Subgroups — C₂×Q₈×D₁₀₄ #16743334

Order 3328.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₅₂	C₂	1	2	yes	maximal, complemented
2	Q₈×D₁₀₄	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	3	yes	maximal, complemented
4	C₂×#11718507	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₁₀₄	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₁₃	—	2	2	no	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	Q₈×D₅₂	—	2	8	no	—
11	Q₈×D₅₂	C₂²	1	16	yes	complemented
12	C₂×#88151	—	2	6	no	—
13	C₂×C₄×D₅₂	C₂²	1	6	yes	—
14	C₂×#88123	C₂²	1	6	yes	—
15	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
18	C₂×#88061	C₂²	1	6	yes	—
19	C₂×#88058	C₂²	1	3	yes	—
20	C₂×#88058	C₂²	1	6	yes	—
21	#11718507	C₂²	1	48	yes	complemented
22	C₄×D₁₀₄	C₂²	1	24	yes	—
23	#11718343	C₂²	1	24	yes	complemented
24	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂×Q₈×D₁₃	—	2	16	no	—
26	C₂×Q₈×D₁₃	—	4	16	no	—
27	C₂²×D₅₂	C₂³	1	2	yes	—
28	C₂²×C₄×D₁₃	—	2	6	no	—
29	C₂²×Dic₂₆	—	2	6	no	—
30	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	D₄	1	4	yes	—
33	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
36	C₂×#6103	—	2	6	no	—
37	C₂×C₄⋊Dic₁₃	—	2	6	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
39	C₂×D₁₀₄	Q₈	1	8	yes	direct product factor
40	C₂×D₁₀₄	C₂³	1	4	yes	—
41	#88151	—	2	48	no	—
42	C₄×D₅₂	—	2	12	no	—
43	C₄×D₅₂	C₂³	1	24	yes	—
44	#88123	—	2	12	no	—
45	#88123	C₂³	1	24	yes	—
46	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
47	C₄×C₈×C₁₃	C₂³	1	12	yes	—
48	#88061	C₂³	1	24	yes	—
49	#88058	C₂³	1	24	yes	—
50	#88058	C₂³	1	12	yes	—
51	C₂×Q₈×D₈	—	13	1	no	maximal, Sylow
52	C₂³×D₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	D₈	1	8	yes	complemented
54	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	2	6	no	—
59	Q₈×D₁₃	—	4	64	no	—
60	C₂×D₅₂	—	2	2	no	—
61	C₂×D₅₂	—	2	4	no	—
62	C₂×D₅₂	C₂×Q₈	1	8	yes	complemented
63	C₂×D₅₂	C₂⁴	1	4	yes	—
64	C₂×C₄×D₁₃	—	2	24	no	—
65	C₂×C₄×D₁₃	—	4	24	no	—
66	C₂×Dic₂₆	—	2	24	no	—
67	C₂×Dic₂₆	—	4	24	no	—
68	C₂×C₈×C₁₃	—	2	6	no	—
69	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	complemented
70	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
71	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
72	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
73	C₄²×C₁₃	C₂×D₄	1	6	yes	—
74	C₄²×C₁₃	C₂⁴	1	6	yes	—
75	#6103	—	2	24	no	—
76	C₄⋊Dic₁₃	—	2	6	no	—
77	C₄⋊Dic₁₃	—	2	24	no	—
78	C₄⋊Dic₁₃	C₂⁴	1	12	yes	—
79	D₁₀₄	C₂×Q₈	1	16	yes	direct product factor
80	C₂×D₄×Q₈	—	13	2	no	—
81	Q₈×D₈	—	13	16	no	—
82	C₂×#500	—	13	3	no	—
83	C₂×#474	—	13	6	no	—
84	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
85	C₂×C₄×D₈	—	13	3	no	—
86	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	8	no	—
88	C₂²×D₁₃	—	2	4	no	—
89	C₂²×D₁₃	—	4	4	no	—
90	C₂²×D₁₃	—	4	4	no	—
91	Q₈×C₁₃	C₂×D₈	1	16	yes	complemented
92	C₂×C₄×C₁₃	—	2	6	no	—
93	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
95	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
97	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
99	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₁₃	—	2	12	no	—
101	C₂×Dic₁₃	—	4	12	no	—
102	D₅₂	—	2	8	no	—
103	D₅₂	—	2	4	no	—
104	D₅₂	C₂²×Q₈	1	8	yes	complemented
105	C₄×D₁₃	—	4	48	no	—
106	Dic₂₆	—	4	48	no	—
107	C₈×C₁₃	—	2	6	no	—
108	C₈×C₁₃	C₂²×Q₈	1	4	yes	complemented
109	C₂³×Q₈	—	26	2	no	—
110	C₂²×D₈	—	13	1	no	—
111	D₄×Q₈	—	13	16	no	—
112	D₄×Q₈	—	26	8	no	—
113	C₂×C₄⋊Q₈	—	13	6	no	—
114	C₂×C₂²⋊Q₈	—	26	6	no	—
115	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
116	C₂×C₄×D₄	—	13	6	no	—
117	#500	—	13	24	no	—
118	#474	—	13	48	no	—
119	C₈×Q₈	C₂×D₁₃	1	8	yes	complemented
120	C₄×D₈	—	13	24	no	—
121	C₂×#107	—	13	3	no	—
122	C₂×#107	—	13	6	no	—
123	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
124	C₂×#102	—	13	6	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
126	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
127	C₂²×C₁₃	C₂²×D₈	1	2	yes	—
128	C₂²×C₁₃	#578	1	2	yes	—
129	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
130	C₂×D₁₃	—	2	8	no	—
131	C₂×D₁₃	—	4	16	no	—
132	C₂×D₁₃	—	4	4	no	—
133	C₂×D₁₃	—	4	4	no	—
134	C₄×C₁₃	—	2	6	no	—
135	C₄×C₁₃	D₄×Q₈	1	2	yes	—
136	C₄×C₁₃	C₂²×D₈	1	12	yes	—
137	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
138	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	Dic₁₃	—	4	12	no	—
140	C₂²×Q₈	—	26	16	no	—
141	C₂²×Q₈	—	26	6	no	—
142	C₂²×Q₈	—	52	16	no	—
143	C₂²×Q₈	D₅₂	1	1	yes	characteristic
144	C₂²×D₄	—	13	2	no	—
145	C₂³×C₄	—	26	6	no	—
146	C₂×D₈	—	13	8	no	—
147	C₂×D₈	—	13	4	no	—
148	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
149	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	13	24	no	—
151	C₄⋊Q₈	—	26	12	no	—
152	C₂²⋊Q₈	—	26	48	no	—
153	C₄×Q₈	D₅₂	1	4	yes	—
154	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
155	C₄×D₄	—	13	24	no	—
156	C₄×D₄	—	26	12	no	—
157	C₂×C₄⋊C₄	—	13	6	no	—
158	C₂×C₄⋊C₄	—	26	6	no	—
159	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
160	C₂×C₂²⋊C₄	—	26	6	no	—
161	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
162	#107	—	13	12	no	—
163	#107	—	13	24	no	—
164	C₄⋊C₈	C₂²×D₁₃	1	12	yes	—
165	#102	—	13	24	no	—
166	C₄×C₈	C₂²×D₁₃	1	12	yes	—
167	C₂×C₁₃	Q₈×D₈	1	4	yes	complemented
168	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
169	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#578	1	1	yes	characteristic
171	D₁₃	—	4	8	no	—
172	C₂⁴	—	26	2	no	—
173	C₂×Q₈	—	26	24	no	—
174	C₂×Q₈	—	52	64	no	—
175	C₂×Q₈	—	52	24	no	—
176	C₂×Q₈	D₁₀₄	1	8	yes	direct product factor
177	C₂×Q₈	C₂×D₅₂	1	4	yes	—
178	C₂×D₄	—	13	8	no	—
179	C₂×D₄	—	13	4	no	—
180	C₂×D₄	—	26	2	no	—
181	C₂×D₄	—	26	4	no	—
182	C₂²×C₄	—	26	6	no	—
183	C₂²×C₄	—	26	24	no	—
184	C₂²×C₄	—	52	24	no	—
185	C₂²×C₄	C₂×D₅₂	1	3	yes	—
186	C₂²×C₄	C₂×D₅₂	1	3	yes	—
187	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
188	D₈	—	13	16	no	—
189	C₂×C₈	—	2	6	no	—
190	C₂×C₈	Q₈×D₁₃	1	4	yes	complemented
191	C₂×C₈	C₂³×D₁₃	1	6	yes	—
192	C₂×C₈	C₂³×D₁₃	1	2	yes	—
193	C₄⋊C₄	—	13	12	no	—
194	C₄⋊C₄	—	26	6	no	—
195	C₄⋊C₄	—	26	24	no	—
196	C₄⋊C₄	C₂×D₅₂	1	12	yes	—
197	C₂²⋊C₄	—	26	24	no	—
198	C₄²	C₂×D₅₂	1	6	yes	—
199	C₄²	C₂³×D₁₃	1	6	yes	—
200	C₁₃	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	26	8	no	—
202	C₂³	—	26	4	no	—
203	C₂³	—	52	4	no	—
204	C₂³	—	52	4	no	—
205	C₂³	C₂²×D₅₂	1	1	yes	characteristic
206	Q₈	—	52	48	no	—
207	Q₈	C₂×D₁₀₄	1	16	yes	direct product factor
208	D₄	—	13	8	no	—
209	D₄	—	26	8	no	—
210	D₄	—	26	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	26	12	no	—
213	C₂×C₄	—	52	12	no	—
214	C₂×C₄	—	52	48	no	—
215	C₂×C₄	C₂×D₁₀₄	1	12	yes	—
216	C₂×C₄	C₂²×D₅₂	1	6	yes	—
217	C₂×C₄	C₂²×D₅₂	1	6	yes	—
218	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
220	C₂×C₄	#88257	1	2	yes	—
221	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₁₃	1	4	yes	complemented
224	C₂²	—	26	8	no	—
225	C₂²	—	52	16	no	—
226	C₂²	—	52	4	no	—
227	C₂²	—	52	4	no	—
228	C₂²	Q₈×D₅₂	1	2	yes	—
229	C₂²	C₂²×D₁₀₄	1	2	yes	—
230	C₂²	#11719458	1	2	yes	—
231	C₂²	C₂³×D₅₂	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	52	12	no	—
234	C₄	Q₈×D₅₂	1	2	yes	—
235	C₄	C₂²×D₁₀₄	1	12	yes	—
236	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
237	C₄	C₂×#88257	1	1	yes	characteristic
238	C₂	—	52	8	no	—
239	C₂	Q₈×D₁₀₄	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₅₂	1	1	yes	characteristic
241	C₂	C₂³×D₁₀₄	1	1	yes	characteristic
242	C₂	C₂×#11719458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₅₂	C₂	semidirect
2. Q₈×D₁₀₄	C₂	direct (N × H)
3. C₂×#11718343	C₂	semidirect
4. C₂×#11718507	C₂	semidirect
6. C₂×C₈×Q₈×C₁₃	C₂	semidirect
11. Q₈×D₅₂	C₂²	semidirect
15. C₈×Q₈×C₁₃	C₂²	semidirect
21. #11718507	C₂²	semidirect
23. #11718343	C₂²	semidirect
39. C₂×D₁₀₄	Q₈	direct (N × H)
53. C₂×Q₈×C₁₃	D₈	semidirect
62. C₂×D₅₂	C₂×Q₈	semidirect
69. C₂×C₈×C₁₃	C₂×Q₈	semidirect
79. D₁₀₄	C₂×Q₈	direct (N × H)
84. C₂×C₈×Q₈	D₁₃	semidirect
91. Q₈×C₁₃	C₂×D₈	semidirect
104. D₅₂	C₂²×Q₈	semidirect
108. C₈×C₁₃	C₂²×Q₈	semidirect
119. C₈×Q₈	C₂×D₁₃	semidirect
167. C₂×C₁₃	Q₈×D₈	semidirect
176. C₂×Q₈	D₁₀₄	direct (N × H)
190. C₂×C₈	Q₈×D₁₃	semidirect
200. C₁₃	C₂×Q₈×D₈	semidirect
207. Q₈	C₂×D₁₀₄	direct (N × H)
223. C₈	C₂×Q₈×D₁₃	semidirect
239. C₂	Q₈×D₁₀₄	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×Q₈×D₅₂	2
Q₈×D₁₀₄	2
C₂×#11718343	2
C₂×#11718507	2
C₂×C₄×D₁₀₄	2
C₂×C₈×Q₈×C₁₃	2
C₂×Q₈×D₈	13

Subgroups — C2×Q8×D104 #16743334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₁₀₄ #16743334