Subgroups — C₂×#26012081 #36366906

Order 22144.

1336 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012081	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19149674	C₂	1	2	yes	maximal, complemented
3	C₂×#19149668	C₂	1	2	yes	maximal, complemented
4	C₂×#19149664	C₂	1	2	yes	maximal, complemented
5	C₂×#19149654	C₂	1	2	yes	maximal, complemented
6	C₂×#19149653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19149649	C₂	1	2	yes	maximal, complemented
8	C₂×#19149639	C₂	1	2	yes	maximal, complemented
9	C₂²×#14583803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₇₃	C₂²	1	1	yes	characteristic
12	C₂²×D₆₉₂	C₂²	1	2	yes	—
13	C₂²×D₆₉₂	C₂²	1	2	yes	—
14	C₂²×D₆₉₂	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₂²⋊D₁₇₃	C₂²	1	2	yes	—
16	C₂³×Dic₁₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇₃	C₂²	1	2	yes	complemented
18	#19149674	C₂²	1	16	yes	complemented
19	#19149668	C₂²	1	16	yes	complemented
20	#19149664	C₂²	1	16	yes	complemented
21	#19149654	C₂²	1	16	yes	complemented
22	#19149653	C₂²	1	8	yes	complemented
23	#19149649	C₂²	1	16	yes	complemented
24	#19149639	C₂²	1	16	yes	complemented
25	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×#14583805	C₂²	1	2	yes	complemented
27	C₂×#14583803	C₂²	1	2	yes	—
28	C₂×#14583803	C₂²	1	2	yes	—
29	C₂×#14583803	C₂²	1	2	yes	—
30	C₂×#14583803	C₂²	1	2	yes	—
31	C₂×#14583803	C₂²	1	8	yes	complemented
32	C₂×#14583803	C₂²	1	2	yes	complemented
33	C₂×#14583803	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	complemented
35	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₇₃	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	complemented
39	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	2	yes	—
41	C₂³×D₁₇₃	—	2	4	no	—
42	C₂³×D₁₇₃	—	4	2	no	—
43	C₂³×D₁₇₃	—	4	2	no	—
44	C₂³×D₁₇₃	C₂³	1	2	yes	—
45	C₂³×D₁₇₃	C₂³	1	2	yes	—
46	C₂³×D₁₇₃	C₂³	1	4	yes	—
47	C₂×D₆₉₂	—	4	4	no	—
48	C₂×D₆₉₂	—	4	4	no	—
49	C₂×D₆₉₂	—	4	2	no	—
50	C₂×D₆₉₂	C₂³	1	8	yes	—
51	C₂×D₆₉₂	C₂³	1	8	yes	—
52	C₂×D₆₉₂	C₂³	1	4	yes	complemented
53	C₂×C₂²⋊D₁₇₃	—	2	8	no	—
54	C₂×C₂²⋊D₁₇₃	C₂³	1	8	yes	—
55	C₂²×Dic₁₇₃	—	4	2	no	—
56	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
57	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
58	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
59	C₂×C₄×D₁₇₃	—	4	4	no	—
60	C₂×C₄×D₁₇₃	C₂³	1	8	yes	complemented
61	C₄⋊Dic₁₇₃	C₂³	1	4	yes	complemented
62	#14583805	C₂³	1	8	yes	complemented
63	#14583803	C₂³	1	8	yes	—
64	#14583803	C₂³	1	8	yes	—
65	#14583803	C₂³	1	8	yes	—
66	#14583803	C₂³	1	8	yes	—
67	#14583803	C₂³	1	8	yes	complemented
68	#14583803	C₂³	1	8	yes	complemented
69	C₄×Dic₁₇₃	C₂³	1	8	yes	complemented
70	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
71	C₂×Q₈×C₁₇₃	—	4	2	no	—
72	C₂×Q₈×C₁₇₃	C₂³	1	4	yes	complemented
73	C₂²×C₄×C₁₇₃	—	4	2	no	—
74	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
79	C₄⋊C₄×C₁₇₃	C₂³	1	8	yes	complemented
80	C₄⋊C₄×C₁₇₃	C₂³	1	4	yes	complemented
81	C₂²⋊C₄×C₁₇₃	C₂³	1	8	yes	—
82	C₂²×D₁₇₃	—	2	4	no	—
83	C₂²×D₁₇₃	—	2	8	no	—
84	C₂²×D₁₇₃	—	2	4	no	—
85	C₂²×D₁₇₃	—	2	4	no	—
86	C₂²×D₁₇₃	—	2	4	no	—
87	C₂²×D₁₇₃	—	2	4	no	—
88	C₂²×D₁₇₃	—	2	4	no	—
89	C₂²×D₁₇₃	—	4	4	no	—
90	C₂²×D₁₇₃	—	4	4	no	—
91	C₂²×D₁₇₃	—	4	16	no	—
92	C₂²×D₁₇₃	C₂⁴	1	4	yes	—
93	C₂²×D₁₇₃	C₂⁴	1	4	yes	—
94	D₆₉₂	—	4	8	no	—
95	D₆₉₂	—	4	8	no	—
96	D₆₉₂	—	4	4	no	—
97	C₂²⋊D₁₇₃	—	2	16	no	—
98	C₂×Dic₁₇₃	—	2	4	no	—
99	C₂×Dic₁₇₃	—	2	4	no	—
100	C₂×Dic₁₇₃	—	4	4	no	—
101	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
102	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
103	C₄×D₁₇₃	—	4	8	no	—
104	C₂³×C₁₇₃	—	2	2	no	—
105	C₂³×C₁₇₃	—	2	2	no	—
106	C₂³×C₁₇₃	#41	1	4	yes	—
107	C₂³×C₁₇₃	C₂⁴	1	2	yes	—
108	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
109	Q₈×C₁₇₃	—	4	4	no	—
110	C₂×C₄×C₁₇₃	—	2	4	no	—
111	C₂×C₄×C₁₇₃	—	2	2	no	—
112	C₂×C₄×C₁₇₃	—	2	4	no	—
113	C₂×C₄×C₁₇₃	—	2	2	no	—
114	C₂×C₄×C₁₇₃	—	4	4	no	—
115	C₂×C₄×C₁₇₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₇₃	C₂⁴	1	4	yes	—
117	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
119	C₂²×C₁₇₃	—	2	8	no	—
120	C₂²×C₁₇₃	—	2	2	no	—
121	C₂²×C₁₇₃	—	2	2	no	—
122	C₂²×C₁₇₃	C₂×#41	1	4	yes	—
123	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
124	C₂²×C₁₇₃	H(5,𝔽₂)	1	2	yes	—
125	C₂²×C₁₇₃	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
127	C₂×D₁₇₃	—	2	4	no	—
128	C₂×D₁₇₃	—	2	4	no	—
129	C₂×D₁₇₃	—	2	4	no	—
130	C₂×D₁₇₃	—	2	4	no	—
131	C₂×D₁₇₃	—	2	4	no	—
132	C₂×D₁₇₃	—	2	4	no	—
133	C₂×D₁₇₃	—	4	16	no	—
134	C₂×D₁₇₃	—	4	8	no	—
135	C₂×D₁₇₃	—	4	8	no	—
136	C₄×C₁₇₃	—	2	2	no	—
137	C₄×C₁₇₃	—	2	2	no	—
138	C₄×C₁₇₃	—	2	4	no	—
139	C₄×C₁₇₃	—	2	4	no	—
140	Dic₁₇₃	—	2	4	no	—
141	Dic₁₇₃	—	2	4	no	—
142	C₂×C₁₇₃	—	2	4	no	—
143	C₂×C₁₇₃	#538	1	4	yes	complemented
144	C₂×C₁₇₃	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₁₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	D₁₇₃	—	4	4	no	—
148	D₁₇₃	—	4	4	no	—
149	C₁₇₃	C₂×#538	1	1	yes	characteristic, complemented, Sylow
150	C₂×#538	—	173	1	no	maximal, Sylow
151	#538	—	173	16	no	—
152	C₂×#126	—	173	2	no	—
153	C₂×#124	—	173	2	no	—
154	C₂×#123	—	173	2	no	—
155	C₂×C₂²⋊Q₈	D₁₇₃	1	1	yes	characteristic, complemented
156	C₂×#121	—	173	1	no	—
157	C₂×#121	—	173	2	no	—
158	C₂×#120	—	173	2	no	—
159	C₂×C₄×D₄	—	173	2	no	—
160	C₂²×C₂²⋊C₄	—	173	1	no	—
161	C₂⁵	—	173	1	no	—
162	C₂²×Q₈	C₂×D₁₇₃	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	173	2	no	—
164	C₂²×D₄	—	173	2	no	—
165	C₂²×D₄	—	173	2	no	—
166	C₂²×D₄	—	173	1	no	—
167	C₂³×C₄	—	173	1	no	—
168	C₂³×C₄	—	173	2	no	—
169	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
170	#126	—	173	16	no	—
171	#124	—	173	16	no	—
172	#123	—	173	16	no	—
173	C₂²⋊Q₈	C₂×D₁₇₃	1	8	yes	complemented
174	#121	—	173	8	no	—
175	#121	—	173	16	no	—
176	#120	—	173	16	no	—
177	C₄×D₄	—	173	16	no	—
178	C₂×C₄⋊C₄	—	173	2	no	—
179	C₂×C₄⋊C₄	—	173	1	no	—
180	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	complemented
181	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic, complemented
182	C₂×C₂²⋊C₄	—	173	2	no	—
183	C₂×C₂²⋊C₄	—	173	2	no	—
184	C₂×C₂²⋊C₄	—	173	8	no	—
185	C₂×C₂²⋊C₄	—	173	2	no	—
186	C₂×C₂²⋊C₄	—	173	2	no	—
187	C₂×C₂²⋊C₄	—	173	2	no	—
188	C₂×C₂²⋊C₄	—	173	2	no	—
189	C₂×C₂²⋊C₄	C₂×D₁₇₃	1	2	yes	—
190	C₂×C₄²	—	173	2	no	—
191	C₂⁴	—	346	4	no	—
192	C₂⁴	—	173	2	no	—
193	C₂⁴	—	173	2	no	—
194	C₂⁴	—	173	4	no	—
195	C₂⁴	—	692	2	no	—
196	C₂⁴	—	692	2	no	—
197	C₂⁴	C₂²×D₁₇₃	1	1	yes	characteristic
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	C₂²×D₁₇₃	1	4	yes	complemented
200	C₂×D₄	—	346	8	no	—
201	C₂×D₄	—	173	8	no	—
202	C₂×D₄	—	173	8	no	—
203	C₂×D₄	—	173	8	no	—
204	C₂×D₄	—	173	4	no	—
205	C₂×D₄	—	692	2	no	—
206	C₂×D₄	—	692	4	no	—
207	C₂×D₄	—	692	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	173	2	no	—
210	C₂²×C₄	—	173	2	no	—
211	C₂²×C₄	—	173	4	no	—
212	C₂²×C₄	—	173	8	no	—
213	C₂²×C₄	—	692	4	no	—
214	C₂²×C₄	—	692	2	no	—
215	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
220	C₄⋊C₄	—	173	8	no	—
221	C₄⋊C₄	—	173	4	no	—
222	C₄⋊C₄	C₂²×D₁₇₃	1	8	yes	complemented
223	C₄⋊C₄	C₂²×D₁₇₃	1	4	yes	complemented
224	C₂²⋊C₄	—	173	8	no	—
225	C₂²⋊C₄	—	173	8	no	—
226	C₂²⋊C₄	—	173	8	no	—
227	C₂²⋊C₄	—	173	8	no	—
228	C₂²⋊C₄	—	173	8	no	—
229	C₂²⋊C₄	—	173	8	no	—
230	C₂²⋊C₄	C₂²×D₁₇₃	1	8	yes	—
231	C₄²	—	173	8	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	346	4	no	—
235	C₂³	—	346	8	no	—
236	C₂³	—	346	4	no	—
237	C₂³	—	346	4	no	—
238	C₂³	—	346	4	no	—
239	C₂³	—	346	4	no	—
240	C₂³	—	346	4	no	—
241	C₂³	—	173	4	no	—
242	C₂³	—	173	4	no	—
243	C₂³	—	692	4	no	—
244	C₂³	—	692	4	no	—
245	C₂³	—	692	16	no	—
246	C₂³	#14583827	1	4	yes	—
247	C₂³	C₂³×D₁₇₃	1	2	yes	—
248	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
249	Q₈	—	4	4	no	—
250	D₄	—	346	16	no	—
251	D₄	—	692	4	no	—
252	D₄	—	692	8	no	—
253	D₄	—	692	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	346	4	no	—
260	C₂×C₄	—	346	4	no	—
261	C₂×C₄	—	173	4	no	—
262	C₂×C₄	—	173	4	no	—
263	C₂×C₄	—	692	8	no	—
264	C₂×C₄	—	692	4	no	—
265	C₂×C₄	C₂³×D₁₇₃	1	4	yes	—
266	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
267	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
268	C₂×C₄	C₂³×D₁₇₃	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	346	4	no	—
273	C₂²	—	346	4	no	—
274	C₂²	—	346	4	no	—
275	C₂²	—	346	4	no	—
276	C₂²	—	346	4	no	—
277	C₂²	—	346	4	no	—
278	C₂²	—	692	16	no	—
279	C₂²	—	692	8	no	—
280	C₂²	—	692	8	no	—
281	C₂²	C₂×#14583827	1	4	yes	—
282	C₂²	C₂×#14583827	1	2	yes	—
283	C₂²	#19149724	1	2	yes	—
284	C₂²	#19149725	1	2	yes	—
285	C₂²	C₂⁴×D₁₇₃	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	346	4	no	—
291	C₄	—	346	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	692	4	no	—
294	C₂	—	692	4	no	—
295	C₂	#26012081	1	4	yes	direct product factor
296	C₂	C₂²×#14583827	1	1	yes	characteristic
297	C₂	C₂×#19149724	1	1	yes	characteristic
298	C₂	C₂×#19149725	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₇₃
commutator, cosocle	C₂²×C₁₇₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₇₃
Fitting	C₂×C₂²⋊Q₈×C₁₇₃	C₂
socle	C₂³×C₁₇₃	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₁₇₃
hypocenter	C₁₇₃	C₂×#538
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #26012081	C₂	direct (N × H)
2. C₂×#19149674	C₂	semidirect
3. C₂×#19149668	C₂	semidirect
4. C₂×#19149664	C₂	semidirect
5. C₂×#19149654	C₂	semidirect
6. C₂×#19149653	C₂	semidirect
7. C₂×#19149649	C₂	semidirect
8. C₂×#19149639	C₂	semidirect
9. C₂²×#14583803	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₁₇₃	C₂	semidirect
14. C₂²×D₆₉₂	C₂²	semidirect
17. C₂²×C₄×D₁₇₃	C₂²	semidirect
18. #19149674	C₂²	semidirect
19. #19149668	C₂²	semidirect
20. #19149664	C₂²	semidirect
21. #19149654	C₂²	semidirect
22. #19149653	C₂²	semidirect
23. #19149649	C₂²	semidirect
24. #19149639	C₂²	semidirect
25. C₂×C₄⋊Dic₁₇₃	C₂²	semidirect
26. C₂×#14583805	C₂²	semidirect
31. C₂×#14583803	C₂²	semidirect
32. C₂×#14583803	C₂²	semidirect
33. C₂×#14583803	C₂²	semidirect
34. C₂×C₄×Dic₁₇₃	C₂²	semidirect
35. C₂²×Q₈×C₁₇₃	C₂²	semidirect
37. C₂²⋊Q₈×C₁₇₃	C₂²	semidirect
38. C₂×C₄⋊C₄×C₁₇₃	C₂²	semidirect
39. C₂×C₄⋊C₄×C₁₇₃	C₂²	semidirect
52. C₂×D₆₉₂	C₂³	semidirect
60. C₂×C₄×D₁₇₃	C₂³	semidirect
61. C₄⋊Dic₁₇₃	C₂³	semidirect
62. #14583805	C₂³	semidirect
67. #14583803	C₂³	semidirect
68. #14583803	C₂³	semidirect
69. C₄×Dic₁₇₃	C₂³	semidirect
72. C₂×Q₈×C₁₇₃	C₂³	semidirect
79. C₄⋊C₄×C₁₇₃	C₂³	semidirect
80. C₄⋊C₄×C₁₇₃	C₂³	semidirect
143. C₂×C₁₇₃	#538	semidirect
149. C₁₇₃	C₂×#538	semidirect
155. C₂×C₂²⋊Q₈	D₁₇₃	semidirect
162. C₂²×Q₈	C₂×D₁₇₃	semidirect
173. C₂²⋊Q₈	C₂×D₁₇₃	semidirect
180. C₂×C₄⋊C₄	C₂×D₁₇₃	semidirect
181. C₂×C₄⋊C₄	C₂×D₁₇₃	semidirect
199. C₂×Q₈	C₂²×D₁₇₃	semidirect
222. C₄⋊C₄	C₂²×D₁₇₃	semidirect
223. C₄⋊C₄	C₂²×D₁₇₃	semidirect
295. C₂	#26012081	direct (N × H)

Maximal subgroups

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

subgroup	index
#26012081	2
C₂×#19149674	2
C₂×#19149668	2
C₂×#19149664	2
C₂×#19149654	2
C₂×#19149653	2
C₂×#19149649	2
C₂×#19149639	2
C₂²×#14583803	2
C₂×C₂²⋊Q₈×C₁₇₃	2
C₂×#538	173

Subgroups — C2×#26012081 #36366906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26012081 #36366906