Subgroups — C₂×#22610913 #33090607

Order 18688.

1206 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988433	C₂	1	1	yes	maximal, characteristic
3	#22610913	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988346	C₂	1	4	yes	maximal
5	C₂×#533×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
8	#17988611	—	2	2	no	—
9	#17988611	C₂²	1	4	yes	—
10	C₂×C₄⋊#10596831	—	2	2	no	—
11	#17988433	—	2	2	no	—
12	#17988433	C₂²	1	4	yes	—
13	C₂×#14556498	—	2	2	no	—
14	#17988346	C₂²	1	16	yes	—
15	#533×C₇₃	—	2	2	no	—
16	#533×C₇₃	—	2	2	no	—
17	#533×C₇₃	C₄	1	4	yes	complemented
18	#533×C₇₃	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈×C₇₃	C₄	1	2	yes	complemented
20	C₂×#125×C₇₃	C₄	1	2	yes	complemented
21	C₂×C₂²⋊Q₈×C₇₃	—	2	2	no	—
22	C₂×C₄×Q₈×C₇₃	—	2	2	no	—
23	C₂×#117×C₇₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
25	C₂³×Dic₇₃	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	8	no	—
28	C₂×#11924012	C₂³	1	4	yes	—
29	C₂²×#10596831	—	2	2	no	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	—	2	4	no	—
32	C₂×C₄×Dic₇₃	Q₈	1	8	yes	complemented
33	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
34	C₄⋊#10596831	—	2	8	no	—
35	#14556498	—	2	8	no	—
36	C₂²×Q₈×C₇₃	—	2	2	no	—
37	C₂³×C₄×C₇₃	D₄	1	2	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₃	—	2	4	no	—
40	C₄⋊Q₈×C₇₃	C₂×C₄	1	8	yes	complemented
41	#125×C₇₃	—	2	4	no	—
42	#125×C₇₃	C₂×C₄	1	8	yes	complemented
43	C₂²⋊Q₈×C₇₃	—	2	8	no	—
44	C₂²⋊Q₈×C₇₃	—	2	8	no	—
45	C₄×Q₈×C₇₃	—	2	8	no	—
46	C₄×Q₈×C₇₃	—	2	8	no	—
47	#117×C₇₃	—	2	2	no	—
48	#117×C₇₃	—	2	2	no	—
49	#117×C₇₃	C₂×C₄	1	4	yes	—
50	#117×C₇₃	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
55	C₂×C₄²×C₇₃	—	2	2	no	—
56	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	D₄	1	2	yes	—
61	C₂×C₄²×C₇₃	D₄	1	2	yes	—
62	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
63	C₂²×Dic₇₃	—	2	8	no	—
64	C₂²×Dic₇₃	—	2	4	no	—
65	C₂²×Dic₇₃	—	4	8	no	—
66	#11924012	—	2	8	no	—
67	#11924012	—	4	4	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	—	4	4	no	—
70	C₄×Dic₇₃	—	2	8	no	—
71	C₄×Dic₇₃	—	2	4	no	—
72	C₄×Dic₇₃	C₂×Q₈	1	8	yes	complemented
73	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₇₃	—	2	8	no	—
75	C₂×Q₈×C₇₃	—	4	8	no	—
76	C₂²×C₄×C₇₃	—	2	4	no	—
77	C₂²×C₄×C₇₃	—	4	4	no	—
78	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇₃	#41	1	4	yes	—
89	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₇₃	—	2	16	no	—
91	C₄⋊C₄×C₇₃	—	2	8	no	—
92	C₄⋊C₄×C₇₃	—	2	4	no	—
93	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄×C₇₃	—	2	4	no	—
95	C₂²⋊C₄×C₇₃	—	2	4	no	—
96	C₄²×C₇₃	—	2	4	no	—
97	C₄²×C₇₃	—	2	4	no	—
98	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₇₃	C₂×D₄	1	4	yes	—
104	C₂³×C₇₃	—	2	4	no	—
105	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₇₃	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₇₃	—	4	16	no	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	—	2	4	no	—
112	C₂×C₄×C₇₃	—	2	8	no	—
113	C₂×C₄×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	4	4	no	—
115	C₂×C₄×C₇₃	—	4	8	no	—
116	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
122	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₇₃	#124	1	2	yes	—
124	C₂×C₄×C₇₃	#124	1	2	yes	—
125	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
131	C₂×Dic₇₃	—	2	8	no	—
132	C₂×Dic₇₃	—	4	8	no	—
133	C₂×Dic₇₃	—	4	16	no	—
134	#10596831	—	4	4	no	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	—	2	8	no	—
138	C₂²×C₇₃	#391	1	2	yes	—
139	C₂²×C₇₃	#391	1	2	yes	—
140	C₂²×C₇₃	#421	1	4	yes	—
141	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
143	C₄×C₇₃	—	2	4	no	—
144	C₄×C₇₃	—	2	4	no	—
145	C₄×C₇₃	—	4	8	no	—
146	C₄×C₇₃	#391	1	2	yes	—
147	C₄×C₇₃	#391	1	2	yes	—
148	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
149	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
150	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	Dic₇₃	—	4	8	no	—
152	C₂×#1995	—	73	1	no	maximal, Sylow
153	C₂×C₇₃	—	2	2	no	—
154	C₂×C₇₃	—	2	2	no	—
155	C₂×C₇₃	#1995	1	4	yes	complemented
156	C₂×C₇₃	C₂×#391	1	1	yes	characteristic
157	C₂×C₇₃	C₂×#421	1	2	yes	—
158	C₂×#533	D₇₃	1	1	yes	characteristic
159	C₂×#423	—	73	1	no	—
160	C₂×#376	—	73	1	no	—
161	#1995	—	73	8	no	—
162	C₂×#339	—	73	4	no	—
163	C₇₃	C₂×#1995	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	73	2	no	—
165	#533	—	2	2	no	—
166	#533	—	2	2	no	—
167	#533	Dic₇₃	1	4	yes	complemented
168	#533	C₂×D₇₃	1	4	yes	—
169	C₂×C₄⋊Q₈	Dic₇₃	1	2	yes	complemented
170	C₂×#125	Dic₇₃	1	2	yes	complemented
171	C₂×C₂²⋊Q₈	—	2	2	no	—
172	C₂×C₄×Q₈	—	2	2	no	—
173	C₂×#117	C₂×D₇₃	1	2	yes	—
174	C₂²×C₄²	—	73	2	no	—
175	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
176	#423	—	73	4	no	—
177	#423	—	146	2	no	—
178	C₂×C₄⋊C₈	—	146	2	no	—
179	#376	—	73	4	no	—
180	#376	—	146	2	no	—
181	C₂×#95	—	146	2	no	—
182	#339	—	73	16	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂³×C₄	—	146	2	no	—
185	C₂³×C₄	C₂²⋊D₇₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
187	C₂×M₁₆	—	73	4	no	—
188	C₂×M₁₆	—	146	8	no	—
189	C₂×M₁₆	—	146	2	no	—
190	C₂²×C₈	—	146	2	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	C₂×Dic₇₃	1	8	yes	complemented
193	#125	—	2	4	no	—
194	#125	C₂×Dic₇₃	1	8	yes	complemented
195	C₂²⋊Q₈	—	2	8	no	—
196	C₂²⋊Q₈	—	2	8	no	—
197	C₄×Q₈	—	2	8	no	—
198	C₄×Q₈	—	2	8	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×Dic₇₃	1	4	yes	—
202	#117	C₂²×D₇₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	73	8	no	—
209	C₂×C₄²	—	73	4	no	—
210	C₂×C₄²	—	146	2	no	—
211	C₂×C₄²	—	146	4	no	—
212	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
213	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
215	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
218	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
219	C₄⋊C₈	—	146	8	no	—
220	#95	—	146	8	no	—
221	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	292	8	no	—
227	C₂²×C₄	—	146	8	no	—
228	C₂²×C₄	—	146	4	no	—
229	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₇₃	1	4	yes	—
232	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₇₃	1	4	yes	—
239	C₂²×C₄	#11924029	1	4	yes	—
240	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
241	M₁₆	—	292	4	no	—
242	M₁₆	—	146	8	no	—
243	C₂×C₈	—	292	4	no	—
244	C₂×C₈	—	146	4	no	—
245	C₄⋊C₄	—	2	16	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×Dic₇₃	1	8	yes	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	73	8	no	—
254	C₄²	—	146	8	no	—
255	C₄²	—	146	4	no	—
256	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
257	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
258	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
259	C₄²	C₂²×Dic₇₃	1	2	yes	—
260	C₄²	C₂²×Dic₇₃	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₇₃	1	4	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
264	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
265	C₂³	#14556567	1	4	yes	—
266	C₂³	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
267	Q₈	—	4	16	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	8	no	—
274	C₂×C₄	—	292	8	no	—
275	C₂×C₄	—	292	16	no	—
276	C₂×C₄	—	146	8	no	—
277	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	4	yes	—
278	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	4	yes	—
279	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
281	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
282	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
283	C₂×C₄	#14556567	1	4	yes	—
284	C₂×C₄	#14556580	1	2	yes	—
285	C₂×C₄	#14556580	1	2	yes	—
286	C₂×C₄	#14556589	1	2	yes	—
287	C₂×C₄	#14556589	1	2	yes	—
288	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
290	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
291	C₂×C₄	C₂×#11924029	1	2	yes	—
292	C₈	—	292	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	8	no	—
296	C₂²	#17988494	1	2	yes	—
297	C₂²	#17988494	1	2	yes	—
298	C₂²	#17988604	1	4	yes	—
299	C₂²	C₂²×C₂²⋊Dic₇₃	1	1	yes	characteristic
300	C₂²	C₂×#14556567	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	292	8	no	—
305	C₄	#17988494	1	2	yes	—
306	C₄	#17988494	1	2	yes	—
307	C₄	C₂×Q₈×Dic₇₃	1	2	yes	—
308	C₄	C₂×#14556580	1	1	yes	characteristic
309	C₄	C₂×#14556589	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	#22610913	1	4	yes	direct product factor
313	C₂	C₂×#17988494	1	1	yes	characteristic
314	C₂	C₂×#17988604	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#533×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#533	D₇₃
hypocenter	C₇₃	C₂×#1995
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #22610913	C₂	direct (N × H)
17. #533×C₇₃	C₄	semidirect
19. C₂×C₄⋊Q₈×C₇₃	C₄	semidirect
20. C₂×#125×C₇₃	C₄	semidirect
32. C₂×C₄×Dic₇₃	Q₈	semidirect
40. C₄⋊Q₈×C₇₃	C₂×C₄	semidirect
42. #125×C₇₃	C₂×C₄	semidirect
72. C₄×Dic₇₃	C₂×Q₈	semidirect
155. C₂×C₇₃	#1995	semidirect
163. C₇₃	C₂×#1995	semidirect
167. #533	Dic₇₃	semidirect
169. C₂×C₄⋊Q₈	Dic₇₃	semidirect
170. C₂×#125	Dic₇₃	semidirect
192. C₄⋊Q₈	C₂×Dic₇₃	semidirect
194. #125	C₂×Dic₇₃	semidirect
312. C₂	#22610913	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
C₂×#17988433	2
#22610913	2
C₂×#17988346	2
C₂×#533×C₇₃	2
C₂×#1995	73

Subgroups — C2×#22610913 #33090607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22610913 #33090607