Subgroups — C₂×#26011507 #36362104

Order 22144.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14583826	C₂	1	1	yes	maximal, characteristic
2	C₂×#19149638	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011507	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19149605	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583826	—	2	1	no	—
7	C₂×#14583826	—	2	2	no	—
8	C₂×#14583826	—	2	1	no	—
9	C₂×#14583826	—	2	2	no	—
10	C₂×#14583826	C₄	1	2	yes	complemented
11	C₂×#14583826	C₄	1	4	yes	complemented
12	C₂×#14583826	C₂²	1	2	yes	—
13	C₂×#14583826	C₂²	1	4	yes	—
14	C₂²×Dic₃₄₆	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₆₉₂	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₇₃	—	2	1	no	—
17	C₂²×C₄×D₁₇₃	—	2	1	no	—
18	#19149638	—	2	2	no	—
19	#19149638	C₂²	1	4	yes	complemented
20	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
21	C₂×C₂²⋊Dic₁₇₃	—	2	1	no	—
22	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	complemented
23	#19149605	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
25	C₂×M₁₆×C₁₇₃	—	2	1	no	—
26	C₂×M₁₆×C₁₇₃	—	2	2	no	—
27	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	complemented
28	C₂×M₁₆×C₁₇₃	C₂²	1	4	yes	complemented
29	C₂²×C₈×C₁₇₃	—	2	1	no	—
30	C₂³×D₁₇₃	—	2	1	no	—
31	#14583826	—	2	4	no	—
32	#14583826	—	2	4	no	—
33	#14583826	—	2	4	no	—
34	#14583826	—	2	4	no	—
35	#14583826	—	2	2	no	—
36	#14583826	—	2	2	no	—
37	#14583826	—	2	4	no	—
38	#14583826	C₂×C₄	1	4	yes	—
39	#14583826	C₂×C₄	1	8	yes	complemented
40	#14583826	C₂³	1	4	yes	—
41	C₂×Dic₃₄₆	—	2	1	no	—
42	C₂×Dic₃₄₆	—	2	2	no	—
43	C₂×Dic₃₄₆	C₂×C₄	1	2	yes	—
44	C₂×Dic₃₄₆	C₂×C₄	1	4	yes	complemented
45	C₂×D₆₉₂	—	2	2	no	—
46	C₂×D₆₉₂	—	2	1	no	—
47	C₂×D₆₉₂	C₂×C₄	1	2	yes	—
48	C₂×D₆₉₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₂²⋊D₁₇₃	—	2	2	no	—
50	C₂×C₂²⋊D₁₇₃	—	2	4	no	—
51	C₂×C₂²⋊D₁₇₃	—	2	2	no	—
52	C₂×C₂²⋊D₁₇₃	—	2	4	no	—
53	C₂²×Dic₁₇₃	—	2	1	no	—
54	C₂²×Dic₁₇₃	—	2	2	no	—
55	C₂×C₄×D₁₇₃	—	2	4	no	—
56	C₂×C₄×D₁₇₃	—	2	2	no	—
57	C₂×C₄×D₁₇₃	—	2	4	no	—
58	C₂×C₄×D₁₇₃	—	2	2	no	—
59	C₄⋊Dic₁₇₃	—	2	4	no	—
60	C₂²⋊Dic₁₇₃	—	2	4	no	—
61	C₄×Dic₁₇₃	—	2	2	no	—
62	C₄×Dic₁₇₃	C₂³	1	4	yes	complemented
63	C₂⁴×C₁₇₃	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
69	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
72	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
74	M₁₆×C₁₇₃	—	2	4	no	—
75	M₁₆×C₁₇₃	—	2	2	no	—
76	M₁₆×C₁₇₃	C₂³	1	4	yes	complemented
77	C₂×C₈×C₁₇₃	—	2	4	no	—
78	C₂×C₈×C₁₇₃	—	2	2	no	—
79	C₂²×D₁₇₃	—	2	2	no	—
80	C₂²×D₁₇₃	—	2	4	no	—
81	C₂²×D₁₇₃	—	4	4	no	—
82	Dic₃₄₆	—	2	2	no	—
83	Dic₃₄₆	—	2	4	no	—
84	Dic₃₄₆	C₂²×C₄	1	4	yes	—
85	D₆₉₂	—	2	2	no	—
86	D₆₉₂	—	2	4	no	—
87	D₆₉₂	C₂²×C₄	1	4	yes	—
88	C₂²⋊D₁₇₃	—	2	4	no	—
89	C₂²⋊D₁₇₃	—	2	8	no	—
90	C₂²⋊D₁₇₃	—	2	4	no	—
91	C₂×Dic₁₇₃	—	2	2	no	—
92	C₂×Dic₁₇₃	—	2	4	no	—
93	C₂×Dic₁₇₃	—	2	4	no	—
94	C₂×Dic₁₇₃	—	4	4	no	—
95	C₄×D₁₇₃	—	2	4	no	—
96	C₄×D₁₇₃	—	2	8	no	—
97	C₄×D₁₇₃	—	2	4	no	—
98	C₂³×C₁₇₃	—	4	2	no	—
99	C₂³×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₇₃	C₂²⋊C₄	1	2	yes	—
102	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₁₇₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
122	C₈×C₁₇₃	—	2	4	no	—
123	C₂²×C₁₇₃	—	2	2	no	—
124	C₂²×C₁₇₃	—	2	2	no	—
125	C₂²×C₁₇₃	—	2	2	no	—
126	C₂²×C₁₇₃	—	4	2	no	—
127	C₂²×C₁₇₃	—	4	2	no	—
128	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₇₃	—	2	4	no	—
134	C₂×D₁₇₃	—	4	8	no	—
135	C₂×D₁₇₃	—	4	4	no	—
136	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
137	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₇₃	—	2	4	no	—
143	Dic₁₇₃	—	4	4	no	—
144	C₂×C₁₇₃	—	2	1	no	—
145	C₂×C₁₇₃	—	2	2	no	—
146	C₂×C₁₇₃	—	2	2	no	—
147	C₂×C₁₇₃	—	2	1	no	—
148	C₂×C₁₇₃	#421	1	2	yes	complemented
149	C₂×C₁₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	D₁₇₃	—	4	4	no	—
151	C₁₇₃	C₂×#421	1	1	yes	characteristic, complemented, Sylow
152	C₂×#421	—	173	1	no	maximal, Sylow
153	C₂²×#41	—	173	1	no	—
154	C₂²×M₁₆	D₁₇₃	1	1	yes	characteristic, complemented
155	C₂×#117	—	173	1	no	—
156	#421	—	173	8	no	—
157	C₂×#104	—	173	4	no	—
158	C₂×#41	—	346	2	no	—
159	C₂×#41	—	346	1	no	—
160	C₂×#41	—	346	1	no	—
161	C₂×#41	—	346	2	no	—
162	C₂×#41	—	173	2	no	—
163	C₂×#41	—	173	4	no	—
164	C₂×#41	—	173	2	no	—
165	C₂×#41	—	173	4	no	—
166	C₂²×Q₈	—	173	1	no	—
167	C₂²×D₄	—	346	1	no	—
168	C₂²×D₄	—	173	1	no	—
169	C₂³×C₄	—	346	1	no	—
170	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	C₂×D₁₇₃	1	2	yes	complemented
174	C₂×M₁₆	C₂×D₁₇₃	1	4	yes	complemented
175	C₂²×C₈	—	2	1	no	—
176	#117	—	346	2	no	—
177	#117	—	173	4	no	—
178	C₂×C₄⋊C₄	—	346	1	no	—
179	C₂×C₂²⋊C₄	—	346	1	no	—
180	C₂×C₄²	—	173	2	no	—
181	#104	—	173	16	no	—
182	C₂⁴	—	346	1	no	—
183	C₂⁴	D₆₉₂	1	1	yes	characteristic
184	#41	—	346	4	no	—
185	#41	—	346	2	no	—
186	#41	—	346	2	no	—
187	#41	—	346	4	no	—
188	#41	—	346	4	no	—
189	#41	—	346	4	no	—
190	#41	—	346	4	no	—
191	#41	—	173	4	no	—
192	#41	—	173	8	no	—
193	#41	—	173	4	no	—
194	C₂×Q₈	—	346	2	no	—
195	C₂×Q₈	—	346	1	no	—
196	C₂×Q₈	—	173	2	no	—
197	C₂×Q₈	—	173	4	no	—
198	C₂×D₄	—	346	2	no	—
199	C₂×D₄	—	346	4	no	—
200	C₂×D₄	—	346	2	no	—
201	C₂×D₄	—	346	4	no	—
202	C₂×D₄	—	346	2	no	—
203	C₂×D₄	—	346	1	no	—
204	C₂×D₄	—	173	2	no	—
205	C₂×D₄	—	173	4	no	—
206	C₂²×C₄	—	346	1	no	—
207	C₂²×C₄	—	346	4	no	—
208	C₂²×C₄	—	346	2	no	—
209	C₂²×C₄	—	346	4	no	—
210	C₂²×C₄	—	346	2	no	—
211	C₂²×C₄	—	346	2	no	—
212	C₂²×C₄	C₄×D₁₇₃	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₁₇₃	1	1	yes	characteristic
214	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
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	D₆₉₂	1	1	yes	characteristic
219	C₂²×C₄	D₆₉₂	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	C₂²×D₁₇₃	1	4	yes	complemented
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₄⋊C₄	—	346	4	no	—
228	C₂²⋊C₄	—	346	4	no	—
229	C₄²	—	346	2	no	—
230	C₄²	—	173	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	346	4	no	—
233	C₂³	—	346	2	no	—
234	C₂³	—	692	4	no	—
235	C₂³	#14583803	1	1	yes	characteristic
236	C₂³	#14583803	1	1	yes	characteristic
237	C₂³	#14583803	1	2	yes	—
238	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
239	C₂³	C₂×D₆₉₂	1	2	yes	—
240	Q₈	—	346	2	no	—
241	Q₈	—	346	4	no	—
242	Q₈	—	173	4	no	—
243	D₄	—	346	4	no	—
244	D₄	—	346	8	no	—
245	D₄	—	346	4	no	—
246	D₄	—	346	2	no	—
247	D₄	—	346	4	no	—
248	D₄	—	173	4	no	—
249	C₂×C₄	—	346	4	no	—
250	C₂×C₄	—	346	2	no	—
251	C₂×C₄	—	346	8	no	—
252	C₂×C₄	—	346	4	no	—
253	C₂×C₄	—	346	4	no	—
254	C₂×C₄	—	346	4	no	—
255	C₂×C₄	—	692	4	no	—
256	C₂×C₄	#14583803	1	1	yes	characteristic
257	C₂×C₄	#14583803	1	2	yes	—
258	C₂×C₄	#14583803	1	2	yes	—
259	C₂×C₄	#14583803	1	1	yes	characteristic
260	C₂×C₄	#14583803	1	2	yes	—
261	C₂×C₄	#14583803	1	2	yes	—
262	C₂×C₄	#14583803	1	2	yes	—
263	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
264	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
266	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
272	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
273	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	346	4	no	—
281	C₂²	—	692	8	no	—
282	C₂²	—	692	4	no	—
283	C₂²	C₂×#14583803	1	2	yes	—
284	C₂²	C₂×#14583803	1	2	yes	—
285	C₂²	C₂×#14583803	1	1	yes	characteristic
286	C₂²	C₂×#14583803	1	1	yes	characteristic
287	C₂²	C₂²×D₆₉₂	1	1	yes	characteristic
288	C₄	—	346	4	no	—
289	C₄	—	692	4	no	—
290	C₄	C₂×#14583803	1	2	yes	—
291	C₄	C₂×#14583803	1	1	yes	characteristic
292	C₄	C₂×#14583803	1	2	yes	—
293	C₄	C₂×#14583803	1	1	yes	characteristic
294	C₄	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
295	C₄	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	692	4	no	—
301	C₂	#26011507	1	2	yes	direct product factor
302	C₂	C₂²×#14583803	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇₃
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	C₂²×M₁₆×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
prime radical	C₂²×#14583826	C₂
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	C₂²×M₁₆	D₁₇₃
hypocenter	C₁₇₃	C₂×#421
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19149638	C₂	semidirect
3. #26011507	C₂	direct (N × H)
4. C₂×#19149605	C₂	semidirect
5. C₂²×M₁₆×C₁₇₃	C₂	semidirect
10. C₂×#14583826	C₄	semidirect
11. C₂×#14583826	C₄	semidirect
14. C₂²×Dic₃₄₆	C₄	semidirect
15. C₂²×D₆₉₂	C₄	semidirect
19. #19149638	C₂²	semidirect
22. C₂×C₄×Dic₁₇₃	C₂²	semidirect
23. #19149605	C₂²	semidirect
27. C₂×M₁₆×C₁₇₃	C₂²	semidirect
28. C₂×M₁₆×C₁₇₃	C₂²	semidirect
39. #14583826	C₂×C₄	semidirect
44. C₂×Dic₃₄₆	C₂×C₄	semidirect
48. C₂×D₆₉₂	C₂×C₄	semidirect
62. C₄×Dic₁₇₃	C₂³	semidirect
76. M₁₆×C₁₇₃	C₂³	semidirect
148. C₂×C₁₇₃	#421	semidirect
151. C₁₇₃	C₂×#421	semidirect
154. C₂²×M₁₆	D₁₇₃	semidirect
173. C₂×M₁₆	C₂×D₁₇₃	semidirect
174. C₂×M₁₆	C₂×D₁₇₃	semidirect
224. M₁₆	C₂²×D₁₇₃	semidirect
301. C₂	#26011507	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×#14583826	2
C₂×#19149638	2
#26011507	2
C₂×#19149605	2
C₂²×M₁₆×C₁₇₃	2
C₂×#421	173

Subgroups — C2×#26011507 #36362104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26011507 #36362104