Subgroups — C₂×#22626143 #34029792

Order 18688.

854 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556664	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989550	C₂	1	1	yes	maximal, characteristic
3	#22626143	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17989474	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#64408	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₇₃⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×#14556670	C₄	1	1	yes	characteristic
9	C₂×#14556664	C₄	1	2	yes	complemented
10	C₂×#14556664	C₄	1	4	yes	complemented
11	C₂×#14556664	C₂²	1	1	yes	characteristic
12	C₂×#14556664	C₂²	1	1	yes	characteristic
13	C₂×#14556664	C₂²	1	4	yes	—
14	C₂²×#11924031	C₄	1	2	yes	complemented
15	#17989550	C₄	1	4	yes	—
16	#17989550	C₂²	1	4	yes	—
17	C₂×#14556645	C₄	1	2	yes	—
18	C₂×C₈×#64408	C₂²	1	2	yes	—
19	#17989474	C₂²	1	8	yes	complemented
20	C₁₆×#64408	C₂²	1	8	yes	complemented
21	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
22	C₂×#14556531	C₄	1	1	yes	characteristic, complemented
23	C₂×#14556530	C₄	1	2	yes	complemented
24	M₃₂×D₇₃	C₄	1	4	yes	complemented
25	M₃₂×D₇₃	C₂²	1	4	yes	—
26	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
27	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
28	C₂×#11924038	C₈	1	2	yes	—
29	D₇₃⋊M₁₆	C₈	1	4	yes	—
30	D₇₃⋊M₁₆	C₂×C₄	1	4	yes	—
31	C₂×#11924037	C₈	1	2	yes	—
32	C₂×#11924036	C₂×C₄	1	2	yes	—
33	#14556670	C₈	1	4	yes	—
34	#14556670	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊#64408	C₈	1	2	yes	—
36	C₂×C₂²⋊#64408	C₈	1	2	yes	—
37	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
38	#14556664	C₂×C₄	1	2	yes	—
39	#14556664	C₂×C₄	1	2	yes	—
40	#14556664	C₂×C₄	1	8	yes	complemented
41	#14556664	C₂³	1	2	yes	—
42	#14556664	C₂³	1	2	yes	—
43	C₂×#11924031	C₂×C₄	1	2	yes	—
44	C₂×#11924031	C₂×C₄	1	2	yes	—
45	C₂×#11924031	C₂×C₄	1	8	yes	complemented
46	#14556645	C₂×C₄	1	8	yes	—
47	C₈×#64408	C₂×C₄	1	4	yes	—
48	C₈×#64408	C₂³	1	4	yes	—
49	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₇₃	C₂³	1	2	yes	—
59	#14556531	C₂×C₄	1	4	yes	complemented
60	#14556530	C₂×C₄	1	8	yes	complemented
61	C₂×#11924003	C₂×C₄	1	2	yes	—
62	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
63	C₁₆×D₇₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
66	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
67	C₂²×#64408	C₂×C₈	1	4	yes	—
68	#11924038	C₂×C₈	1	8	yes	—
69	#11924037	C₂×C₈	1	8	yes	—
70	#11924036	C₂×C₈	1	4	yes	—
71	#11924036	C₂²×C₄	1	4	yes	—
72	C₂×#10596833	C₂×C₈	1	4	yes	—
73	C₄⋊#64408	C₂×C₈	1	8	yes	—
74	C₂²⋊#64408	C₂×C₈	1	8	yes	—
75	C₄×#64408	C₂×C₈	1	4	yes	—
76	C₄×#64408	C₂²×C₄	1	4	yes	—
77	#11924031	C₂²×C₄	1	4	yes	—
78	#11924031	C₂²×C₄	1	4	yes	—
79	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₄²	1	2	yes	—
82	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
89	C₂×#10596831	C₄²	1	1	yes	characteristic
90	C₂×#10596831	C₄²	1	2	yes	—
91	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596831	C₂²×C₄	1	2	yes	—
93	C₈×D₇₃	C₄²	1	2	yes	—
94	C₈×D₇₃	C₄²	1	2	yes	—
95	C₈×D₇₃	C₄²	1	2	yes	—
96	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₈×D₇₃	C₂²×C₄	1	2	yes	—
98	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₈×D₇₃	C₂²×C₄	1	2	yes	—
100	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₇₃	C₂²×C₄	1	2	yes	—
102	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
103	#11924003	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₄²	1	2	yes	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
110	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
111	C₂²×D₇₃	—	2	4	no	—
112	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
113	C₂²×D₇₃	C₄×C₈	1	2	yes	—
114	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
115	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×#64408	—	2	8	no	—
117	C₂×#64408	C₂²×C₈	1	8	yes	—
118	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
123	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
126	C₄×D₇₃	C₄×C₈	1	2	yes	—
127	C₄×D₇₃	C₄×C₈	1	2	yes	—
128	C₄×D₇₃	C₄×C₈	1	2	yes	—
129	C₄×D₇₃	C₂×C₄²	1	2	yes	—
130	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
131	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₄×D₇₃	C₂²×C₈	1	2	yes	—
133	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₄×D₇₃	C₂²×C₈	1	2	yes	—
135	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
136	#10596833	C₂²×C₈	1	8	yes	—
137	C₈×C₇₃	C₂×C₄²	1	2	yes	—
138	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
139	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
140	#10596831	C₂×C₄²	1	1	yes	characteristic
141	#10596831	C₂×C₄²	1	2	yes	—
142	#10596831	C₂³×C₄	1	1	yes	characteristic
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
145	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	2	4	no	—
148	C₂×D₇₃	—	2	4	no	—
149	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
150	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
151	C₂×D₇₃	#504	1	4	yes	—
152	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
153	#64408	—	2	8	no	—
154	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
155	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
156	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
157	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
158	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
159	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
160	C₂×#2108	—	73	1	no	maximal, Sylow
161	C₂×C₇₃	—	2	2	no	—
162	C₂×C₇₃	#2108	1	2	yes	complemented
163	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
164	D₇₃	—	2	2	no	—
165	D₇₃	#2108	1	2	yes	complemented
166	D₇₃	C₂×#504	1	2	yes	—
167	C₂²×M₃₂	—	73	1	no	—
168	C₂³×C₁₆	—	73	1	no	—
169	C₂×#405	—	73	1	no	—
170	#2108	—	73	8	no	—
171	C₂×#346	—	73	2	no	—
172	C₂×C₄×C₁₆	—	73	2	no	—
173	C₇₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	73	1	no	—
175	C₂³×C₈	—	73	1	no	—
176	C₂×#117	—	73	1	no	—
177	C₂×M₃₂	—	73	1	no	—
178	C₂×M₃₂	—	73	4	no	—
179	C₂×M₃₂	—	73	4	no	—
180	C₂×M₃₂	—	73	2	no	—
181	C₂×M₃₂	#64408	1	1	yes	characteristic, complemented
182	C₂²×C₁₆	—	73	2	no	—
183	C₂²×C₁₆	—	73	2	no	—
184	C₂²×C₁₆	—	73	4	no	—
185	C₂²×C₁₆	—	73	1	no	—
186	C₂²×C₁₆	—	73	1	no	—
187	C₂²×C₁₆	—	73	4	no	—
188	C₂²×C₁₆	—	73	2	no	—
189	#405	—	73	4	no	—
190	#405	—	73	4	no	—
191	C₂×#97	—	73	2	no	—
192	C₂×C₄×C₈	—	73	2	no	—
193	#346	—	73	8	no	—
194	C₄×C₁₆	—	73	8	no	—
195	C₂³×C₄	—	73	1	no	—
196	C₂×M₁₆	—	73	2	no	—
197	C₂×M₁₆	—	73	4	no	—
198	C₂×M₁₆	—	73	4	no	—
199	C₂×M₁₆	—	73	2	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	1	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	2	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
211	#117	—	73	4	no	—
212	#117	—	73	4	no	—
213	C₂×C₄⋊C₄	—	73	2	no	—
214	C₂×C₂²⋊C₄	—	73	2	no	—
215	C₂×C₄²	—	73	2	no	—
216	M₃₂	—	73	4	no	—
217	M₃₂	—	73	8	no	—
218	M₃₂	C₂×#64408	1	4	yes	complemented
219	C₂×C₁₆	—	73	2	no	—
220	C₂×C₁₆	—	73	2	no	—
221	C₂×C₁₆	—	73	2	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₂×C₁₆	—	73	8	no	—
224	C₂×C₁₆	—	73	2	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	73	8	no	—
227	C₂×C₁₆	—	73	2	no	—
228	C₂×C₁₆	—	73	4	no	—
229	C₂×C₁₆	—	73	4	no	—
230	C₂×C₁₆	C₂×#64408	1	2	yes	—
231	#97	—	73	8	no	—
232	C₄×C₈	—	73	4	no	—
233	C₄×C₈	—	73	4	no	—
234	C₂⁴	—	73	1	no	—
235	C₂²×C₄	—	73	1	no	—
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	2	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	2	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	2	no	—
244	C₂²×C₄	—	73	4	no	—
245	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
246	M₁₆	—	73	8	no	—
247	M₁₆	—	73	8	no	—
248	C₂×C₈	—	73	4	no	—
249	C₂×C₈	—	73	4	no	—
250	C₂×C₈	—	73	4	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	1	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	C₄×#64408	1	1	yes	characteristic
266	C₂×C₈	C₄×#64408	1	2	yes	—
267	C₂×C₈	C₂²×#64408	1	2	yes	—
268	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
269	C₄⋊C₄	—	73	8	no	—
270	C₂²⋊C₄	—	73	8	no	—
271	C₄²	—	73	4	no	—
272	C₄²	—	73	4	no	—
273	C₁₆	—	73	4	no	—
274	C₁₆	—	73	4	no	—
275	C₁₆	—	73	4	no	—
276	C₁₆	C₂²×#64408	1	4	yes	—
277	C₂³	—	73	1	no	—
278	C₂³	—	73	2	no	—
279	C₂³	—	73	2	no	—
280	C₂³	—	73	1	no	—
281	C₂³	—	146	4	no	—
282	C₂³	C₈×#64408	1	1	yes	characteristic
283	C₂×C₄	—	73	1	no	—
284	C₂×C₄	—	73	2	no	—
285	C₂×C₄	—	73	2	no	—
286	C₂×C₄	—	73	1	no	—
287	C₂×C₄	—	73	1	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	2	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	2	no	—
292	C₂×C₄	—	73	1	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	2	no	—
296	C₂×C₄	—	73	2	no	—
297	C₂×C₄	—	73	8	no	—
298	C₂×C₄	—	146	8	no	—
299	C₂×C₄	C₈×#64408	1	2	yes	—
300	C₂×C₄	C₈×#64408	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
302	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
303	C₈	—	73	8	no	—
304	C₈	—	73	1	no	—
305	C₈	—	73	1	no	—
306	C₈	—	73	2	no	—
307	C₈	C₂×C₄×#64408	1	2	yes	—
308	C₈	C₂×C₄×#64408	1	1	yes	characteristic
309	C₈	C₂³×#64408	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	73	4	no	—
312	C₂²	—	73	1	no	—
313	C₂²	—	73	1	no	—
314	C₂²	—	73	2	no	—
315	C₂²	—	146	4	no	—
316	C₂²	—	146	4	no	—
317	C₂²	—	146	2	no	—
318	C₂²	C₂×C₈×#64408	1	2	yes	—
319	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
320	C₄	—	73	1	no	—
321	C₄	—	73	1	no	—
322	C₄	—	73	2	no	—
323	C₄	—	146	8	no	—
324	C₄	C₂×C₈×#64408	1	1	yes	characteristic
325	C₄	C₂×C₈×#64408	1	2	yes	—
326	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	73	2	no	—
329	C₂	—	73	2	no	—
330	C₂	—	146	2	no	—
331	C₂	#22626143	1	2	yes	direct product factor
332	C₂	C₂²×C₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₈	C₂³×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	C₂×#2108
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22626143	C₂	direct (N × H)
4. C₂×#17989474	C₂	semidirect
5. C₂×C₁₆×#64408	C₂	semidirect
9. C₂×#14556664	C₄	semidirect
10. C₂×#14556664	C₄	semidirect
14. C₂²×#11924031	C₄	semidirect
19. #17989474	C₂²	semidirect
20. C₁₆×#64408	C₂²	semidirect
22. C₂×#14556531	C₄	semidirect
23. C₂×#14556530	C₄	semidirect
24. M₃₂×D₇₃	C₄	semidirect
27. C₂×M₃₂×C₇₃	C₄	semidirect
40. #14556664	C₂×C₄	semidirect
45. C₂×#11924031	C₂×C₄	semidirect
59. #14556531	C₂×C₄	semidirect
60. #14556530	C₂×C₄	semidirect
65. M₃₂×C₇₃	C₂×C₄	semidirect
162. C₂×C₇₃	#2108	semidirect
165. D₇₃	#2108	semidirect
173. C₇₃	C₂×#2108	semidirect
181. C₂×M₃₂	#64408	semidirect
218. M₃₂	C₂×#64408	semidirect
331. C₂	#22626143	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×#14556664	2
C₂×#17989550	2
#22626143	2
C₂×#17989474	2
C₂×C₁₆×#64408	2
C₂×M₃₂×D₇₃	2
C₂×#2108	73

Subgroups — C2×#22626143 #34029792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626143 #34029792