Subgroups — C₄×D₄×#89137 #28379466

Order 13952.

924 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×#89137	C₂	1	2	yes	maximal
2	C₂²×C₄×#89137	C₂	1	2	yes	maximal, complemented
3	C₄⋊C₄×#89137	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×#89137	C₂	1	4	yes	maximal, complemented
6	C₄×C₂²⋊#89137	C₂	1	2	yes	maximal, complemented
7	C₄²×#89137	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×#89137	C₂²	1	4	yes	—
10	D₄×#89137	C₄	1	16	yes	direct product factor
11	C₂×C₄⋊#89137	C₂²	1	2	yes	—
12	C₂×C₂²⋊#89137	C₂²	1	4	yes	—
13	C₂×C₄×#89137	—	2	8	no	—
14	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
15	C₂×C₄×#89137	C₂²	1	2	yes	—
16	C₂×C₄×#89137	C₂²	1	2	yes	—
17	C₂×C₄×#89137	C₂²	1	4	yes	—
18	C₂×C₄×#89137	C₂²	1	2	yes	—
19	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
20	(C₂×#89137)⋊C₄	C₂²	1	4	yes	—
21	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
22	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
24	C₄×D₄₃₆	C₄	1	1	yes	characteristic, complemented
25	D₄×Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
26	#16812661	C₄	1	1	yes	characteristic, complemented
27	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	complemented
28	#16812659	C₄	1	2	yes	complemented
29	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×D₁₀₉	C₂²	1	2	yes	—
31	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
33	C₂²×#89137	—	2	8	no	—
34	C₂²×#89137	—	2	8	no	—
35	C₂²×#89137	C₂×C₄	1	16	yes	complemented
36	C₂²×#89137	C₂³	1	2	yes	—
37	C₂²×#89137	C₂³	1	4	yes	—
38	C₂²×#89137	C₂³	1	2	yes	—
39	C₄⋊#89137	C₂×C₄	1	8	yes	complemented
40	C₂²⋊#89137	C₂×C₄	1	16	yes	complemented
41	C₄×#89137	—	2	8	no	—
42	C₄×#89137	—	2	4	no	—
43	C₄×#89137	—	2	4	no	—
44	C₄×#89137	—	2	2	no	—
45	C₄×#89137	C₂×C₄	1	8	yes	complemented
46	C₄×#89137	D₄	1	4	yes	direct product factor
47	C₂³×D₁₀₉	C₂³	1	2	yes	—
48	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
50	D₄×D₁₀₉	C₂×C₄	1	8	yes	—
51	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	complemented
52	C₂×C₄×D₁₀₉	—	2	4	no	—
53	C₂×C₄×D₁₀₉	—	2	4	no	—
54	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	complemented
55	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	complemented
56	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
58	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
61	#13182008	C₂×C₄	1	2	yes	complemented
62	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	complemented
63	#13182006	C₂×C₄	1	2	yes	complemented
64	#13182006	C₂×C₄	1	2	yes	complemented
65	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
66	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
67	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
70	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
71	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
72	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂²×D₁₀₉	—	2	2	no	—
75	C₂²×D₁₀₉	—	2	2	no	—
76	C₂²×D₁₀₉	—	2	4	no	—
77	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₉	C₂²×C₄	1	8	yes	—
79	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₂×#89137	—	2	4	no	—
81	C₂×#89137	—	2	2	no	—
82	C₂×#89137	—	2	2	no	—
83	C₂×#89137	—	2	4	no	—
84	C₂×#89137	—	2	32	no	—
85	C₂×#89137	C₂²×C₄	1	4	yes	—
86	C₂×#89137	C₂²×C₄	1	8	yes	—
87	C₂×#89137	C₂²×C₄	1	4	yes	—
88	C₂×#89137	C₂×D₄	1	4	yes	—
89	C₂×#89137	#41	1	4	yes	—
90	D₄×C₁₀₉	C₄²	1	4	yes	complemented
91	C₂×C₄×C₁₀₉	—	2	4	no	—
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₁₀₉	C₄²	1	8	yes	complemented
97	C₂×Dic₁₀₉	—	2	4	no	—
98	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
102	D₄₃₆	C₄²	1	4	yes	complemented
103	C₄×D₁₀₉	—	2	2	no	—
104	C₄×D₁₀₉	—	2	2	no	—
105	C₄×D₁₀₉	—	2	2	no	—
106	C₄×D₁₀₉	—	2	4	no	—
107	C₄×D₁₀₉	—	2	4	no	—
108	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
109	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
110	C₄×D₁₀₉	#41	1	2	yes	—
111	C₂²×C₁₀₉	—	2	2	no	—
112	C₂²×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	C₂×C₄²	1	4	yes	complemented
114	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₀₉	—	2	2	no	—
116	C₂×D₁₀₉	—	2	2	no	—
117	C₂×D₁₀₉	—	2	8	no	—
118	C₂×D₁₀₉	—	2	8	no	—
119	C₂×D₁₀₉	C₂×C₄²	1	4	yes	complemented
120	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
121	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₁₀₉	C₂×#41	1	2	yes	—
123	C₂×D₁₀₉	C₂×#41	1	1	yes	characteristic
124	#89137	—	2	8	no	—
125	#89137	—	2	4	no	—
126	#89137	C₄×D₄	1	8	yes	direct product factor
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	C₂×C₄²	1	2	yes	complemented
130	C₄×C₁₀₉	C₄×D₄	1	2	yes	complemented
131	Dic₁₀₉	—	2	2	no	—
132	Dic₁₀₉	—	2	1	no	—
133	Dic₁₀₉	C₂×C₄²	1	2	yes	complemented
134	Dic₁₀₉	C₄×D₄	1	2	yes	complemented
135	C₂×C₁₀₉	—	2	4	no	—
136	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
137	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
139	D₁₀₉	—	2	4	no	—
140	D₁₀₉	C₂×C₄×D₄	1	2	yes	—
141	D₁₀₉	C₄×#41	1	2	yes	—
142	C₄²×D₄	—	109	1	no	maximal, Sylow
143	C₁₀₉	C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×D₄	—	109	2	no	—
145	C₂×C₄×D₄	—	109	1	no	—
146	C₂²×C₄²	—	109	2	no	—
147	C₄×C₄⋊C₄	—	109	1	no	—
148	C₄×C₄⋊C₄	—	109	2	no	—
149	C₄×C₂²⋊C₄	—	109	2	no	—
150	C₄×C₂²⋊C₄	—	109	4	no	—
151	C₄³	—	109	1	no	—
152	C₂²×D₄	—	109	1	no	—
153	C₂³×C₄	—	109	4	no	—
154	C₂³×C₄	—	109	2	no	—
155	C₄×D₄	—	109	16	no	—
156	C₄×D₄	—	109	1	no	—
157	C₄×D₄	—	109	2	no	—
158	C₄×D₄	—	109	1	no	—
159	C₄×D₄	—	109	2	no	—
160	C₄×D₄	—	109	1	no	—
161	C₄×D₄	#89137	1	1	yes	characteristic, direct product factor
162	C₂×C₄⋊C₄	—	109	2	no	—
163	C₂×C₄⋊C₄	—	109	1	no	—
164	C₂×C₂²⋊C₄	—	109	4	no	—
165	C₂×C₂²⋊C₄	—	109	2	no	—
166	C₂×C₄²	—	109	2	no	—
167	C₂×C₄²	—	109	1	no	—
168	C₂×C₄²	—	109	1	no	—
169	C₂×C₄²	—	109	2	no	—
170	C₂×C₄²	—	109	1	no	—
171	C₂×C₄²	—	109	4	no	—
172	C₂×C₄²	—	109	2	no	—
173	C₂×C₄²	—	218	8	no	—
174	#95	—	109	4	no	—
175	#95	—	109	2	no	—
176	C₂⁴	—	109	2	no	—
177	C₂×D₄	—	109	2	no	—
178	C₂×D₄	—	109	8	no	—
179	C₂×D₄	—	109	1	no	—
180	C₂×D₄	C₂×#89137	1	1	yes	characteristic
181	C₂²×C₄	—	109	2	no	—
182	C₂²×C₄	—	109	1	no	—
183	C₂²×C₄	—	109	1	no	—
184	C₂²×C₄	—	109	16	no	—
185	C₂²×C₄	—	109	2	no	—
186	C₂²×C₄	—	109	4	no	—
187	C₂²×C₄	—	109	2	no	—
188	C₂²×C₄	—	109	2	no	—
189	C₂²×C₄	—	109	1	no	—
190	C₂²×C₄	—	109	2	no	—
191	C₂²×C₄	—	109	2	no	—
192	C₂²×C₄	—	218	4	no	—
193	C₂²×C₄	—	218	4	no	—
194	C₂²×C₄	—	218	8	no	—
195	C₂²×C₄	—	218	8	no	—
196	C₂²×C₄	C₂×#89137	1	2	yes	complemented
197	C₄⋊C₄	—	109	8	no	—
198	C₄⋊C₄	—	109	2	no	—
199	C₄⋊C₄	—	109	1	no	—
200	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic, complemented
201	C₂²⋊C₄	—	109	16	no	—
202	C₂²⋊C₄	—	109	2	no	—
203	C₂²⋊C₄	—	109	2	no	—
204	C₂²⋊C₄	—	109	2	no	—
205	C₂²⋊C₄	C₂×#89137	1	2	yes	complemented
206	C₄²	—	109	8	no	—
207	C₄²	—	109	4	no	—
208	C₄²	—	109	1	no	—
209	C₄²	—	109	1	no	—
210	C₄²	—	109	1	no	—
211	C₄²	—	218	4	no	—
212	C₄²	—	218	2	no	—
213	C₄²	—	218	8	no	—
214	C₄²	—	218	4	no	—
215	C₄²	C₂×#89137	1	1	yes	characteristic, complemented
216	C₂³	—	109	1	no	—
217	C₂³	—	109	8	no	—
218	C₂³	—	109	2	no	—
219	C₂³	—	218	2	no	—
220	C₂³	—	218	2	no	—
221	C₂³	—	218	4	no	—
222	C₂³	C₂²×#89137	1	2	yes	—
223	D₄	—	109	8	no	—
224	D₄	—	109	4	no	—
225	D₄	C₄×#89137	1	4	yes	direct product factor
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	109	4	no	—
228	C₂×C₄	—	109	4	no	—
229	C₂×C₄	—	109	4	no	—
230	C₂×C₄	—	109	1	no	—
231	C₂×C₄	—	109	1	no	—
232	C₂×C₄	—	109	4	no	—
233	C₂×C₄	—	109	2	no	—
234	C₂×C₄	—	109	2	no	—
235	C₂×C₄	—	109	4	no	—
236	C₂×C₄	—	109	8	no	—
237	C₂×C₄	—	109	1	no	—
238	C₂×C₄	—	109	2	no	—
239	C₂×C₄	—	218	4	no	—
240	C₂×C₄	—	218	2	no	—
241	C₂×C₄	—	218	2	no	—
242	C₂×C₄	—	218	2	no	—
243	C₂×C₄	—	218	4	no	—
244	C₂×C₄	—	218	32	no	—
245	C₂×C₄	—	218	4	no	—
246	C₂×C₄	—	218	2	no	—
247	C₂×C₄	—	218	2	no	—
248	C₂×C₄	—	218	4	no	—
249	C₂×C₄	—	218	4	no	—
250	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
251	C₂×C₄	C₂²×#89137	1	2	yes	—
252	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
253	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	109	2	no	—
257	C₂²	—	109	1	no	—
258	C₂²	—	109	1	no	—
259	C₂²	—	109	2	no	—
260	C₂²	—	109	4	no	—
261	C₂²	—	218	8	no	—
262	C₂²	—	218	8	no	—
263	C₂²	—	218	2	no	—
264	C₂²	—	218	2	no	—
265	C₂²	C₂×C₄×#89137	1	4	yes	complemented
266	C₂²	C₂³×#89137	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	109	8	no	—
270	C₄	—	109	2	no	—
271	C₄	—	109	2	no	—
272	C₄	—	218	8	no	—
273	C₄	—	218	4	no	—
274	C₄	—	218	2	no	—
275	C₄	—	218	1	no	—
276	C₄	C₂×C₄×#89137	1	2	yes	complemented
277	C₄	D₄×#89137	1	2	yes	direct product factor
278	C₂	—	2	4	no	—
279	C₂	—	109	2	no	—
280	C₂	—	109	2	no	—
281	C₂	—	218	4	no	—
282	C₂	C₂²×C₄×#89137	1	1	yes	characteristic
283	C₂	C₂×D₄×#89137	1	1	yes	characteristic
284	C₂	#41×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂²	C₂³×#89137
Fitting	C₄×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×D₄×D₁₀₉	C₂²
cosocle	C₂²×D₁₀₉	C₂⁴
hypercenter	C₄×D₄	#89137
hypocenter	C₁₀₉	C₄²×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×#89137	C₂	semidirect
3. C₄⋊C₄×#89137	C₂	semidirect
4. C₄×C₄⋊#89137	C₂	semidirect
5. C₂²⋊C₄×#89137	C₂	semidirect
6. C₄×C₂²⋊#89137	C₂	semidirect
7. C₄²×#89137	C₂	semidirect
10. D₄×#89137	C₄	direct (N × H)
24. C₄×D₄₃₆	C₄	semidirect
25. D₄×Dic₁₀₉	C₄	semidirect
26. #16812661	C₄	semidirect
27. C₄×C₂²⋊D₁₀₉	C₄	semidirect
28. #16812659	C₄	semidirect
32. C₄×D₄×C₁₀₉	C₄	semidirect
35. C₂²×#89137	C₂×C₄	semidirect
39. C₄⋊#89137	C₂×C₄	semidirect
40. C₂²⋊#89137	C₂×C₄	semidirect
45. C₄×#89137	C₂×C₄	semidirect
46. C₄×#89137	D₄	direct (N × H)
51. C₂²×Dic₁₀₉	C₂×C₄	semidirect
54. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
55. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
60. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
61. #13182008	C₂×C₄	semidirect
62. C₂²⋊Dic₁₀₉	C₂×C₄	semidirect
63. #13182006	C₂×C₄	semidirect
64. #13182006	C₂×C₄	semidirect
65. C₄×Dic₁₀₉	C₂×C₄	semidirect
66. C₄×Dic₁₀₉	C₂×C₄	semidirect
67. C₄×Dic₁₀₉	C₂×C₄	semidirect
69. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
70. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
71. C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
72. C₄²×C₁₀₉	C₂×C₄	semidirect
90. D₄×C₁₀₉	C₄²	semidirect
96. C₂²⋊D₁₀₉	C₄²	semidirect
102. D₄₃₆	C₄²	semidirect
113. C₂²×C₁₀₉	C₂×C₄²	semidirect
119. C₂×D₁₀₉	C₂×C₄²	semidirect
126. #89137	C₄×D₄	direct (N × H)
129. C₄×C₁₀₉	C₂×C₄²	semidirect
130. C₄×C₁₀₉	C₄×D₄	semidirect
133. Dic₁₀₉	C₂×C₄²	semidirect
134. Dic₁₀₉	C₄×D₄	semidirect
143. C₁₀₉	C₄²×D₄	semidirect
161. C₄×D₄	#89137	direct (N × H)
196. C₂²×C₄	C₂×#89137	semidirect
200. C₄⋊C₄	C₂×#89137	semidirect
205. C₂²⋊C₄	C₂×#89137	semidirect
215. C₄²	C₂×#89137	semidirect
225. D₄	C₄×#89137	direct (N × H)
265. C₂²	C₂×C₄×#89137	semidirect
276. C₄	C₂×C₄×#89137	semidirect
277. C₄	D₄×#89137	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×D₄×#89137	2
C₂²×C₄×#89137	2
C₄⋊C₄×#89137	2
C₄×C₄⋊#89137	2
C₂²⋊C₄×#89137	2
C₄×C₂²⋊#89137	2
C₄²×#89137	2
C₄×D₄×D₁₀₉	2
C₄²×D₄	109

Subgroups — C4×D4×#89137 #28379466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×#89137 #28379466