Subgroups — C₂×#23750177 #34057977

Order 19072.

1168 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	#23750177	C₂	1	16	yes	maximal, direct product factor
2	C₂×#18011910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#18011909	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#18011901	C₂	1	2	yes	maximal, complemented
5	C₂×#18011890	C₂	1	1	yes	maximal, characteristic
6	C₂×#18011888	C₂	1	2	yes	maximal
7	C₂×Q₈×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#18011876	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#18011876	C₂	1	2	yes	maximal, complemented
10	C₂×#18011867	C₂	1	2	yes	maximal
11	C₂²×C₄×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
15	C₂³×Dic₁₄₉	C₂²	1	1	yes	characteristic
16	C₂³×Dic₁₄₉	C₂²	1	1	yes	characteristic
17	#18011910	C₂²	1	8	yes	complemented
18	#18011909	C₂²	1	8	yes	complemented
19	#18011901	C₂²	1	16	yes	complemented
20	#18011890	C₂²	1	8	yes	—
21	#18011888	C₂²	1	16	yes	—
22	Q₈×Dic₁₄₉	C₂²	1	8	yes	complemented
23	#18011876	C₂²	1	8	yes	complemented
24	#18011876	C₂²	1	16	yes	complemented
25	#18011867	C₂²	1	16	yes	—
26	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
28	C₂×#14558715	C₂²	1	2	yes	—
29	C₂×#14558715	C₂²	1	2	yes	—
30	C₂×#14558715	C₂²	1	2	yes	—
31	C₂×#14558715	C₂²	1	2	yes	—
32	C₂×#14558715	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₄₉	C₂²	1	8	yes	—
37	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
41	C₂²×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₄₉	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
47	C₂×Dic₂₉₈	—	2	8	no	—
48	C₂×Dic₂₉₈	—	2	4	no	—
49	C₂×Dic₂₉₈	C₂³	1	4	yes	—
50	C₂×Dic₂₉₈	C₂³	1	8	yes	—
51	C₂²×Dic₁₄₉	—	2	4	no	—
52	C₂²×Dic₁₄₉	Q₈	1	8	yes	complemented
53	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
55	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
57	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
60	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
62	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
63	#14558715	C₂³	1	8	yes	—
64	#14558715	C₂³	1	8	yes	—
65	#14558715	C₂³	1	8	yes	—
66	#14558715	C₂³	1	8	yes	—
67	#14558715	C₂³	1	8	yes	—
68	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
69	C₄×Dic₁₄₉	C₂³	1	4	yes	—
70	C₄×Dic₁₄₉	C₂³	1	4	yes	—
71	C₄×Dic₁₄₉	C₂³	1	4	yes	—
72	C₄×Dic₁₄₉	C₂³	1	4	yes	—
73	C₄×Dic₁₄₉	C₂³	1	8	yes	—
74	C₄×Dic₁₄₉	C₂³	1	8	yes	—
75	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
76	C₂×Q₈×C₁₄₉	—	2	4	no	—
77	C₂×Q₈×C₁₄₉	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₄₉	—	2	4	no	—
79	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
83	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
85	C₄⋊C₄×C₁₄₉	C₂³	1	8	yes	—
86	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
87	Dic₂₉₈	—	2	8	no	—
88	Dic₂₉₈	—	2	16	no	—
89	C₂×Dic₁₄₉	—	2	2	no	—
90	C₂×Dic₁₄₉	—	2	4	no	—
91	C₂×Dic₁₄₉	—	2	4	no	—
92	C₂×Dic₁₄₉	—	2	8	no	—
93	C₂×Dic₁₄₉	C₂×Q₈	1	4	yes	—
94	C₂×Dic₁₄₉	C₂×Q₈	1	4	yes	—
95	C₂×Dic₁₄₉	C₂×Q₈	1	16	yes	complemented
96	C₂×Dic₁₄₉	#41	1	4	yes	—
97	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
98	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
99	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
100	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
101	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
102	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
103	C₂³×C₁₄₉	—	2	2	no	—
104	C₂³×C₁₄₉	—	2	2	no	—
105	C₂³×C₁₄₉	C₂×Q₈	1	4	yes	—
106	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
107	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
108	Q₈×C₁₄₉	—	2	8	no	—
109	C₂×C₄×C₁₄₉	—	2	4	no	—
110	C₂×C₄×C₁₄₉	—	2	2	no	—
111	C₂×C₄×C₁₄₉	—	2	4	no	—
112	C₂×C₄×C₁₄₉	—	2	8	no	—
113	C₂×C₄×C₁₄₉	#41	1	4	yes	—
114	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
117	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
118	C₂²×C₁₄₉	—	2	8	no	—
119	C₂²×C₁₄₉	—	2	2	no	—
120	C₂²×C₁₄₉	—	2	2	no	—
121	C₂²×C₁₄₉	C₂²×Q₈	1	4	yes	—
122	C₂²×C₁₄₉	C₂²×Q₈	1	2	yes	—
123	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
124	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
125	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
126	C₄×C₁₄₉	—	2	2	no	—
127	C₄×C₁₄₉	—	2	4	no	—
128	C₄×C₁₄₉	—	2	4	no	—
129	C₄×C₁₄₉	C₂×#41	1	4	yes	—
130	Dic₁₄₉	—	2	2	no	—
131	Dic₁₄₉	—	2	4	no	—
132	Dic₁₄₉	—	2	4	no	—
133	Dic₁₄₉	C₂²×Q₈	1	4	yes	—
134	Dic₁₄₉	C₂²×Q₈	1	4	yes	—
135	Dic₁₄₉	C₂×#41	1	4	yes	—
136	C₂×C₁₄₉	—	2	4	no	—
137	C₂×C₁₄₉	#533	1	4	yes	complemented
138	C₂×C₁₄₉	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
141	C₁₄₉	C₂×#533	1	1	yes	characteristic, complemented, Sylow
142	C₂×#533	—	149	1	no	maximal, Sylow
143	#533	—	149	16	no	—
144	C₂×C₄⋊Q₈	—	149	1	no	—
145	C₂×C₄⋊Q₈	—	149	1	no	—
146	C₂×#125	—	149	2	no	—
147	C₂×C₂²⋊Q₈	—	149	1	no	—
148	C₂×C₂²⋊Q₈	—	149	2	no	—
149	C₂×C₂²⋊Q₈	D₁₄₉	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	149	1	no	—
151	C₂×C₄×Q₈	—	149	1	no	—
152	C₂×C₄×Q₈	—	149	2	no	—
153	C₂×#117	—	149	2	no	—
154	C₂²×C₄²	—	149	1	no	—
155	C₂²×Q₈	—	149	1	no	—
156	C₂²×Q₈	—	149	2	no	—
157	C₂²×Q₈	C₂×D₁₄₉	1	1	yes	characteristic
158	C₂³×C₄	—	149	1	no	—
159	C₂³×C₄	—	149	1	no	—
160	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
161	C₄⋊Q₈	—	149	8	no	—
162	C₄⋊Q₈	—	149	8	no	—
163	#125	—	149	16	no	—
164	C₂²⋊Q₈	—	149	8	no	—
165	C₂²⋊Q₈	—	149	16	no	—
166	C₂²⋊Q₈	C₂×D₁₄₉	1	8	yes	—
167	C₄×Q₈	—	149	8	no	—
168	C₄×Q₈	—	149	8	no	—
169	C₄×Q₈	—	149	16	no	—
170	#117	—	149	16	no	—
171	C₂×C₄⋊C₄	—	149	2	no	—
172	C₂×C₄⋊C₄	—	149	2	no	—
173	C₂×C₄⋊C₄	—	149	2	no	—
174	C₂×C₄⋊C₄	—	149	2	no	—
175	C₂×C₄⋊C₄	—	149	2	no	—
176	C₂×C₄⋊C₄	—	149	2	no	—
177	C₂×C₄⋊C₄	—	149	1	no	—
178	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	149	2	no	—
181	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
182	C₂×C₄²	—	149	2	no	—
183	C₂×C₄²	—	149	2	no	—
184	C₂×C₄²	—	149	1	no	—
185	C₂×C₄²	—	149	1	no	—
186	C₂×C₄²	—	149	1	no	—
187	C₂×C₄²	—	149	1	no	—
188	C₂×C₄²	—	149	8	no	—
189	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	298	8	no	—
192	C₂×Q₈	—	298	4	no	—
193	C₂×Q₈	—	149	4	no	—
194	C₂×Q₈	—	149	8	no	—
195	C₂×Q₈	C₂²×D₁₄₉	1	4	yes	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	298	4	no	—
198	C₂²×C₄	—	149	4	no	—
199	C₂²×C₄	—	149	1	no	—
200	C₂²×C₄	—	149	2	no	—
201	C₂²×C₄	—	149	2	no	—
202	C₂²×C₄	—	149	8	no	—
203	C₂²×C₄	—	149	4	no	—
204	C₂²×C₄	—	149	1	no	—
205	C₂²×C₄	—	149	1	no	—
206	C₂²×C₄	—	149	1	no	—
207	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
208	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
211	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
212	C₄⋊C₄	—	149	8	no	—
213	C₄⋊C₄	—	149	8	no	—
214	C₄⋊C₄	—	149	8	no	—
215	C₄⋊C₄	—	149	8	no	—
216	C₄⋊C₄	—	149	8	no	—
217	C₄⋊C₄	—	149	8	no	—
218	C₄⋊C₄	—	149	4	no	—
219	C₄⋊C₄	C₂²×D₁₄₉	1	8	yes	—
220	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
221	C₂²⋊C₄	—	149	8	no	—
222	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
223	C₄²	—	149	4	no	—
224	C₄²	—	149	4	no	—
225	C₄²	—	149	4	no	—
226	C₄²	—	149	4	no	—
227	C₄²	—	149	8	no	—
228	C₄²	—	149	8	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	Q₈×D₁₄₉	1	4	yes	—
232	C₂³	C₂³×D₁₄₉	1	2	yes	—
233	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	Q₈	—	298	16	no	—
236	Q₈	—	298	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	298	2	no	—
242	C₂×C₄	—	298	4	no	—
243	C₂×C₄	—	298	4	no	—
244	C₂×C₄	—	298	8	no	—
245	C₂×C₄	—	149	2	no	—
246	C₂×C₄	—	149	4	no	—
247	C₂×C₄	—	149	4	no	—
248	C₂×C₄	—	149	4	no	—
249	C₂×C₄	—	149	4	no	—
250	C₂×C₄	—	149	16	no	—
251	C₂×C₄	—	149	4	no	—
252	C₂×C₄	—	149	2	no	—
253	C₂×C₄	—	149	2	no	—
254	C₂×C₄	—	149	2	no	—
255	C₂×C₄	#14558735	1	4	yes	—
256	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
257	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
258	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
259	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
260	C₂²	—	2	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×Q₈×D₁₄₉	1	4	yes	—
264	C₂²	C₂×Q₈×D₁₄₉	1	2	yes	—
265	C₂²	#41×D₁₄₉	1	2	yes	—
266	C₂²	C₂×#14558735	1	2	yes	—
267	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	298	2	no	—
272	C₄	—	298	4	no	—
273	C₄	—	298	4	no	—
274	C₄	—	149	4	no	—
275	C₄	—	149	4	no	—
276	C₄	—	149	4	no	—
277	C₄	C₂×#14558735	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	#23750177	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₄₉	1	1	yes	characteristic
281	C₂	C₂×#41×D₁₄₉	1	1	yes	characteristic
282	C₂	C₂²×#14558735	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting	C₂×C₂²⋊Q₈×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂⁴×C₁₄₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₄₉
hypocenter	C₁₄₉	C₂×#533
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #23750177	C₂	direct (N × H)
2. C₂×#18011910	C₂	semidirect
3. C₂×#18011909	C₂	semidirect
4. C₂×#18011901	C₂	semidirect
7. C₂×Q₈×Dic₁₄₉	C₂	semidirect
8. C₂×#18011876	C₂	semidirect
9. C₂×#18011876	C₂	semidirect
17. #18011910	C₂²	semidirect
18. #18011909	C₂²	semidirect
19. #18011901	C₂²	semidirect
22. Q₈×Dic₁₄₉	C₂²	semidirect
23. #18011876	C₂²	semidirect
24. #18011876	C₂²	semidirect
52. C₂²×Dic₁₄₉	Q₈	semidirect
95. C₂×Dic₁₄₉	C₂×Q₈	semidirect
137. C₂×C₁₄₉	#533	semidirect
141. C₁₄₉	C₂×#533	semidirect
279. C₂	#23750177	direct (N × H)

Maximal subgroups

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

subgroup	index
#23750177	2
C₂×#18011910	2
C₂×#18011909	2
C₂×#18011901	2
C₂×#18011890	2
C₂×#18011888	2
C₂×Q₈×Dic₁₄₉	2
C₂×#18011876	2
C₂×#18011867	2
C₂²×C₄×Dic₁₄₉	2
C₂×C₂²⋊Q₈×C₁₄₉	2
C₂×#533	149

Subgroups — C2×#23750177 #34057977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750177 #34057977