Subgroups — C₂×#20295691 #28376505

Order 13952.

1568 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
2	#20295691	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16812698	C₂	1	2	yes	maximal, complemented
4	C₂×#16812677	C₂	1	4	yes	maximal, complemented
5	C₂×#16812675	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#16812661	C₂	1	2	yes	maximal
9	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
11	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
12	C₂×Q₈×D₁₀₉	—	2	16	no	—
13	C₂×Q₈×D₁₀₉	C₂²	1	16	yes	complemented
14	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
17	#16812698	C₂²	1	16	yes	complemented
18	#16812677	C₂²	1	32	yes	complemented
19	#16812675	C₂²	1	16	yes	complemented
20	C₄×Dic₂₁₈	C₂²	1	8	yes	complemented
21	C₄×D₄₃₆	C₂²	1	8	yes	—
22	#16812661	C₂²	1	16	yes	—
23	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×#13182008	C₂²	1	2	yes	—
26	C₂×#13182008	C₂²	1	4	yes	—
27	C₂×#13182006	C₂²	1	4	yes	—
28	C₂×#13182006	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
31	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂³×D₁₀₉	C₂³	1	2	yes	—
36	C₂×Dic₂₁₈	—	2	16	no	—
37	C₂×Dic₂₁₈	D₄	1	8	yes	complemented
38	C₂×Dic₂₁₈	C₂³	1	16	yes	—
39	C₂×Dic₂₁₈	C₂³	1	4	yes	—
40	Q₈×D₁₀₉	—	2	64	no	—
41	C₂×D₄₃₆	Q₈	1	8	yes	complemented
42	C₂×D₄₃₆	C₂³	1	4	yes	—
43	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
45	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
46	C₂×C₄×D₁₀₉	—	2	8	no	—
47	C₂×C₄×D₁₀₉	—	2	16	no	—
48	C₂×C₄×D₁₀₉	C₂³	1	16	yes	—
49	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
50	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
51	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
52	#13182008	C₂³	1	8	yes	—
53	#13182008	C₂³	1	16	yes	—
54	#13182006	C₂³	1	16	yes	—
55	#13182006	C₂³	1	8	yes	—
56	C₄×Dic₁₀₉	C₂³	1	8	yes	—
57	C₂×Q₈×C₁₀₉	—	2	8	no	—
58	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
59	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
65	C₄²×C₁₀₉	C₂³	1	4	yes	—
66	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
67	C₂²×D₁₀₉	—	2	4	no	—
68	C₂²×D₁₀₉	—	2	4	no	—
69	C₂²×D₁₀₉	C₂×Q₈	1	8	yes	complemented
70	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
71	Q₈×C₁₀₉	—	2	16	no	—
72	C₂×C₄×C₁₀₉	—	2	8	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	complemented
76	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
78	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
80	C₂×Dic₁₀₉	—	2	4	no	—
81	C₂×Dic₁₀₉	—	2	4	no	—
82	C₂×Dic₁₀₉	C₂×D₄	1	8	yes	—
83	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
84	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
85	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
86	D₄₃₆	C₂×Q₈	1	16	yes	complemented
87	C₄×D₁₀₉	—	2	16	no	—
88	C₄×D₁₀₉	—	2	32	no	—
89	Dic₂₁₈	—	2	32	no	—
90	Dic₂₁₈	C₂×D₄	1	16	yes	complemented
91	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
92	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
93	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
94	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
95	C₂×D₁₀₉	—	2	16	no	—
96	C₂×D₁₀₉	—	2	4	no	—
97	C₂×D₁₀₉	—	2	4	no	—
98	C₂×D₁₀₉	C₂²×Q₈	1	8	yes	complemented
99	C₄×C₁₀₉	—	2	8	no	—
100	C₄×C₁₀₉	—	2	2	no	—
101	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
102	C₄×C₁₀₉	C₂²×Q₈	1	4	yes	complemented
103	Dic₁₀₉	—	2	4	no	—
104	Dic₁₀₉	—	2	4	no	—
105	Dic₁₀₉	C₂²×D₄	1	8	yes	—
106	C₂×C₁₀₉	D₄×Q₈	1	4	yes	complemented
107	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
108	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
109	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
110	D₁₀₉	—	2	8	no	—
111	C₂×D₄×Q₈	—	109	1	no	maximal, Sylow
112	C₁₀₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
113	C₂³×Q₈	—	109	2	no	—
114	D₄×Q₈	—	109	16	no	—
115	C₂×C₄⋊Q₈	—	109	2	no	—
116	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
117	C₂×C₂²⋊Q₈	—	109	2	no	—
118	C₂×C₂²⋊Q₈	—	109	4	no	—
119	C₂×C₄×Q₈	—	109	1	no	—
120	C₂×C₄×D₄	—	109	2	no	—
121	C₂×C₄×D₄	—	109	1	no	—
122	C₂²×Q₈	—	109	16	no	—
123	C₂²×Q₈	—	109	1	no	—
124	C₂²×Q₈	—	109	4	no	—
125	C₂²×Q₈	—	218	16	no	—
126	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
127	C₂²×D₄	—	109	1	no	—
128	C₂³×C₄	—	109	4	no	—
129	C₂³×C₄	—	109	2	no	—
130	C₄⋊Q₈	—	109	16	no	—
131	C₄⋊Q₈	C₂×D₁₀₉	1	8	yes	complemented
132	C₂²⋊Q₈	—	109	16	no	—
133	C₂²⋊Q₈	—	109	32	no	—
134	C₄×Q₈	—	109	8	no	—
135	C₄×D₄	—	109	16	no	—
136	C₄×D₄	—	109	8	no	—
137	C₂×C₄⋊C₄	—	109	1	no	—
138	C₂×C₄⋊C₄	—	109	2	no	—
139	C₂×C₄⋊C₄	—	109	1	no	—
140	C₂×C₄⋊C₄	—	109	4	no	—
141	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
143	C₂×C₂²⋊C₄	—	109	4	no	—
144	C₂×C₂²⋊C₄	—	109	2	no	—
145	C₂×C₄²	—	109	2	no	—
146	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
147	C₂⁴	—	109	2	no	—
148	C₂×Q₈	—	2	8	no	—
149	C₂×Q₈	—	109	8	no	—
150	C₂×Q₈	—	109	4	no	—
151	C₂×Q₈	—	109	16	no	—
152	C₂×Q₈	—	218	16	no	—
153	C₂×Q₈	—	218	64	no	—
154	C₂×Q₈	C₂²×D₁₀₉	1	8	yes	—
155	C₂×D₄	—	109	8	no	—
156	C₂×D₄	—	109	4	no	—
157	C₂²×C₄	—	109	2	no	—
158	C₂²×C₄	—	109	2	no	—
159	C₂²×C₄	—	109	2	no	—
160	C₂²×C₄	—	109	16	no	—
161	C₂²×C₄	—	109	8	no	—
162	C₂²×C₄	—	218	16	no	—
163	C₂²×C₄	—	218	8	no	—
164	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
166	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
168	C₄⋊C₄	—	109	8	no	—
169	C₄⋊C₄	—	109	16	no	—
170	C₄⋊C₄	—	109	4	no	—
171	C₄⋊C₄	—	109	4	no	—
172	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
173	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
174	C₂²⋊C₄	—	109	16	no	—
175	C₂²⋊C₄	—	109	8	no	—
176	C₄²	—	109	8	no	—
177	C₄²	C₂²×D₁₀₉	1	4	yes	—
178	C₂³	—	109	8	no	—
179	C₂³	—	109	4	no	—
180	C₂³	—	218	4	no	—
181	C₂³	—	218	4	no	—
182	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
183	Q₈	—	2	16	no	—
184	Q₈	—	109	16	no	—
185	Q₈	—	218	32	no	—
186	D₄	—	109	16	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	109	8	no	—
190	C₂×C₄	—	109	4	no	—
191	C₂×C₄	—	109	4	no	—
192	C₂×C₄	—	109	4	no	—
193	C₂×C₄	—	218	32	no	—
194	C₂×C₄	—	218	4	no	—
195	C₂×C₄	—	218	4	no	—
196	C₂×C₄	—	218	16	no	—
197	C₂×C₄	D₄×D₁₀₉	1	4	yes	—
198	C₂×C₄	Q₈×D₁₀₉	1	4	yes	complemented
199	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
200	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
201	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
202	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
203	C₂²	—	109	8	no	—
204	C₂²	—	218	16	no	—
205	C₂²	—	218	4	no	—
206	C₂²	—	218	4	no	—
207	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
208	C₂²	C₂×Q₈×D₁₀₉	1	2	yes	—
209	C₂²	#16812747	1	2	yes	—
210	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	—	2	2	no	—
213	C₄	—	109	8	no	—
214	C₄	—	218	4	no	—
215	C₄	—	218	4	no	—
216	C₄	C₂×D₄×D₁₀₉	1	4	yes	—
217	C₄	C₂×Q₈×D₁₀₉	1	4	yes	complemented
218	C₂	—	218	8	no	—
219	C₂	#20295691	1	4	yes	direct product factor
220	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
221	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
222	C₂	C₂×#16812747	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂²×D₄₃₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₀₉	C₂	semidirect
2. #20295691	C₂	direct (N × H)
3. C₂×#16812698	C₂	semidirect
4. C₂×#16812677	C₂	semidirect
5. C₂×#16812675	C₂	semidirect
6. C₂×C₄×Dic₂₁₈	C₂	semidirect
9. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
13. C₂×Q₈×D₁₀₉	C₂²	semidirect
17. #16812698	C₂²	semidirect
18. #16812677	C₂²	semidirect
19. #16812675	C₂²	semidirect
20. C₄×Dic₂₁₈	C₂²	semidirect
31. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
37. C₂×Dic₂₁₈	D₄	semidirect
41. C₂×D₄₃₆	Q₈	semidirect
69. C₂²×D₁₀₉	C₂×Q₈	semidirect
75. C₂×C₄×C₁₀₉	C₂×Q₈	semidirect
86. D₄₃₆	C₂×Q₈	semidirect
90. Dic₂₁₈	C₂×D₄	semidirect
98. C₂×D₁₀₉	C₂²×Q₈	semidirect
102. C₄×C₁₀₉	C₂²×Q₈	semidirect
106. C₂×C₁₀₉	D₄×Q₈	semidirect
112. C₁₀₉	C₂×D₄×Q₈	semidirect
116. C₂×C₄⋊Q₈	D₁₀₉	semidirect
131. C₄⋊Q₈	C₂×D₁₀₉	semidirect
198. C₂×C₄	Q₈×D₁₀₉	semidirect
217. C₄	C₂×Q₈×D₁₀₉	semidirect
219. C₂	#20295691	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₀₉	2
#20295691	2
C₂×#16812698	2
C₂×#16812677	2
C₂×#16812675	2
C₂×C₄×Dic₂₁₈	2
C₂×C₄×D₄₃₆	2
C₂×#16812661	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×D₄×Q₈	109

Subgroups — C2×#20295691 #28376505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295691 #28376505