Subgroups — C₂×C₄×C₂²⋊#10597010 #34061119

Order 19072.

1360 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊#10597010	C₂	1	2	yes	maximal
2	C₂²×C₄×#10597010	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊#10597010	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×#10597010)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×D₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10597010	C₂²	1	4	yes	—
7	C₂×C₂²⋊#10597010	C₄	1	16	yes	direct product factor
8	C₂×C₂²⋊#10597010	C₂²	1	8	yes	—
9	C₂×C₄×#10597010	—	2	8	no	—
10	C₂×C₄×#10597010	C₂²	1	8	yes	complemented
11	(C₂×#10597010)⋊C₄	C₂²	1	8	yes	complemented
12	C₂⁴×D₁₄₉	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₄₉	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₄×D₁₄₉	—	2	8	no	—
15	C₂²×C₄×D₁₄₉	C₄	1	4	yes	complemented
16	C₂²×C₄×D₁₄₉	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₄×D₁₄₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₄₉	C₂²	1	4	yes	—
20	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₁₄₉	C₄	1	1	yes	characteristic, complemented
22	C₂²×#10597010	—	2	8	no	—
23	C₂²×#10597010	—	2	8	no	—
24	C₂²×#10597010	C₂×C₄	1	16	yes	complemented
25	C₂²×#10597010	C₂³	1	8	yes	—
26	C₂²⋊#10597010	C₂×C₄	1	32	yes	direct product factor
27	C₄×#10597010	—	2	16	no	—
28	C₂³×D₁₄₉	—	2	4	no	—
29	C₂³×D₁₄₉	—	2	4	no	—
30	C₂³×D₁₄₉	C₂×C₄	1	8	yes	—
31	C₂³×D₁₄₉	C₂×C₄	1	2	yes	—
32	C₂³×D₁₄₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₁₄₉	C₂³	1	2	yes	—
34	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₄₉	—	2	4	no	—
36	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₄₉	C₂×C₄	1	4	yes	complemented
39	C₂×C₄×D₁₄₉	—	2	4	no	—
40	C₂×C₄×D₁₄₉	—	2	4	no	—
41	C₂×C₄×D₁₄₉	—	2	16	no	—
42	C₂×C₄×D₁₄₉	—	2	16	no	—
43	C₂×C₄×D₁₄₉	—	2	4	no	—
44	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	complemented
47	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	complemented
48	C₂×C₄×D₁₄₉	D₄	1	8	yes	—
49	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
50	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
51	C₂⁴×C₁₄₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₄₉	—	2	4	no	—
53	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
56	C₂×#10597010	—	2	32	no	—
57	C₂×#10597010	—	2	8	no	—
58	C₂×#10597010	—	2	8	no	—
59	C₂×#10597010	C₂²×C₄	1	16	yes	complemented
60	C₂²×D₁₄₉	—	2	2	no	—
61	C₂²×D₁₄₉	—	2	2	no	—
62	C₂²×D₁₄₉	—	2	4	no	—
63	C₂²×D₁₄₉	—	2	4	no	—
64	C₂²×D₁₄₉	—	2	32	no	—
65	C₂²×D₁₄₉	—	2	4	no	—
66	C₂²×D₁₄₉	—	2	4	no	—
67	C₂²×D₁₄₉	C₄²	1	8	yes	complemented
68	C₂²×D₁₄₉	C₄²	1	4	yes	complemented
69	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₄₉	C₂²×C₄	1	8	yes	—
71	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₁₄₉	C₂²×C₄	1	4	yes	—
73	C₂²×D₁₄₉	C₂×D₄	1	4	yes	—
74	C₂²×D₁₄₉	#41	1	4	yes	—
75	C₂²×D₁₄₉	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₁₄₉	—	2	2	no	—
77	C₂×Dic₁₄₉	—	2	8	no	—
78	C₂×Dic₁₄₉	C₂²⋊C₄	1	4	yes	complemented
79	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
81	C₄×D₁₄₉	—	2	4	no	—
82	C₄×D₁₄₉	—	2	4	no	—
83	C₄×D₁₄₉	—	2	8	no	—
84	C₄×D₁₄₉	—	2	8	no	—
85	C₄×D₁₄₉	C₂²⋊C₄	1	8	yes	complemented
86	C₄×D₁₄₉	C₂×D₄	1	8	yes	—
87	C₂³×C₁₄₉	—	2	2	no	—
88	C₂³×C₁₄₉	—	2	2	no	—
89	C₂³×C₁₄₉	C₄²	1	4	yes	complemented
90	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₄₉	—	2	2	no	—
93	C₂×C₄×C₁₄₉	—	2	8	no	—
94	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	4	yes	complemented
95	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
97	C₂²×C₁₄₉	—	2	2	no	—
98	C₂²×C₁₄₉	—	2	2	no	—
99	C₂²×C₁₄₉	—	2	8	no	—
100	C₂²×C₁₄₉	C₂×C₄²	1	2	yes	—
101	C₂²×C₁₄₉	C₂×C₄²	1	4	yes	complemented
102	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂²×C₁₄₉	#117	1	2	yes	—
104	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₁₄₉	—	2	2	no	—
106	C₂×D₁₄₉	—	2	2	no	—
107	C₂×D₁₄₉	—	2	8	no	—
108	C₂×D₁₄₉	—	2	16	no	—
109	C₂×D₁₄₉	—	2	16	no	—
110	C₂×D₁₄₉	C₂×C₄²	1	2	yes	—
111	C₂×D₁₄₉	C₂×C₄²	1	4	yes	complemented
112	C₂×D₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×D₁₄₉	#117	1	2	yes	—
114	C₂×D₁₄₉	C₄×D₄	1	16	yes	—
115	C₂×D₁₄₉	C₂³×C₄	1	2	yes	—
116	C₂×D₁₄₉	C₂²×D₄	1	2	yes	—
117	C₂×D₁₄₉	C₂×#41	1	2	yes	—
118	#10597010	—	2	16	no	—
119	C₄×C₁₄₉	—	2	2	no	—
120	C₄×C₁₄₉	C₂×C₂²⋊C₄	1	4	yes	complemented
121	Dic₁₄₉	—	2	2	no	—
122	Dic₁₄₉	C₂×C₂²⋊C₄	1	4	yes	complemented
123	C₂×C₁₄₉	—	2	4	no	—
124	C₂×C₁₄₉	C₄×C₂²⋊C₄	1	4	yes	complemented
125	C₂×C₁₄₉	C₂²×C₄²	1	1	yes	characteristic
126	C₂×C₁₄₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₁₄₉	C₂×#117	1	1	yes	characteristic
128	D₁₄₉	—	2	4	no	—
129	D₁₄₉	C₄×C₂²⋊C₄	1	4	yes	complemented
130	D₁₄₉	C₂×C₄×D₄	1	4	yes	—
131	C₁₄₉	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄×C₂²⋊C₄	—	149	1	no	maximal, Sylow
133	C₂⁴×C₄	—	149	1	no	—
134	C₂²×C₂²⋊C₄	—	149	2	no	—
135	C₂²×C₄²	—	149	2	no	—
136	C₄×C₂²⋊C₄	—	149	8	no	—
137	C₂×#95	—	149	2	no	—
138	C₂⁵	—	149	1	no	—
139	C₂³×C₄	—	298	8	no	—
140	C₂³×C₄	—	149	4	no	—
141	C₂³×C₄	—	149	1	no	—
142	C₂³×C₄	—	149	1	no	—
143	C₂³×C₄	—	149	1	no	—
144	C₂³×C₄	—	149	1	no	—
145	C₂³×C₄	—	149	4	no	—
146	C₂³×C₄	—	149	4	no	—
147	C₂³×C₄	—	149	1	no	—
148	C₂³×C₄	#10597010	1	1	yes	characteristic, complemented
149	C₂×C₂²⋊C₄	—	149	16	no	—
150	C₂×C₂²⋊C₄	—	149	8	no	—
151	C₂×C₄²	—	298	8	no	—
152	C₂×C₄²	—	149	8	no	—
153	#95	—	149	8	no	—
154	C₂⁴	—	298	4	no	—
155	C₂⁴	—	298	4	no	—
156	C₂⁴	—	149	8	no	—
157	C₂⁴	—	149	2	no	—
158	C₂⁴	—	149	1	no	—
159	C₂⁴	—	149	2	no	—
160	C₂⁴	—	149	1	no	—
161	C₂⁴	C₂×#10597010	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	298	4	no	—
164	C₂²×C₄	—	298	4	no	—
165	C₂²×C₄	—	298	16	no	—
166	C₂²×C₄	—	298	16	no	—
167	C₂²×C₄	—	298	8	no	—
168	C₂²×C₄	—	298	8	no	—
169	C₂²×C₄	—	298	4	no	—
170	C₂²×C₄	—	298	4	no	—
171	C₂²×C₄	—	149	16	no	—
172	C₂²×C₄	—	149	8	no	—
173	C₂²×C₄	—	149	1	no	—
174	C₂²×C₄	—	149	4	no	—
175	C₂²×C₄	—	149	2	no	—
176	C₂²×C₄	—	149	4	no	—
177	C₂²×C₄	—	149	4	no	—
178	C₂²×C₄	—	149	2	no	—
179	C₂²×C₄	—	149	1	no	—
180	C₂²×C₄	—	149	8	no	—
181	C₂²×C₄	—	149	2	no	—
182	C₂²×C₄	—	149	2	no	—
183	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
184	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
185	C₂²×C₄	C₂×#10597010	1	4	yes	complemented
186	C₂²⋊C₄	—	149	32	no	—
187	C₄²	—	298	16	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	298	4	no	—
191	C₂³	—	298	4	no	—
192	C₂³	—	298	4	no	—
193	C₂³	—	298	4	no	—
194	C₂³	—	298	32	no	—
195	C₂³	—	298	2	no	—
196	C₂³	—	298	2	no	—
197	C₂³	—	149	4	no	—
198	C₂³	—	149	8	no	—
199	C₂³	—	149	8	no	—
200	C₂³	—	149	2	no	—
201	C₂³	—	149	4	no	—
202	C₂³	—	149	4	no	—
203	C₂³	—	149	4	no	—
204	C₂³	—	149	1	no	—
205	C₂³	—	149	1	no	—
206	C₂³	C₄×#10597010	1	4	yes	complemented
207	C₂³	C₂²×#10597010	1	2	yes	—
208	C₂³	C₂²×#10597010	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	298	4	no	—
212	C₂×C₄	—	298	4	no	—
213	C₂×C₄	—	298	8	no	—
214	C₂×C₄	—	298	8	no	—
215	C₂×C₄	—	298	8	no	—
216	C₂×C₄	—	298	32	no	—
217	C₂×C₄	—	298	2	no	—
218	C₂×C₄	—	298	8	no	—
219	C₂×C₄	—	298	8	no	—
220	C₂×C₄	—	149	16	no	—
221	C₂×C₄	—	149	2	no	—
222	C₂×C₄	—	149	4	no	—
223	C₂×C₄	—	149	2	no	—
224	C₂×C₄	—	149	8	no	—
225	C₂×C₄	—	149	8	no	—
226	C₂×C₄	C₂²⋊#10597010	1	4	yes	direct product factor
227	C₂×C₄	C₂²×#10597010	1	2	yes	—
228	C₂×C₄	C₂²×#10597010	1	2	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	8	no	—
232	C₂²	—	298	16	no	—
233	C₂²	—	298	16	no	—
234	C₂²	—	298	2	no	—
235	C₂²	—	298	2	no	—
236	C₂²	—	298	8	no	—
237	C₂²	—	149	4	no	—
238	C₂²	—	149	2	no	—
239	C₂²	—	149	2	no	—
240	C₂²	—	149	2	no	—
241	C₂²	—	149	2	no	—
242	C₂²	—	149	2	no	—
243	C₂²	—	149	2	no	—
244	C₂²	—	149	16	no	—
245	C₂²	C₂×C₄×#10597010	1	2	yes	—
246	C₂²	C₂×C₄×#10597010	1	4	yes	complemented
247	C₂²	C₂×C₂²⋊#10597010	1	2	yes	—
248	C₂²	#18011990	1	2	yes	—
249	C₂²	C₂³×#10597010	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	298	16	no	—
252	C₄	—	298	2	no	—
253	C₄	—	149	4	no	—
254	C₄	C₂×C₂²⋊#10597010	1	4	yes	direct product factor
255	C₂	—	2	4	no	—
256	C₂	—	298	4	no	—
257	C₂	—	149	4	no	—
258	C₂	—	149	4	no	—
259	C₂	C₄×C₂²⋊#10597010	1	4	yes	direct product factor
260	C₂	C₂²×C₄×#10597010	1	1	yes	characteristic
261	C₂	C₂²×C₂²⋊#10597010	1	1	yes	characteristic
262	C₂	C₂×#18011990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10597010
commutator	C₂×C₁₄₉	C₂²×C₄²
Frattini	C₂²	C₂³×#10597010
Fitting	C₂³×C₄×C₁₄₉	C₄
socle	C₂³×C₁₄₉	C₂²×C₄
prime radical	C₂⁴×D₁₄₉	C₂²
cosocle	C₂²×D₁₄₉	C₂⁴
hypercenter	C₂³×C₄	#10597010
hypocenter	C₁₄₉	C₂×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×#10597010	C₂	semidirect
3. C₄×C₂²⋊#10597010	C₂	direct (N × H)
4. C₂×(C₂×#10597010)⋊C₄	C₂	semidirect
7. C₂×C₂²⋊#10597010	C₄	direct (N × H)
10. C₂×C₄×#10597010	C₂²	semidirect
11. (C₂×#10597010)⋊C₄	C₂²	semidirect
13. C₂³×Dic₁₄₉	C₄	semidirect
15. C₂²×C₄×D₁₄₉	C₄	semidirect
16. C₂²×C₄×D₁₄₉	C₄	semidirect
17. C₂²×C₄×D₁₄₉	C₄	semidirect
21. C₂³×C₄×C₁₄₉	C₄	semidirect
24. C₂²×#10597010	C₂×C₄	semidirect
26. C₂²⋊#10597010	C₂×C₄	direct (N × H)
38. C₂²×Dic₁₄₉	C₂×C₄	semidirect
46. C₂×C₄×D₁₄₉	C₂×C₄	semidirect
47. C₂×C₄×D₁₄₉	C₂×C₄	semidirect
55. C₂²×C₄×C₁₄₉	C₂×C₄	semidirect
59. C₂×#10597010	C₂²×C₄	semidirect
67. C₂²×D₁₄₉	C₄²	semidirect
68. C₂²×D₁₄₉	C₄²	semidirect
78. C₂×Dic₁₄₉	C₂²⋊C₄	semidirect
85. C₄×D₁₄₉	C₂²⋊C₄	semidirect
89. C₂³×C₁₄₉	C₄²	semidirect
94. C₂×C₄×C₁₄₉	C₂²⋊C₄	semidirect
101. C₂²×C₁₄₉	C₂×C₄²	semidirect
111. C₂×D₁₄₉	C₂×C₄²	semidirect
120. C₄×C₁₄₉	C₂×C₂²⋊C₄	semidirect
122. Dic₁₄₉	C₂×C₂²⋊C₄	semidirect
124. C₂×C₁₄₉	C₄×C₂²⋊C₄	semidirect
129. D₁₄₉	C₄×C₂²⋊C₄	semidirect
131. C₁₄₉	C₂×C₄×C₂²⋊C₄	semidirect
148. C₂³×C₄	#10597010	semidirect
185. C₂²×C₄	C₂×#10597010	semidirect
206. C₂³	C₄×#10597010	semidirect
226. C₂×C₄	C₂²⋊#10597010	direct (N × H)
246. C₂²	C₂×C₄×#10597010	semidirect
254. C₄	C₂×C₂²⋊#10597010	direct (N × H)
259. C₂	C₄×C₂²⋊#10597010	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×C₂²⋊#10597010	2
C₂²×C₄×#10597010	2
C₄×C₂²⋊#10597010	2
C₂×(C₂×#10597010)⋊C₄	2
C₂³×C₄×D₁₄₉	2
C₂×C₄×C₂²⋊C₄	149

Subgroups — C2×C4×C22⋊#10597010 #34061119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂²⋊#10597010 #34061119