Subgroups — C₂²×C₄×D₃₄₄ #25960600

Order 11008.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₃₄₄	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063247	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₇₂	C₂²	1	2	yes	—
8	C₂³×C₄×D₄₃	—	2	2	no	—
9	C₂²×D₃₄₄	C₄	1	16	yes	direct product factor
10	C₂²×D₃₄₄	C₂²	1	12	yes	—
11	C₂×C₄×D₁₇₂	—	2	12	no	—
12	C₂×C₄×D₁₇₂	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂²×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	C₄×D₃₄₄	C₂²	1	64	yes	direct product factor
16	C₂×#13063259	C₂²	1	12	yes	complemented
17	C₂×#13063247	C₂²	1	24	yes	complemented
18	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₄₃	—	2	2	no	—
23	C₂²×D₁₇₂	—	2	6	no	—
24	C₂²×D₁₇₂	—	2	8	no	—
25	C₂²×D₁₇₂	C₂×C₄	1	16	yes	complemented
26	C₂²×D₁₇₂	C₂³	1	12	yes	—
27	C₂³×Dic₄₃	—	2	2	no	—
28	C₂²×C₄×D₄₃	—	2	24	no	—
29	C₂²×C₄×D₄₃	—	4	16	no	—
30	C₂×D₃₄₄	C₂×C₄	1	96	yes	direct product factor
31	C₂×D₃₄₄	C₂³	1	16	yes	—
32	C₄×D₁₇₂	—	2	48	no	—
33	C₄×D₁₇₂	C₂³	1	32	yes	complemented
34	C₂×C₄⋊Dic₄₃	—	2	6	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	24	no	—
37	#13063259	C₂³	1	16	yes	complemented
38	#13063247	C₂³	1	32	yes	complemented
39	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	—	2	4	no	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
44	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
45	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
46	C₂×C₄²×C₄₃	D₄	1	6	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
48	C₄×C₈×C₄₃	C₂³	1	16	yes	complemented
49	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂³×D₄₃	—	2	16	no	—
51	C₂³×D₄₃	—	2	12	no	—
52	C₂³×D₄₃	—	4	8	no	—
53	C₂³×D₄₃	—	4	8	no	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	D₈	1	8	yes	complemented
56	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₄₃	#41	1	4	yes	—
61	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
62	C₂²×Dic₄₃	—	2	12	no	—
63	C₂²×Dic₄₃	—	4	8	no	—
64	C₂×D₁₇₂	—	2	48	no	—
65	C₂×D₁₇₂	—	2	12	no	—
66	C₂×D₁₇₂	—	2	24	no	—
67	C₂×D₁₇₂	C₂²×C₄	1	48	yes	complemented
68	C₂×D₁₇₂	C₂⁴	1	8	yes	—
69	C₂×C₄×D₄₃	—	2	32	no	—
70	C₂×C₄×D₄₃	—	4	96	no	—
71	C₂×C₈×C₄₃	—	2	12	no	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	24	yes	complemented
73	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
74	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
75	C₄²×C₄₃	C₂×D₄	1	12	yes	—
76	C₄²×C₄₃	C₂⁴	1	4	yes	—
77	(C₂×D₄₃)⋊C₄	—	2	32	no	—
78	C₄⋊Dic₄₃	—	2	12	no	—
79	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
80	D₃₄₄	C₂²×C₄	1	64	yes	direct product factor
81	C₂³×C₄₃	C₄×D₄	1	4	yes	—
82	C₂³×C₄₃	C₂×D₈	1	4	yes	—
83	C₂³×C₄₃	#135	1	4	yes	—
84	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
85	C₂²×D₄₃	—	2	48	no	—
86	C₂²×D₄₃	—	2	8	no	—
87	C₂²×D₄₃	—	4	24	no	—
88	C₂²×D₄₃	—	4	24	no	—
89	C₂²×D₄₃	—	4	64	no	—
90	C₂×C₄×C₄₃	—	2	12	no	—
91	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
92	C₂×C₄×C₄₃	C₂×D₈	1	24	yes	complemented
93	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
94	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
95	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
97	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
98	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
99	C₂×Dic₄₃	—	2	8	no	—
100	C₂×Dic₄₃	—	4	24	no	—
101	D₁₇₂	—	2	48	no	—
102	D₁₇₂	—	2	8	no	—
103	D₁₇₂	C₂³×C₄	1	16	yes	complemented
104	C₄×D₄₃	—	4	64	no	—
105	C₈×C₄₃	—	2	4	no	—
106	C₈×C₄₃	C₂³×C₄	1	8	yes	complemented
107	C₂²×C₄×D₈	—	43	1	no	maximal, Sylow
108	C₂²×C₄₃	C₄×D₈	1	16	yes	complemented
109	C₂²×C₄₃	C₂×C₄×D₄	1	6	yes	—
110	C₂²×C₄₃	C₂²×D₈	1	6	yes	—
111	C₂²×C₄₃	C₂×#135	1	6	yes	—
112	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
113	C₂×D₄₃	—	2	16	no	—
114	C₂×D₄₃	—	4	96	no	—
115	C₂×D₄₃	—	4	8	no	—
116	C₂×D₄₃	—	4	8	no	—
117	C₄×C₄₃	—	2	4	no	—
118	C₄×C₄₃	C₂×C₄×D₄	1	6	yes	—
119	C₄×C₄₃	C₂²×D₈	1	8	yes	complemented
120	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
122	Dic₄₃	—	4	8	no	—
123	C₂³×D₈	—	43	1	no	—
124	C₂²×C₄×D₄	—	43	2	no	—
125	C₂×C₄×D₈	—	43	24	no	—
126	C₂²×#107	—	43	1	no	—
127	C₂²×#102	—	43	2	no	—
128	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
129	C₂×C₄₃	C₂×C₄×D₈	1	12	yes	complemented
130	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
131	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
132	C₂×C₄₃	C₂²×#135	1	1	yes	characteristic
133	D₄₃	—	4	16	no	—
134	C₂³×D₄	—	43	2	no	—
135	C₂⁴×C₄	—	86	2	no	—
136	C₂²×D₈	—	43	12	no	—
137	C₂²×D₈	—	43	16	no	—
138	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
139	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
140	C₂×C₄×D₄	—	43	24	no	—
141	C₂×C₄×D₄	—	86	12	no	—
142	C₂²×C₄⋊C₄	—	43	2	no	—
143	C₂²×C₂²⋊C₄	—	86	2	no	—
144	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
145	C₄×D₈	—	43	64	no	—
146	C₂×#107	—	43	12	no	—
147	C₂×#102	—	43	24	no	—
148	C₂×C₄×C₈	C₂×D₄₃	1	12	yes	complemented
149	C₄₃	C₂²×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
150	C₂⁵	—	86	2	no	—
151	C₂²×D₄	—	43	16	no	—
152	C₂²×D₄	—	43	12	no	—
153	C₂²×D₄	—	86	6	no	—
154	C₂²×D₄	—	86	8	no	—
155	C₂³×C₄	—	86	2	no	—
156	C₂³×C₄	—	86	24	no	—
157	C₂³×C₄	—	172	16	no	—
158	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
159	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
160	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
161	C₂×D₈	—	43	96	no	—
162	C₂×D₈	—	43	16	no	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	C₄×D₄₃	1	8	yes	complemented
165	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
166	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
167	C₄×D₄	—	43	32	no	—
168	C₄×D₄	—	86	48	no	—
169	C₂×C₄⋊C₄	—	43	12	no	—
170	C₂×C₄⋊C₄	—	86	6	no	—
171	C₂×C₂²⋊C₄	—	86	24	no	—
172	C₂×C₄²	D₁₇₂	1	6	yes	—
173	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
174	#107	—	43	16	no	—
175	#102	—	43	32	no	—
176	C₄×C₈	C₂²×D₄₃	1	16	yes	complemented
177	C₂⁴	—	86	16	no	—
178	C₂⁴	—	86	12	no	—
179	C₂⁴	—	172	8	no	—
180	C₂⁴	—	172	8	no	—
181	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
182	C₂×D₄	—	43	48	no	—
183	C₂×D₄	—	43	8	no	—
184	C₂×D₄	—	86	48	no	—
185	C₂×D₄	—	86	12	no	—
186	C₂×D₄	—	86	24	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	86	12	no	—
189	C₂²×C₄	—	86	32	no	—
190	C₂²×C₄	—	172	8	no	—
191	C₂²×C₄	—	172	96	no	—
192	C₂²×C₄	D₃₄₄	1	8	yes	direct product factor
193	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
194	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
195	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
196	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
197	C₂²×C₄	#10621898	1	4	yes	—
198	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
199	D₈	—	43	64	no	—
200	C₂×C₈	—	2	12	no	—
201	C₂×C₈	C₂×C₄×D₄₃	1	24	yes	complemented
202	C₂×C₈	C₂³×D₄₃	1	4	yes	—
203	C₂×C₈	C₂³×D₄₃	1	4	yes	—
204	C₄⋊C₄	—	43	8	no	—
205	C₄⋊C₄	—	86	12	no	—
206	C₂²⋊C₄	—	86	32	no	—
207	C₄²	C₂×D₁₇₂	1	12	yes	—
208	C₄²	C₂³×D₄₃	1	4	yes	—
209	C₂³	—	86	48	no	—
210	C₂³	—	86	8	no	—
211	C₂³	—	172	24	no	—
212	C₂³	—	172	24	no	—
213	C₂³	—	172	64	no	—
214	C₂³	C₄×D₁₇₂	1	4	yes	—
215	C₂³	C₂×D₃₄₄	1	4	yes	—
216	C₂³	#13063347	1	4	yes	—
217	C₂³	C₂²×D₁₇₂	1	3	yes	—
218	D₄	—	43	16	no	—
219	D₄	—	86	48	no	—
220	D₄	—	86	8	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	86	8	no	—
223	C₂×C₄	—	172	24	no	—
224	C₂×C₄	—	172	64	no	—
225	C₂×C₄	C₄×D₁₇₂	1	12	yes	—
226	C₂×C₄	C₂×D₃₄₄	1	24	yes	direct product factor
227	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
228	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
229	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
230	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
231	C₂×C₄	C₂×#10621898	1	6	yes	—
232	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	C₂²×C₄×D₄₃	1	8	yes	complemented
235	C₂²	—	86	16	no	—
236	C₂²	—	172	8	no	—
237	C₂²	—	172	96	no	—
238	C₂²	—	172	8	no	—
239	C₂²	C₄×D₃₄₄	1	16	yes	direct product factor
240	C₂²	C₂×C₄×D₁₇₂	1	6	yes	—
241	C₂²	C₂²×D₃₄₄	1	6	yes	—
242	C₂²	C₂×#13063347	1	6	yes	—
243	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	172	8	no	—
246	C₄	C₂×C₄×D₁₇₂	1	6	yes	—
247	C₄	C₂²×D₃₄₄	1	8	yes	direct product factor
248	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
249	C₄	C₂²×#10621898	1	1	yes	characteristic
250	C₂	—	172	16	no	—
251	C₂	C₂×C₄×D₃₄₄	1	12	yes	direct product factor
252	C₂	C₂²×C₄×D₁₇₂	1	1	yes	characteristic
253	C₂	C₂³×D₃₄₄	1	1	yes	characteristic
254	C₂	C₂²×#13063347	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₁₇₂
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂³×D₃₄₄	C₂
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂²×C₄×D₈
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₁₇₂	C₂	semidirect
3. C₂×C₄×D₃₄₄	C₂	direct (N × H)
4. C₂²×#13063259	C₂	semidirect
5. C₂²×#13063247	C₂	semidirect
6. C₂²×C₄×C₈×C₄₃	C₂	semidirect
9. C₂²×D₃₄₄	C₄	direct (N × H)
12. C₂×C₄×D₁₇₂	C₂²	semidirect
15. C₄×D₃₄₄	C₂²	direct (N × H)
16. C₂×#13063259	C₂²	semidirect
17. C₂×#13063247	C₂²	semidirect
21. C₂×C₄×C₈×C₄₃	C₂²	semidirect
25. C₂²×D₁₇₂	C₂×C₄	semidirect
30. C₂×D₃₄₄	C₂×C₄	direct (N × H)
33. C₄×D₁₇₂	C₂³	semidirect
37. #13063259	C₂³	semidirect
38. #13063247	C₂³	semidirect
43. C₂²×C₈×C₄₃	C₂×C₄	semidirect
48. C₄×C₈×C₄₃	C₂³	semidirect
55. C₂²×C₄×C₄₃	D₈	semidirect
67. C₂×D₁₇₂	C₂²×C₄	semidirect
72. C₂×C₈×C₄₃	C₂²×C₄	semidirect
80. D₃₄₄	C₂²×C₄	direct (N × H)
92. C₂×C₄×C₄₃	C₂×D₈	semidirect
103. D₁₇₂	C₂³×C₄	semidirect
106. C₈×C₄₃	C₂³×C₄	semidirect
108. C₂²×C₄₃	C₄×D₈	semidirect
119. C₄×C₄₃	C₂²×D₈	semidirect
128. C₂²×C₄×C₈	D₄₃	semidirect
129. C₂×C₄₃	C₂×C₄×D₈	semidirect
148. C₂×C₄×C₈	C₂×D₄₃	semidirect
149. C₄₃	C₂²×C₄×D₈	semidirect
164. C₂²×C₈	C₄×D₄₃	semidirect
176. C₄×C₈	C₂²×D₄₃	semidirect
192. C₂²×C₄	D₃₄₄	direct (N × H)
201. C₂×C₈	C₂×C₄×D₄₃	semidirect
226. C₂×C₄	C₂×D₃₄₄	direct (N × H)
234. C₈	C₂²×C₄×D₄₃	semidirect
239. C₂²	C₄×D₃₄₄	direct (N × H)
247. C₄	C₂²×D₃₄₄	direct (N × H)
251. C₂	C₂×C₄×D₃₄₄	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×D₃₄₄	2
C₂²×C₄×D₁₇₂	2
C₂×C₄×D₃₄₄	2
C₂²×#13063259	2
C₂²×#13063247	2
C₂²×C₄×C₈×C₄₃	2
C₂²×C₄×D₈	43

Subgroups — C22×C4×D344 #25960600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₃₄₄ #25960600