. Q₈×Dic₄₃	C₂³	semidirect
50. C₄×D₁₇₂	C₂³	semidirect
51. #13063287	C₂³	semidirect
53. #13063283	C₂³	semidirect
54. #13063282	C₂³	semidirect
58. C₄⋊C₄×D₄₃	C₂³	semidirect
64. C₄²×D₄₃	C₂³	semidirect
65. C₂²×Q₈×C₄₃	C₂×C₄	semidirect
69. C₄×Q₈×C₄₃	C₂³	semidirect
80. C₂×Q₈×C₄₃	C₂²×C₄	semidirect
91. #10621902	C₂²×C₄	semidirect
92. #10621902	C₂²×C₄	semidirect
94. C₂×D₁₇₂	C₂²×C₄	semidirect
99. C₂×C₄×D₄₃	C₂²×C₄	semidirect
159. C₂×#519	D₄₃	semidirect
164. C₂×C₄₃	#3428	semidirect
177. #519	C₂×D₄₃	semidirect
181. C₂×C₄×Q₈	C₂×D₄₃	semidirect
193. C₄₃	C₂×#3428	semidirect
200. C₂²×Q₈	C₄×D₄₃	semidirect
214. C₄×Q₈	C₂²×D₄₃	semidirect
242. C₂×Q₈	C₂×C₄×D₄₃	semidirect
316. C₂	#19147797	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#10621902	2
#19147797	2
C₂×#14583381	2
C₂×#14583377	2
C₂×#14583373	2
C₂×C₄×#10621902	2
C₂×#117×D₄₃	2
C₂×C₄₃×#519	2
C₂×#3428	43

Subgroups — C2×#19147797 #26003194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147797 #26003194