Subgroups — C₂×#14272044 #16835199

Order 3712.

388 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90800	C₂	1	1	yes	maximal, characteristic
2	C₂²×#90800	C₂	1	1	yes	maximal, characteristic
3	#14272044	C₂	1	4	yes	maximal, direct product factor
4	C₂×#11745239	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₉	C₂²	1	1	yes	characteristic
6	C₂×#90800	—	2	1	no	—
7	C₂×#90800	—	2	1	no	—
8	C₂×#90800	—	2	1	no	—
9	C₂×#90800	—	2	1	no	—
10	C₂×#90800	—	2	1	no	—
11	C₂×#90800	—	2	1	no	—
12	C₂×#90800	—	2	1	no	—
13	C₂×#90800	C₄	1	2	yes	complemented
14	C₂×#90800	C₄	1	2	yes	complemented
15	C₂×#90800	C₄	1	2	yes	complemented
16	C₂×#90800	C₂²	1	2	yes	—
17	C₂×#90800	C₂²	1	2	yes	—
18	C₂²×#6599	—	2	1	no	—
19	C₂²×#6599	C₄	1	2	yes	complemented
20	C₂×C₂²⋊C₄×C₂₉	C₄	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₂₉	C₄	1	1	yes	characteristic
22	C₂×#90779	—	2	1	no	—
23	C₂×C₄×Dic₂₉	—	2	1	no	—
24	#11745239	—	2	1	no	—
25	#11745239	—	2	1	no	—
26	#11745239	C₄	1	2	yes	—
27	#11745239	C₂²	1	2	yes	—
28	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₉	—	2	1	no	—
30	C₂²×Dic₂₉	—	2	1	no	—
31	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
37	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
39	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
40	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
41	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
42	C₂²×Dic₂₉	Q₈	1	1	yes	characteristic
43	C₂²×Dic₂₉	Q₈	1	1	yes	characteristic
44	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
45	#90800	—	2	2	no	—
46	#90800	—	2	2	no	—
47	#90800	—	2	2	no	—
48	#90800	—	2	2	no	—
49	#90800	—	2	2	no	—
50	#90800	—	2	2	no	—
51	#90800	—	2	2	no	—
52	#90800	C₂×C₄	1	4	yes	complemented
53	C₂×#6599	—	2	2	no	—
54	C₂×#6599	—	2	2	no	—
55	C₂×#6599	—	2	2	no	—
56	C₂×#6599	—	2	2	no	—
57	C₂×#6599	—	2	2	no	—
58	C₂×#6599	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₂₉	—	2	1	no	—
60	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₂₉	—	2	1	no	—
62	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
63	#90779	—	2	2	no	—
64	#90779	—	2	2	no	—
65	C₄×Dic₂₉	—	2	2	no	—
66	C₄×Dic₂₉	—	2	2	no	—
67	C₂³×C₂₉	—	4	2	no	—
68	C₂³×C₂₉	C₄²	1	1	yes	characteristic
69	C₂³×C₂₉	C₄²	1	1	yes	characteristic
70	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₂₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₂₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	—	4	2	no	—
77	C₂×Dic₂₉	—	2	2	no	—
78	C₂×Dic₂₉	—	4	2	no	—
79	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
80	C₂×Dic₂₉	C₄²	1	1	yes	characteristic
81	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
94	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
95	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
96	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
97	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
98	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₂₉	C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₂₉	C₂×Q₈	1	1	yes	characteristic
107	#6599	—	2	4	no	—
108	#6599	—	2	4	no	—
109	C₂×#344	—	29	1	no	maximal, Sylow
110	C₂²×C₂₉	—	2	2	no	—
111	C₂²×C₂₉	—	2	2	no	—
112	C₂²×C₂₉	—	2	2	no	—
113	C₂²×C₂₉	—	4	1	no	—
114	C₂²×C₂₉	—	4	1	no	—
115	C₂²×C₂₉	—	4	1	no	—
116	C₂²×C₂₉	—	4	1	no	—
117	C₂²×C₂₉	#95	1	1	yes	characteristic
118	C₂²×C₂₉	#95	1	1	yes	characteristic
119	C₂²×C₂₉	#95	1	1	yes	characteristic
120	C₂²×C₂₉	#95	1	1	yes	characteristic
121	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
122	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₂₉	—	4	2	no	—
125	Dic₂₉	—	4	2	no	—
126	Dic₂₉	#95	1	1	yes	characteristic
127	Dic₂₉	#95	1	1	yes	characteristic
128	Dic₂₉	#95	1	1	yes	characteristic
129	Dic₂₉	#95	1	1	yes	characteristic
130	Dic₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	Dic₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	Dic₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
133	Dic₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×M₁₆	—	29	1	no	—
135	C₂²×M₁₆	—	29	1	no	—
136	C₂×#117	—	29	1	no	—
137	#344	—	29	4	no	—
138	C₂×C₂₉	—	2	1	no	—
139	C₂×C₂₉	—	2	1	no	—
140	C₂×C₂₉	—	2	1	no	—
141	C₂×C₂₉	—	2	1	no	—
142	C₂×C₂₉	—	2	1	no	—
143	C₂×C₂₉	—	2	1	no	—
144	C₂×C₂₉	#344	1	2	yes	complemented
145	C₂×C₂₉	C₂×#95	1	1	yes	characteristic
146	C₂³×C₄	—	29	1	no	—
147	C₂×M₁₆	—	29	2	no	—
148	C₂×M₁₆	—	29	2	no	—
149	C₂×M₁₆	—	29	2	no	—
150	C₂×M₁₆	—	29	2	no	—
151	C₂×M₁₆	—	29	2	no	—
152	C₂×M₁₆	—	58	1	no	—
153	C₂×M₁₆	—	58	1	no	—
154	C₂×M₁₆	—	58	1	no	—
155	C₂×M₁₆	—	58	1	no	—
156	C₂×M₁₆	—	58	1	no	—
157	C₂×M₁₆	—	58	1	no	—
158	C₂×M₁₆	—	58	1	no	—
159	C₂²×C₈	—	29	2	no	—
160	C₂²×C₈	—	58	1	no	—
161	#117	—	29	2	no	—
162	#117	—	29	2	no	—
163	#117	—	58	1	no	—
164	#117	—	58	1	no	—
165	C₂×C₄⋊C₄	—	58	1	no	—
166	C₂×C₂²⋊C₄	—	29	1	no	—
167	C₂×C₂²⋊C₄	#1181	1	1	yes	characteristic
168	C₂×C₄²	—	58	1	no	—
169	C₂₉	C₂×#344	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	C₂×#1181	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	29	1	no	—
173	C₂²×C₄	—	29	1	no	—
174	C₂²×C₄	—	29	1	no	—
175	C₂²×C₄	—	29	1	no	—
176	C₂²×C₄	—	29	1	no	—
177	C₂²×C₄	—	29	1	no	—
178	C₂²×C₄	—	29	1	no	—
179	C₂²×C₄	—	29	1	no	—
180	C₂²×C₄	—	29	1	no	—
181	C₂²×C₄	—	29	1	no	—
182	C₂²×C₄	—	29	1	no	—
183	C₂²×C₄	—	29	1	no	—
184	C₂²×C₄	—	29	1	no	—
185	C₂²×C₄	—	29	1	no	—
186	C₂²×C₄	—	58	1	no	—
187	M₁₆	—	29	4	no	—
188	M₁₆	—	58	2	no	—
189	M₁₆	—	58	2	no	—
190	M₁₆	—	58	2	no	—
191	M₁₆	—	58	2	no	—
192	M₁₆	—	58	2	no	—
193	M₁₆	—	58	2	no	—
194	M₁₆	—	58	2	no	—
195	C₂×C₈	—	29	4	no	—
196	C₂×C₈	—	58	2	no	—
197	C₂×C₈	—	58	2	no	—
198	C₂×C₈	—	58	2	no	—
199	C₂×C₈	—	58	2	no	—
200	C₂×C₈	—	58	2	no	—
201	C₄⋊C₄	—	58	2	no	—
202	C₄⋊C₄	—	58	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	29	2	no	—
205	C₂²⋊C₄	—	58	1	no	—
206	C₂²⋊C₄	C₂×#1181	1	2	yes	—
207	C₄²	—	58	2	no	—
208	C₄²	—	58	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₄×#1181	1	1	yes	characteristic
211	C₂³	C₄×#1181	1	1	yes	characteristic
212	C₂³	C₄⋊#1181	1	1	yes	characteristic
213	C₂³	C₄⋊#1181	1	1	yes	characteristic
214	C₂³	C₂²⋊#1181	1	1	yes	characteristic
215	C₂³	C₂²⋊#1181	1	1	yes	characteristic
216	C₂³	C₂²×#1181	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	29	1	no	—
220	C₂×C₄	—	29	1	no	—
221	C₂×C₄	—	29	1	no	—
222	C₂×C₄	—	29	1	no	—
223	C₂×C₄	—	29	1	no	—
224	C₂×C₄	—	29	1	no	—
225	C₂×C₄	—	29	1	no	—
226	C₂×C₄	—	29	1	no	—
227	C₂×C₄	—	29	1	no	—
228	C₂×C₄	—	29	1	no	—
229	C₂×C₄	—	29	1	no	—
230	C₂×C₄	—	29	1	no	—
231	C₂×C₄	—	29	1	no	—
232	C₂×C₄	—	29	1	no	—
233	C₂×C₄	—	29	1	no	—
234	C₂×C₄	—	29	1	no	—
235	C₂×C₄	—	29	1	no	—
236	C₂×C₄	—	29	1	no	—
237	C₂×C₄	—	29	1	no	—
238	C₂×C₄	—	29	1	no	—
239	C₂×C₄	—	29	1	no	—
240	C₂×C₄	—	29	1	no	—
241	C₂×C₄	—	29	1	no	—
242	C₂×C₄	—	29	1	no	—
243	C₂×C₄	—	29	1	no	—
244	C₂×C₄	—	29	1	no	—
245	C₂×C₄	—	29	1	no	—
246	C₂×C₄	—	29	1	no	—
247	C₂×C₄	—	58	2	no	—
248	C₂×C₄	—	116	2	no	—
249	C₈	—	58	4	no	—
250	C₈	—	58	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	(C₂×#1181)⋊C₄	1	1	yes	characteristic
259	C₂²	(C₂×#1181)⋊C₄	1	1	yes	characteristic
260	C₂²	(C₂×#1181)⋊C₄	1	1	yes	characteristic
261	C₂²	(C₂×#1181)⋊C₄	1	1	yes	characteristic
262	C₂²	C₂×C₄×#1181	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊#1181	1	1	yes	characteristic
264	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	29	1	no	—
267	C₄	—	29	1	no	—
268	C₄	—	29	1	no	—
269	C₄	—	29	1	no	—
270	C₄	—	29	1	no	—
271	C₄	—	29	1	no	—
272	C₄	—	29	1	no	—
273	C₄	—	29	1	no	—
274	C₄	—	116	2	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	#14272044	1	2	yes	direct product factor
282	C₂	C₂×(C₂×#1181)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#1181)⋊C₄
commutator	C₂²×C₂₉	C₂×C₄²
Frattini	C₂³	C₂²×#1181
Fitting	C₂×C₂²⋊C₄×C₂₉	C₄
socle	C₂²×C₂₉	#95
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂²×Dic₂₉	C₂³
hypercenter	C₂×C₂²⋊C₄	#1181
hypocenter	C₂₉	C₂×#344
LCS	G ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #14272044	C₂	direct (N × H)
13. C₂×#90800	C₄	semidirect
14. C₂×#90800	C₄	semidirect
15. C₂×#90800	C₄	semidirect
19. C₂²×#6599	C₄	semidirect
52. #90800	C₂×C₄	semidirect
58. C₂×#6599	C₂×C₄	semidirect
144. C₂×C₂₉	#344	semidirect
169. C₂₉	C₂×#344	semidirect
281. C₂	#14272044	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×#90800	2
#14272044	2
C₂×#11745239	2
C₂×#344	29

Subgroups — C2×#14272044 #16835199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14272044 #16835199