Subgroups — C₂×#22578024 #31794167

Order 17536.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578024	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984083	C₂	1	2	yes	maximal, complemented
4	C₂×#17984076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17984074	C₂	1	2	yes	maximal, complemented
6	C₂×#17984070	C₂	1	2	yes	maximal, complemented
7	C₂×#17984067	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#17984066	C₂	1	2	yes	maximal, complemented
9	C₂×#17984062	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
15	C₂²×D₅₄₈	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
18	#17984088	C₂²	1	8	yes	complemented
19	#17984083	C₂²	1	16	yes	complemented
20	#17984076	C₂²	1	8	yes	complemented
21	#17984074	C₂²	1	16	yes	complemented
22	#17984070	C₂²	1	16	yes	complemented
23	#17984067	C₂²	1	8	yes	complemented
24	#17984066	C₂²	1	16	yes	complemented
25	#17984062	C₂²	1	8	yes	—
26	C₄×Dic₂₇₄	C₂²	1	8	yes	complemented
27	C₄×D₅₄₈	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
31	C₂×#14555489	C₂²	1	2	yes	—
32	C₂×#14555489	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₁₃₇	C₂²	1	2	yes	—
34	C₂×#14555487	C₂²	1	2	yes	—
35	C₂×#14555487	C₂²	1	2	yes	—
36	C₂×#14555487	C₂²	1	2	yes	—
37	C₂×#14555487	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
39	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
40	#117×C₁₃₇	C₂²	1	8	yes	complemented
41	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
47	C₂³×D₁₃₇	C₂³	1	2	yes	—
48	C₂×Dic₂₇₄	—	2	4	no	—
49	C₂×Dic₂₇₄	—	4	4	no	—
50	C₂×Dic₂₇₄	C₂³	1	4	yes	—
51	C₂×Dic₂₇₄	C₂³	1	8	yes	—
52	C₂×D₅₄₈	—	2	4	no	—
53	C₂×D₅₄₈	C₂³	1	4	yes	—
54	C₂×C₂²⋊D₁₃₇	—	4	4	no	—
55	C₂×C₂²⋊D₁₃₇	C₂³	1	8	yes	—
56	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
57	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
58	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
59	C₂×C₄×D₁₃₇	—	4	4	no	—
60	C₂×C₄×D₁₃₇	C₂³	1	8	yes	—
61	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
62	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
63	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
64	#14555489	C₂³	1	8	yes	—
65	#14555489	C₂³	1	8	yes	—
66	C₂²⋊Dic₁₃₇	C₂³	1	8	yes	—
67	#14555487	C₂³	1	8	yes	—
68	#14555487	C₂³	1	8	yes	—
69	#14555487	C₂³	1	8	yes	—
70	#14555487	C₂³	1	8	yes	—
71	C₄×Dic₁₃₇	C₂³	1	8	yes	—
72	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	—	2	4	no	—
74	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
82	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₁₃₇	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₁₃₇	C₂³	1	4	yes	—
85	C₄²×C₁₃₇	C₂³	1	4	yes	—
86	C₄²×C₁₃₇	C₂³	1	4	yes	—
87	C₂²×D₁₃₇	—	2	4	no	—
88	C₂²×D₁₃₇	—	2	4	no	—
89	C₂²×D₁₃₇	—	2	4	no	—
90	C₂²×D₁₃₇	C₂⁴	1	4	yes	—
91	Dic₂₇₄	—	2	8	no	—
92	Dic₂₇₄	—	4	8	no	—
93	D₅₄₈	—	2	8	no	—
94	C₂²⋊D₁₃₇	—	4	8	no	—
95	C₂×Dic₁₃₇	—	2	4	no	—
96	C₂×Dic₁₃₇	—	2	4	no	—
97	C₂×Dic₁₃₇	—	2	4	no	—
98	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
99	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
100	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
101	C₄×D₁₃₇	—	4	8	no	—
102	C₂³×C₁₃₇	—	2	2	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	—	2	2	no	—
105	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
106	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	2	8	no	—
113	C₂×C₄×C₁₃₇	#41	1	4	yes	—
114	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	2	2	no	—
123	C₂²×C₁₃₇	—	4	4	no	—
124	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
125	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
127	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₃₇	—	2	4	no	—
129	C₂×D₁₃₇	—	2	4	no	—
130	C₂×D₁₃₇	—	2	4	no	—
131	C₂×D₁₃₇	—	4	8	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	2	2	no	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	2	2	no	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	C₂×#41	1	4	yes	—
138	Dic₁₃₇	—	2	4	no	—
139	Dic₁₃₇	—	2	4	no	—
140	Dic₁₃₇	—	2	4	no	—
141	C₂×C₁₃₇	—	4	2	no	—
142	C₂×C₁₃₇	#542	1	4	yes	complemented
143	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
144	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	D₁₃₇	—	4	4	no	—
147	C₁₃₇	C₂×#542	1	1	yes	characteristic, complemented, Sylow
148	C₂×#542	—	137	1	no	maximal, Sylow
149	#542	—	137	16	no	—
150	C₂×C₄⋊Q₈	—	137	1	no	—
151	C₂×#126	—	137	2	no	—
152	C₂×#124	—	137	2	no	—
153	C₂×#124	—	137	1	no	—
154	C₂×#123	—	137	2	no	—
155	C₂×C₂²⋊Q₈	—	137	1	no	—
156	C₂×C₂²⋊Q₈	—	137	2	no	—
157	C₂×#121	—	137	1	no	—
158	C₂×C₄×Q₈	—	137	1	no	—
159	C₂×C₄×D₄	—	137	1	no	—
160	C₂×#117	D₁₃₇	1	1	yes	characteristic, complemented
161	C₂²×Q₈	—	137	1	no	—
162	C₂²×Q₈	—	137	2	no	—
163	C₂²×D₄	—	137	2	no	—
164	C₂²×D₄	—	137	1	no	—
165	C₂³×C₄	—	137	2	no	—
166	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	137	8	no	—
168	#126	—	137	16	no	—
169	#124	—	137	16	no	—
170	#124	—	137	8	no	—
171	#123	—	137	16	no	—
172	C₂²⋊Q₈	—	137	8	no	—
173	C₂²⋊Q₈	—	137	16	no	—
174	#121	—	137	8	no	—
175	C₄×Q₈	—	137	8	no	—
176	C₄×D₄	—	137	8	no	—
177	#117	C₂×D₁₃₇	1	8	yes	complemented
178	C₂×C₄⋊C₄	—	137	1	no	—
179	C₂×C₄⋊C₄	—	137	1	no	—
180	C₂×C₄⋊C₄	—	137	2	no	—
181	C₂×C₄⋊C₄	—	137	2	no	—
182	C₂×C₄⋊C₄	—	137	2	no	—
183	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	137	2	no	—
186	C₂×C₂²⋊C₄	—	137	2	no	—
187	C₂×C₂²⋊C₄	—	137	2	no	—
188	C₂×C₂²⋊C₄	—	137	2	no	—
189	C₂×C₂²⋊C₄	—	137	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
192	C₂×C₄²	—	137	2	no	—
193	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
195	C₂⁴	—	137	2	no	—
196	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
197	C₂×Q₈	—	274	4	no	—
198	C₂×Q₈	—	548	4	no	—
199	C₂×Q₈	—	137	4	no	—
200	C₂×Q₈	—	137	8	no	—
201	C₂×D₄	—	274	4	no	—
202	C₂×D₄	—	548	4	no	—
203	C₂×D₄	—	137	8	no	—
204	C₂×D₄	—	137	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	548	4	no	—
207	C₂²×C₄	—	137	2	no	—
208	C₂²×C₄	—	137	2	no	—
209	C₂²×C₄	—	137	2	no	—
210	C₂²×C₄	—	137	8	no	—
211	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
212	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
218	C₄⋊C₄	—	137	8	no	—
219	C₄⋊C₄	—	137	8	no	—
220	C₄⋊C₄	—	137	4	no	—
221	C₄⋊C₄	—	137	4	no	—
222	C₄⋊C₄	—	137	8	no	—
223	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
224	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
225	C₂²⋊C₄	—	137	8	no	—
226	C₂²⋊C₄	—	137	8	no	—
227	C₂²⋊C₄	—	137	8	no	—
228	C₂²⋊C₄	—	137	8	no	—
229	C₂²⋊C₄	—	137	8	no	—
230	C₂²⋊C₄	C₂²×D₁₃₇	1	4	yes	—
231	C₂²⋊C₄	C₂²×D₁₃₇	1	4	yes	—
232	C₄²	—	137	8	no	—
233	C₄²	C₂²×D₁₃₇	1	4	yes	—
234	C₄²	C₂²×D₁₃₇	1	4	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	274	4	no	—
239	C₂³	—	274	4	no	—
240	C₂³	—	274	4	no	—
241	C₂³	—	137	4	no	—
242	C₂³	C₂³×D₁₃₇	1	2	yes	—
243	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
244	Q₈	—	274	8	no	—
245	Q₈	—	548	8	no	—
246	D₄	—	274	8	no	—
247	D₄	—	548	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	274	4	no	—
255	C₂×C₄	—	274	4	no	—
256	C₂×C₄	—	274	4	no	—
257	C₂×C₄	—	548	8	no	—
258	C₂×C₄	—	137	4	no	—
259	C₂×C₄	—	137	4	no	—
260	C₂×C₄	—	137	4	no	—
261	C₂×C₄	#14555510	1	4	yes	—
262	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
263	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
264	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
265	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
266	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
267	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	274	4	no	—
273	C₂²	—	274	4	no	—
274	C₂²	—	274	4	no	—
275	C₂²	—	548	8	no	—
276	C₂²	C₂×#14555510	1	2	yes	—
277	C₂²	#17984134	1	2	yes	—
278	C₂²	#17984135	1	2	yes	—
279	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	274	4	no	—
286	C₄	—	274	4	no	—
287	C₄	—	274	4	no	—
288	C₄	C₂×#14555510	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	548	4	no	—
291	C₂	#22578024	1	4	yes	direct product factor
292	C₂	C₂²×#14555510	1	1	yes	characteristic
293	C₂	C₂×#17984134	1	1	yes	characteristic
294	C₂	C₂×#17984135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×#117×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂×#17984062	C₂
hypercenter	C₂×#117	D₁₃₇
hypocenter	C₁₃₇	C₂×#542
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22578024	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
3. C₂×#17984083	C₂	semidirect
4. C₂×#17984076	C₂	semidirect
5. C₂×#17984074	C₂	semidirect
6. C₂×#17984070	C₂	semidirect
7. C₂×#17984067	C₂	semidirect
8. C₂×#17984066	C₂	semidirect
10. C₂×C₄×Dic₂₇₄	C₂	semidirect
11. C₂×C₄×D₅₄₈	C₂	semidirect
12. C₂×#117×C₁₃₇	C₂	semidirect
18. #17984088	C₂²	semidirect
19. #17984083	C₂²	semidirect
20. #17984076	C₂²	semidirect
21. #17984074	C₂²	semidirect
22. #17984070	C₂²	semidirect
23. #17984067	C₂²	semidirect
24. #17984066	C₂²	semidirect
26. C₄×Dic₂₇₄	C₂²	semidirect
27. C₄×D₅₄₈	C₂²	semidirect
40. #117×C₁₃₇	C₂²	semidirect
142. C₂×C₁₃₇	#542	semidirect
147. C₁₃₇	C₂×#542	semidirect
160. C₂×#117	D₁₃₇	semidirect
177. #117	C₂×D₁₃₇	semidirect
291. C₂	#22578024	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578024	2
C₂×#17984088	2
C₂×#17984083	2
C₂×#17984076	2
C₂×#17984074	2
C₂×#17984070	2
C₂×#17984067	2
C₂×#17984066	2
C₂×#17984062	2
C₂×C₄×Dic₂₇₄	2
C₂×C₄×D₅₄₈	2
C₂×#117×C₁₃₇	2
C₂×#542	137

Subgroups — C2×#22578024 #31794167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578024 #31794167