Subgroups — C₂×#22626287 #34030101

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556662	C₂	1	1	yes	maximal, characteristic
2	C₂×D₇₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#22626287	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17989516	C₂	1	2	yes	maximal, complemented
5	C₂×#17989515	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924038	—	2	1	no	—
8	C₂×D₇₃⋊M₁₆	—	2	1	no	—
9	C₂×#14556662	—	2	1	no	—
10	C₂×#14556662	—	2	2	no	—
11	C₂×#14556662	C₂²	1	1	yes	characteristic
12	C₂×#14556662	C₂²	1	1	yes	characteristic
13	C₂×#14556662	C₂²	1	2	yes	—
14	C₂×#14556662	C₂²	1	4	yes	—
15	D₇₃⋊#108	—	2	2	no	—
16	D₇₃⋊#108	C₂²	1	4	yes	—
17	C₂×#14556661	—	2	1	no	—
18	#17989516	C₂²	1	8	yes	complemented
19	#17989515	C₂²	1	8	yes	complemented
20	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
21	C₂×#14556531	C₄	1	1	yes	characteristic
22	C₂×#14556530	C₄	1	2	yes	—
23	M₃₂×D₇₃	C₄	1	4	yes	—
24	M₃₂×D₇₃	C₂²	1	4	yes	—
25	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
26	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
27	C₂×#11924038	—	2	1	no	—
28	C₂×#11924038	—	2	1	no	—
29	C₂×#11924038	—	2	4	no	—
30	C₂×#11924038	—	2	2	no	—
31	C₂×#11924038	—	4	1	no	—
32	C₂×#11924038	—	4	2	no	—
33	D₇₃⋊M₁₆	—	2	4	no	—
34	D₇₃⋊M₁₆	—	4	2	no	—
35	C₂×#11924037	—	4	1	no	—
36	C₂×#11924036	—	4	1	no	—
37	C₂²×#10596833	—	4	1	no	—
38	#14556662	—	2	1	no	—
39	#14556662	—	2	1	no	—
40	#14556662	—	2	4	no	—
41	#14556662	—	2	2	no	—
42	#14556662	C₂³	1	2	yes	—
43	#14556662	C₂³	1	4	yes	—
44	#14556662	C₂³	1	2	yes	—
45	#14556661	—	2	4	no	—
46	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
47	C₂²×#10596831	Q₈	1	1	yes	characteristic
48	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	2	yes	—
52	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
53	C₂×C₈×D₇₃	Q₈	1	2	yes	—
54	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₇₃	C₂³	1	2	yes	—
56	#14556531	C₂×C₄	1	4	yes	—
57	#14556530	C₂×C₄	1	8	yes	—
58	C₂×#11924003	C₂×C₄	1	2	yes	—
59	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
60	C₁₆×D₇₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
63	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
64	#11924038	—	2	2	no	—
65	#11924038	—	2	2	no	—
66	#11924038	—	2	4	no	—
67	#11924038	—	4	1	no	—
68	#11924038	—	4	1	no	—
69	#11924038	—	4	4	no	—
70	#11924038	—	4	2	no	—
71	#11924037	—	4	4	no	—
72	#11924036	—	4	4	no	—
73	C₂×#10596833	—	4	2	no	—
74	C₂×#10596833	—	4	4	no	—
75	C₂×#10596833	—	4	1	no	—
76	C₂×#10596833	—	4	1	no	—
77	C₂³×D₇₃	Q₁₆	1	1	yes	characteristic
78	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₇₃	D₈	1	2	yes	—
83	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
84	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
85	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×#10596831	C₄⋊C₄	1	2	yes	—
88	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
89	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
90	C₂×#10596831	C₂×Q₈	1	2	yes	—
91	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
92	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
93	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
94	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₈×D₇₃	C₂²×C₄	1	2	yes	—
96	C₈×D₇₃	C₂×D₄	1	2	yes	—
97	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
98	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
99	C₈×D₇₃	C₂×Q₈	1	2	yes	—
100	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
101	#11924003	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
107	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
108	C₂³×C₇₃	#107	1	1	yes	characteristic
109	C₂²×D₇₃	—	4	2	no	—
110	C₂²×D₇₃	#107	1	1	yes	characteristic
111	C₂²×D₇₃	#107	1	2	yes	—
112	C₂²×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×D₇₃	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₇₃	#107	1	2	yes	—
115	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×Dic₇₃	#107	1	2	yes	—
119	C₂×Dic₇₃	#107	1	1	yes	characteristic
120	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
122	C₄×D₇₃	#107	1	2	yes	—
123	C₄×D₇₃	#107	1	2	yes	—
124	C₄×D₇₃	#107	1	2	yes	—
125	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₄×D₇₃	C₂×D₈	1	2	yes	—
128	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
129	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
130	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
131	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
132	#10596833	—	4	4	no	—
133	#10596833	—	4	2	no	—
134	#10596833	—	4	2	no	—
135	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
139	#10596831	C₂×C₄⋊C₄	1	2	yes	—
140	#10596831	C₂²×Q₈	1	1	yes	characteristic
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	C₂×#107	1	2	yes	—
143	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
144	C₂×D₇₃	—	2	2	no	—
145	C₂×D₇₃	—	2	2	no	—
146	C₂×D₇₃	—	4	2	no	—
147	C₂×D₇₃	—	4	2	no	—
148	C₂×D₇₃	C₂×#107	1	2	yes	—
149	C₂×D₇₃	C₂×#107	1	1	yes	characteristic
150	C₂×D₇₃	C₂²×Q₁₆	1	1	yes	characteristic
151	C₄×C₇₃	C₂×#107	1	2	yes	—
152	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
153	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	Dic₇₃	C₂×#107	1	1	yes	characteristic
155	Dic₇₃	C₂×#107	1	2	yes	—
156	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×#2161	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	2	no	—
159	C₂×C₇₃	#2161	1	2	yes	complemented
160	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
161	D₇₃	—	2	1	no	—
162	D₇₃	—	2	2	no	—
163	D₇₃	—	2	1	no	—
164	C₂²×M₃₂	—	73	1	no	—
165	C₂²×#108	—	73	1	no	—
166	C₂²×#108	—	73	1	no	—
167	#2161	—	73	8	no	—
168	C₂×#368	—	73	2	no	—
169	C₂×#368	—	73	2	no	—
170	C₇₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
171	C₂²×M₁₆	—	146	1	no	—
172	C₂²×M₁₆	—	146	1	no	—
173	C₂³×C₈	—	73	1	no	—
174	C₂×M₃₂	—	73	1	no	—
175	C₂×M₃₂	—	73	4	no	—
176	C₂×M₃₂	—	73	4	no	—
177	C₂×M₃₂	—	73	2	no	—
178	C₂×M₃₂	#64408	1	1	yes	characteristic
179	C₂²×C₁₆	—	73	2	no	—
180	C₂×#108	—	73	1	no	—
181	C₂×#108	—	73	1	no	—
182	C₂×#108	—	73	2	no	—
183	C₂×#108	—	73	4	no	—
184	C₂×#108	—	73	4	no	—
185	C₂×#108	—	146	1	no	—
186	C₂×#108	—	146	2	no	—
187	C₂×#108	—	146	2	no	—
188	C₂×#108	—	146	1	no	—
189	#368	—	73	8	no	—
190	#368	—	73	8	no	—
191	C₂³×C₄	—	73	1	no	—
192	C₂×M₁₆	—	292	1	no	—
193	C₂×M₁₆	—	292	2	no	—
194	C₂×M₁₆	—	292	1	no	—
195	C₂×M₁₆	—	292	2	no	—
196	C₂×M₁₆	—	146	4	no	—
197	C₂×M₁₆	—	146	1	no	—
198	C₂×M₁₆	—	146	1	no	—
199	C₂×M₁₆	—	146	2	no	—
200	C₂×M₁₆	—	146	4	no	—
201	C₂²×C₈	—	292	1	no	—
202	C₂²×C₈	—	292	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	C₂²×C₈	—	73	1	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	73	1	no	—
212	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
213	M₃₂	—	73	4	no	—
214	M₃₂	—	73	8	no	—
215	M₃₂	C₂×#64408	1	4	yes	—
216	C₂×C₁₆	—	73	2	no	—
217	C₂×C₁₆	—	73	4	no	—
218	C₂×C₁₆	—	73	4	no	—
219	C₂×C₁₆	C₂×#64408	1	2	yes	—
220	#108	—	73	2	no	—
221	#108	—	73	4	no	—
222	#108	—	73	2	no	—
223	#108	—	146	4	no	—
224	#108	—	146	1	no	—
225	#108	—	146	1	no	—
226	#108	—	146	2	no	—
227	#108	—	146	4	no	—
228	C₂⁴	—	73	1	no	—
229	C₂²×C₄	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	2	no	—
232	C₂²×C₄	—	73	2	no	—
233	C₂²×C₄	—	73	1	no	—
234	C₂²×C₄	—	73	2	no	—
235	C₂²×C₄	—	73	1	no	—
236	C₂²×C₄	—	73	2	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
239	M₁₆	—	292	4	no	—
240	M₁₆	—	292	4	no	—
241	M₁₆	—	292	1	no	—
242	M₁₆	—	292	1	no	—
243	M₁₆	—	292	2	no	—
244	M₁₆	—	146	2	no	—
245	M₁₆	—	146	2	no	—
246	M₁₆	—	146	4	no	—
247	C₂×C₈	—	292	4	no	—
248	C₂×C₈	—	292	2	no	—
249	C₂×C₈	—	292	1	no	—
250	C₂×C₈	—	292	1	no	—
251	C₂×C₈	—	292	4	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	73	1	no	—
254	C₂×C₈	—	73	2	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	1	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	C₄⋊#64408	1	2	yes	—
267	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
268	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
269	C₂×C₈	C₂²×#64408	1	2	yes	—
270	C₁₆	—	73	4	no	—
271	C₁₆	C₂²×#64408	1	4	yes	—
272	C₂³	—	292	2	no	—
273	C₂³	—	73	2	no	—
274	C₂³	—	73	2	no	—
275	C₂³	—	73	1	no	—
276	C₂³	—	73	1	no	—
277	C₂³	#14556660	1	1	yes	characteristic
278	C₂×C₄	—	73	2	no	—
279	C₂×C₄	—	73	2	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
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	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	2	no	—
290	C₂×C₄	—	73	2	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	#14556660	1	2	yes	—
293	C₂×C₄	#14556660	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
296	C₈	—	292	4	no	—
297	C₈	—	292	2	no	—
298	C₈	—	292	2	no	—
299	C₈	—	73	1	no	—
300	C₈	—	73	2	no	—
301	C₈	—	73	1	no	—
302	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
303	C₈	C₂×C₄⋊#64408	1	2	yes	—
304	C₈	C₂³×#64408	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	292	2	no	—
307	C₂²	—	292	2	no	—
308	C₂²	—	73	1	no	—
309	C₂²	—	73	2	no	—
310	C₂²	—	73	1	no	—
311	C₂²	—	146	2	no	—
312	C₂²	—	146	2	no	—
313	C₂²	C₂×#14556660	1	2	yes	—
314	C₂²	C₂×#14556660	1	1	yes	characteristic
315	C₄	—	73	1	no	—
316	C₄	—	73	2	no	—
317	C₄	—	73	1	no	—
318	C₄	C₂×#14556660	1	2	yes	—
319	C₄	C₂×#14556660	1	1	yes	characteristic
320	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	146	1	no	—
323	C₂	—	146	1	no	—
324	C₂	—	146	2	no	—
325	C₂	#22626287	1	2	yes	direct product factor
326	C₂	C₂²×#14556660	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556660
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×D₇₃	Q₁₆
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	C₂×#2161
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22626287	C₂	direct (N × H)
4. C₂×#17989516	C₂	semidirect
5. C₂×#17989515	C₂	semidirect
18. #17989516	C₂²	semidirect
19. #17989515	C₂²	semidirect
159. C₂×C₇₃	#2161	semidirect
170. C₇₃	C₂×#2161	semidirect
325. C₂	#22626287	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×#14556662	2
C₂×D₇₃⋊#108	2
#22626287	2
C₂×#17989516	2
C₂×#17989515	2
C₂×M₃₂×D₇₃	2
C₂×#2161	73

Subgroups — C2×#22626287 #34030101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626287 #34030101