Subgroups — C₂×#20296037 #28379684

Order 13952.

536 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×#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	#20296037	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812768	C₂	1	2	yes	maximal, complemented
5	C₂×#16812767	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#89137	—	2	1	no	—
8	#16812778	—	2	2	no	—
9	#16812778	C₂²	1	4	yes	—
10	C₂×C₄⋊#89137	—	2	1	no	—
11	C₂×C₄⋊#89137	—	2	2	no	—
12	C₂×C₄⋊#89137	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#89137	C₂²	1	2	yes	—
14	C₂×C₄⋊#89137	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#89137	C₂²	1	4	yes	—
16	C₂×C₂²⋊#89137	—	2	1	no	—
17	C₂×C₄×#89137	—	2	1	no	—
18	#16812768	C₂²	1	8	yes	complemented
19	#16812767	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×#13182013	C₄	1	1	yes	characteristic, complemented
22	C₂×#13182012	C₄	1	2	yes	complemented
23	M₁₆×D₁₀₉	C₄	1	4	yes	complemented
24	M₁₆×D₁₀₉	C₂²	1	4	yes	—
25	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
26	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
27	C₂²×#89137	—	2	1	no	—
28	C₂²×#89137	—	2	1	no	—
29	C₂²×#89137	—	2	4	no	—
30	C₂²×#89137	—	2	2	no	—
31	C₂²×#89137	—	4	4	no	—
32	C₄⋊#89137	—	2	4	no	—
33	C₄⋊#89137	—	2	2	no	—
34	C₄⋊#89137	—	2	1	no	—
35	C₄⋊#89137	—	2	1	no	—
36	C₄⋊#89137	C₂³	1	2	yes	—
37	C₄⋊#89137	C₂³	1	4	yes	—
38	C₄⋊#89137	C₂³	1	2	yes	—
39	C₂²⋊#89137	—	2	4	no	—
40	C₄×#89137	—	2	4	no	—
41	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
42	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
47	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
48	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
51	#13182013	C₂×C₄	1	4	yes	complemented
52	#13182012	C₂×C₄	1	8	yes	complemented
53	C₂×#11724784	C₂×C₄	1	2	yes	complemented
54	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
55	C₈×D₁₀₉	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
58	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
59	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×D₁₀₉	—	2	2	no	—
61	C₂²×D₁₀₉	—	2	2	no	—
62	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
65	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂×#89137	—	2	2	no	—
67	C₂×#89137	—	2	2	no	—
68	C₂×#89137	—	2	4	no	—
69	C₂×#89137	—	4	8	no	—
70	C₂×#89137	—	4	2	no	—
71	C₂×#89137	—	4	4	no	—
72	C₂×#89137	—	4	2	no	—
73	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
78	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
80	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
81	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
82	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
83	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
84	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
86	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
87	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
89	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
90	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₀₉	C₂²×C₄	1	4	yes	complemented
92	#11724784	C₂²×C₄	1	4	yes	complemented
93	C₂²×C₁₀₉	—	2	2	no	—
94	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×D₁₀₉	—	2	2	no	—
97	C₂×D₁₀₉	—	2	2	no	—
98	C₂×D₁₀₉	—	2	2	no	—
99	C₂×D₁₀₉	—	2	4	no	—
100	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×D₁₀₉	#137	1	2	yes	—
104	C₂×D₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	#89137	—	4	2	no	—
106	#89137	—	4	4	no	—
107	#89137	—	4	2	no	—
108	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
111	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
113	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₁₀₉	—	2	2	no	—
115	C₂×C₁₀₉	#428	1	2	yes	complemented
116	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	D₁₀₉	—	2	2	no	—
118	D₁₀₉	#428	1	2	yes	complemented
119	D₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	D₁₀₉	C₂×#137	1	1	yes	characteristic
121	C₂×#428	—	109	1	no	maximal, Sylow
122	C₁₀₉	C₂×#428	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	—	109	1	no	—
124	C₂×#117	—	109	1	no	—
125	C₂²×C₄⋊C₄	—	109	1	no	—
126	#428	—	109	8	no	—
127	C₂×#107	—	109	2	no	—
128	C₂×#106	—	109	2	no	—
129	C₂³×C₄	—	109	1	no	—
130	C₂³×C₄	—	218	1	no	—
131	C₂×M₁₆	—	109	1	no	—
132	C₂×M₁₆	—	109	4	no	—
133	C₂×M₁₆	—	109	4	no	—
134	C₂×M₁₆	—	109	2	no	—
135	C₂×M₁₆	#89137	1	1	yes	characteristic, complemented
136	C₂²×C₈	—	109	2	no	—
137	#117	—	109	4	no	—
138	#117	—	218	2	no	—
139	C₂×C₄⋊C₄	—	109	2	no	—
140	C₂×C₄⋊C₄	—	109	4	no	—
141	C₂×C₄⋊C₄	—	109	1	no	—
142	C₂×C₄⋊C₄	—	109	1	no	—
143	C₂×C₄⋊C₄	—	218	1	no	—
144	C₂×C₄⋊C₄	—	218	2	no	—
145	C₂×C₂²⋊C₄	—	218	1	no	—
146	C₂×C₄²	—	218	1	no	—
147	#107	—	109	8	no	—
148	#106	—	109	8	no	—
149	C₂⁴	—	109	1	no	—
150	C₂²×C₄	—	109	1	no	—
151	C₂²×C₄	—	109	1	no	—
152	C₂²×C₄	—	109	2	no	—
153	C₂²×C₄	—	109	2	no	—
154	C₂²×C₄	—	109	1	no	—
155	C₂²×C₄	—	109	2	no	—
156	C₂²×C₄	—	109	1	no	—
157	C₂²×C₄	—	109	1	no	—
158	C₂²×C₄	—	109	2	no	—
159	C₂²×C₄	—	436	4	no	—
160	C₂²×C₄	—	218	1	no	—
161	C₂²×C₄	—	218	1	no	—
162	C₂²×C₄	—	218	2	no	—
163	C₂²×C₄	—	218	4	no	—
164	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
165	M₁₆	—	109	4	no	—
166	M₁₆	—	109	8	no	—
167	M₁₆	C₂×#89137	1	4	yes	complemented
168	C₂×C₈	—	109	2	no	—
169	C₂×C₈	—	109	4	no	—
170	C₂×C₈	—	109	4	no	—
171	C₂×C₈	C₂×#89137	1	2	yes	complemented
172	C₄⋊C₄	—	109	2	no	—
173	C₄⋊C₄	—	109	4	no	—
174	C₄⋊C₄	—	109	2	no	—
175	C₄⋊C₄	—	218	4	no	—
176	C₄⋊C₄	—	218	2	no	—
177	C₄⋊C₄	—	218	1	no	—
178	C₄⋊C₄	—	218	1	no	—
179	C₂²⋊C₄	—	218	4	no	—
180	C₄²	—	218	4	no	—
181	C₂³	—	109	1	no	—
182	C₂³	—	109	2	no	—
183	C₂³	—	109	2	no	—
184	C₂³	—	109	1	no	—
185	C₂³	—	218	2	no	—
186	C₂³	—	218	2	no	—
187	C₂³	C₄⋊#89137	1	1	yes	characteristic
188	C₂×C₄	—	109	2	no	—
189	C₂×C₄	—	109	2	no	—
190	C₂×C₄	—	109	1	no	—
191	C₂×C₄	—	109	1	no	—
192	C₂×C₄	—	109	2	no	—
193	C₂×C₄	—	109	1	no	—
194	C₂×C₄	—	109	2	no	—
195	C₂×C₄	—	109	2	no	—
196	C₂×C₄	—	109	2	no	—
197	C₂×C₄	—	109	1	no	—
198	C₂×C₄	—	109	2	no	—
199	C₂×C₄	—	109	1	no	—
200	C₂×C₄	—	109	1	no	—
201	C₂×C₄	—	109	2	no	—
202	C₂×C₄	—	436	4	no	—
203	C₂×C₄	—	436	2	no	—
204	C₂×C₄	—	436	2	no	—
205	C₂×C₄	—	436	8	no	—
206	C₂×C₄	—	218	2	no	—
207	C₂×C₄	—	218	2	no	—
208	C₂×C₄	—	218	4	no	—
209	C₂×C₄	C₄⋊#89137	1	2	yes	—
210	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
211	C₂×C₄	C₂²×#89137	1	2	yes	—
212	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
213	C₈	—	109	4	no	—
214	C₈	C₂²×#89137	1	4	yes	complemented
215	C₂²	—	2	2	no	—
216	C₂²	—	109	2	no	—
217	C₂²	—	109	2	no	—
218	C₂²	—	109	1	no	—
219	C₂²	—	109	1	no	—
220	C₂²	—	109	2	no	—
221	C₂²	—	218	2	no	—
222	C₂²	—	218	2	no	—
223	C₂²	—	218	4	no	—
224	C₂²	—	218	2	no	—
225	C₂²	C₂×C₄⋊#89137	1	2	yes	—
226	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
227	C₄	—	109	1	no	—
228	C₄	—	109	2	no	—
229	C₄	—	109	1	no	—
230	C₄	—	436	4	no	—
231	C₄	—	436	2	no	—
232	C₄	—	436	2	no	—
233	C₄	C₂×C₄⋊#89137	1	2	yes	—
234	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
235	C₄	C₂³×#89137	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	109	1	no	—
238	C₂	—	109	1	no	—
239	C₂	—	109	2	no	—
240	C₂	—	218	2	no	—
241	C₂	#20296037	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	C₂×#428
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20296037	C₂	direct (N × H)
4. C₂×#16812768	C₂	semidirect
5. C₂×#16812767	C₂	semidirect
18. #16812768	C₂²	semidirect
19. #16812767	C₂²	semidirect
21. C₂×#13182013	C₄	semidirect
22. C₂×#13182012	C₄	semidirect
23. M₁₆×D₁₀₉	C₄	semidirect
26. C₂×M₁₆×C₁₀₉	C₄	semidirect
51. #13182013	C₂×C₄	semidirect
52. #13182012	C₂×C₄	semidirect
53. C₂×#11724784	C₂×C₄	semidirect
54. C₈×D₁₀₉	C₂×C₄	semidirect
57. M₁₆×C₁₀₉	C₂×C₄	semidirect
58. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
91. C₈×C₁₀₉	C₂²×C₄	semidirect
92. #11724784	C₂²×C₄	semidirect
115. C₂×C₁₀₉	#428	semidirect
118. D₁₀₉	#428	semidirect
122. C₁₀₉	C₂×#428	semidirect
135. C₂×M₁₆	#89137	semidirect
167. M₁₆	C₂×#89137	semidirect
171. C₂×C₈	C₂×#89137	semidirect
214. C₈	C₂²×#89137	semidirect
241. C₂	#20296037	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×#16812778	2
C₂²×C₄⋊#89137	2
#20296037	2
C₂×#16812768	2
C₂×#16812767	2
C₂×M₁₆×D₁₀₉	2
C₂×#428	109

Subgroups — C2×#20296037 #28379684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20296037 #28379684