Subgroups — #34032684

Order 18688.

2074 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626900	C₂	1	6	yes	maximal, complemented
2	#22626899	C₂	1	8	yes	maximal, complemented
3	#22626898	C₂	1	2	yes	maximal, complemented
4	#22626896	C₂	1	8	yes	maximal, complemented
5	C₂×#17989643	C₂	1	6	yes	maximal, complemented
6	C₂×#131×D₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989646	C₂²	1	48	yes	complemented
8	#17989645	C₂²	1	16	yes	complemented
9	C₂×#14556666	C₂²	1	12	yes	complemented
10	#17989643	C₂²	1	4	yes	—
11	#17989643	C₂²	1	12	yes	complemented
12	#17989643	C₂²	1	12	yes	complemented
13	#17989643	C₂²	1	12	yes	complemented
14	C₂×#14556665	C₂²	1	12	yes	complemented
15	C₂×#14556664	C₂²	1	12	yes	complemented
16	C₂×#41×D₇₃	C₄	1	1	yes	characteristic
17	C₂×#14556599	C₂²	1	3	yes	—
18	C₂×#14556598	C₂²	1	1	yes	characteristic
19	C₂×#14556597	C₄	1	3	yes	—
20	#131×D₇₃	C₄	1	4	yes	—
21	#131×D₇₃	C₄	1	4	yes	—
22	#131×D₇₃	C₂²	1	4	yes	—
23	#131×D₇₃	C₂²	1	4	yes	—
24	C₂×M₁₆×D₇₃	C₄	1	3	yes	—
25	C₂²×C₈×D₇₃	C₂²	1	3	yes	—
26	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
27	#14556666	C₂³	1	8	yes	—
28	#14556666	C₂³	1	24	yes	complemented
29	#14556665	C₂³	1	8	yes	—
30	#14556665	C₂³	1	24	yes	complemented
31	#14556664	C₂³	1	8	yes	—
32	#14556664	C₂³	1	24	yes	complemented
33	C₂×#11924031	C₂³	1	24	yes	complemented
34	C₂×#11924029	C₈	1	1	yes	characteristic
35	C₂×#11924028	C₂×C₄	1	3	yes	—
36	C₂×#11924027	C₈	1	3	yes	—
37	#41×D₇₃	C₈	1	4	yes	—
38	#41×D₇₃	C₈	1	4	yes	—
39	#41×D₇₃	C₂×C₄	1	4	yes	—
40	#41×D₇₃	C₂×C₄	1	4	yes	—
41	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
42	C₂×D₄×D₇₃	C₈	1	3	yes	—
43	C₂²×C₄×D₇₃	C₂×C₄	1	3	yes	—
44	#14556599	C₂×C₄	1	12	yes	—
45	#14556599	C₂³	1	12	yes	—
46	#14556598	C₂×C₄	1	4	yes	—
47	#14556598	C₂³	1	4	yes	—
48	#14556597	C₂×C₄	1	12	yes	—
49	#14556597	C₂×C₄	1	12	yes	—
50	C₂×#11924012	C₂×C₄	1	3	yes	—
51	C₂×#11924011	C₂×C₄	1	6	yes	—
52	M₁₆×D₇₃	C₂×C₄	1	12	yes	—
53	M₁₆×D₇₃	C₂×C₄	1	12	yes	—
54	C₂²×#10596831	C₂³	1	3	yes	—
55	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
56	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
57	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
58	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
59	C₂×C₈×D₇₃	C₂³	1	6	yes	—
60	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₇₃	C₂³	1	3	yes	—
62	C₂×C₈×D₇₃	C₂³	1	6	yes	—
63	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#131×C₇₃	C₂×C₄	1	4	yes	—
65	#131×C₇₃	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
67	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
68	#11924031	C₂⁴	1	16	yes	—
69	C₂³×D₇₃	C₂×C₈	1	3	yes	—
70	#11924029	C₂×C₈	1	4	yes	—
71	#11924029	C₂×C₈	1	4	yes	—
72	#11924028	C₂×C₈	1	12	yes	—
73	#11924028	C₂²×C₄	1	12	yes	—
74	#11924027	C₂×C₈	1	12	yes	—
75	#11924027	C₂×C₈	1	12	yes	—
76	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
77	Q₈×D₇₃	C₂×C₈	1	4	yes	—
78	Q₈×D₇₃	C₂×C₈	1	4	yes	—
79	C₂×D₂₉₂	C₂×C₈	1	3	yes	—
80	C₂×C₂²⋊D₇₃	C₂×C₈	1	6	yes	—
81	D₄×D₇₃	C₂×C₈	1	12	yes	—
82	D₄×D₇₃	C₂×C₈	1	12	yes	—
83	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
84	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
85	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	—
86	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	—
87	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
88	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂²×C₄	1	6	yes	—
90	C₂×C₄×D₇₃	C₂²×C₄	1	6	yes	—
91	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
92	#11924012	C₂²×C₄	1	12	yes	—
93	#11924011	C₂²×C₄	1	24	yes	—
94	C₂×#10596831	C₂²×C₄	1	3	yes	—
95	C₂×#10596831	C₂²×C₄	1	6	yes	—
96	C₂×#10596831	C₂⁴	1	1	yes	characteristic
97	C₂×#10596831	C₂⁴	1	6	yes	—
98	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₇₃	C₂²×C₄	1	6	yes	—
101	C₈×D₇₃	C₂²×C₄	1	6	yes	—
102	C₈×D₇₃	C₂²×C₄	1	6	yes	—
103	C₈×D₇₃	C₂²×C₄	1	6	yes	—
104	C₈×D₇₃	C₂²×C₄	1	6	yes	—
105	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
106	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
107	C₈×D₇₃	C₂⁴	1	6	yes	—
108	#41×C₇₃	C₂×C₈	1	4	yes	—
109	#41×C₇₃	C₂²×C₄	1	4	yes	—
110	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
111	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
112	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
113	M₁₆×C₇₃	C₂²×C₄	1	12	yes	—
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
116	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
117	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
118	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
119	C₂²×D₇₃	—	4	6	no	—
120	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₇₃	C₂²×C₈	1	6	yes	—
122	C₂²×D₇₃	C₂²×C₈	1	6	yes	—
123	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
124	Q₈×C₇₃	C₂²×C₈	1	4	yes	—
125	D₄×C₇₃	C₂²×C₈	1	12	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
128	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂³×C₄	1	6	yes	—
130	C₂²⋊D₇₃	C₂²×C₈	1	24	yes	—
131	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
133	C₂×Dic₇₃	C₂²×C₈	1	6	yes	—
134	C₂×Dic₇₃	C₂²×C₈	1	6	yes	—
135	D₂₉₂	C₂²×C₈	1	12	yes	—
136	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₇₃	C₂²×C₈	1	6	yes	—
139	C₄×D₇₃	C₂²×C₈	1	6	yes	—
140	C₄×D₇₃	C₂²×C₈	1	6	yes	—
141	C₄×D₇₃	C₂²×C₈	1	6	yes	—
142	C₄×D₇₃	C₂²×C₈	1	6	yes	—
143	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
144	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
145	C₄×D₇₃	C₂³×C₄	1	6	yes	—
146	Dic₁₄₆	C₂²×C₈	1	12	yes	—
147	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
148	C₈×C₇₃	C₂³×C₄	1	6	yes	—
149	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
150	#10596831	C₂³×C₄	1	1	yes	characteristic
151	#10596831	C₂³×C₄	1	6	yes	—
152	#10596831	C₂⁵	1	1	yes	characteristic
153	C₂²×C₇₃	—	4	3	no	—
154	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
155	C₂²×C₇₃	C₂³×C₈	1	6	yes	—
156	C₂×D₇₃	—	2	2	no	—
157	C₂×D₇₃	—	2	12	no	—
158	C₂×D₇₃	—	4	6	no	—
159	C₂×D₇₃	—	4	3	no	—
160	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
161	C₂×D₇₃	C₂³×C₈	1	6	yes	—
162	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₇₃	C₂³×C₈	1	6	yes	—
164	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
165	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
166	Dic₇₃	C₂³×C₈	1	6	yes	—
167	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
168	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
169	C₂×#3407	—	73	1	no	maximal, Sylow
170	C₂×C₇₃	—	2	1	no	—
171	C₂×C₇₃	—	2	6	no	—
172	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
173	D₇₃	—	2	1	no	—
174	D₇₃	—	2	6	no	—
175	D₇₃	#3407	1	2	yes	complemented
176	C₂²×#131	—	73	1	no	—
177	#3407	—	73	2	no	—
178	#3407	—	73	8	no	—
179	#3407	—	73	6	no	—
180	C₂×#504	—	73	8	no	—
181	C₂²×M₃₂	—	73	6	no	—
182	C₇₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
183	C₂²×#41	—	73	1	no	—
184	C₂×#131	—	73	1	no	—
185	C₂×#131	—	73	4	no	—
186	C₂×#131	—	73	4	no	—
187	C₂×#131	—	73	4	no	—
188	C₂×#131	—	73	4	no	—
189	C₂×#131	—	73	3	no	—
190	C₂×#131	—	73	3	no	—
191	C₂×#131	#64408	1	1	yes	characteristic
192	C₂²×M₁₆	—	73	3	no	—
193	C₂³×C₈	—	73	3	no	—
194	#504	—	73	16	no	—
195	#504	—	73	48	no	—
196	C₂×M₃₂	—	73	12	no	—
197	C₂×M₃₂	—	73	4	no	—
198	C₂×M₃₂	—	73	12	no	—
199	C₂×M₃₂	—	73	12	no	—
200	C₂×M₃₂	—	73	12	no	—
201	C₂×M₃₂	—	73	12	no	—
202	C₂²×C₁₆	—	73	12	no	—
203	C₂×#41	—	73	4	no	—
204	C₂×#41	—	73	4	no	—
205	C₂×#41	—	73	4	no	—
206	C₂×#41	—	73	4	no	—
207	C₂×#41	—	73	1	no	—
208	C₂×#41	—	73	3	no	—
209	C₂×#41	—	73	3	no	—
210	C₂×#41	C₂×#64408	1	1	yes	characteristic
211	C₂²×Q₈	—	73	1	no	—
212	C₂²×D₄	—	73	3	no	—
213	C₂³×C₄	—	73	3	no	—
214	#131	—	73	4	no	—
215	#131	—	73	4	no	—
216	#131	—	73	12	no	—
217	#131	—	73	12	no	—
218	#131	—	73	12	no	—
219	#131	—	73	12	no	—
220	#131	C₂×#64408	1	4	yes	—
221	#131	C₂×#64408	1	4	yes	—
222	C₂×M₁₆	—	73	3	no	—
223	C₂×M₁₆	—	73	12	no	—
224	C₂×M₁₆	—	73	12	no	—
225	C₂×M₁₆	—	73	6	no	—
226	C₂×M₁₆	C₂×#64408	1	3	yes	—
227	C₂²×C₈	—	73	3	no	—
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	73	3	no	—
230	C₂²×C₈	—	73	3	no	—
231	C₂²×C₈	—	73	6	no	—
232	C₂²×C₈	—	73	6	no	—
233	C₂²×C₈	—	73	6	no	—
234	C₂²×C₈	—	73	6	no	—
235	C₂²×C₈	—	73	3	no	—
236	C₂²×C₈	C₂×#64408	1	3	yes	—
237	M₃₂	—	73	8	no	—
238	M₃₂	—	73	8	no	—
239	M₃₂	—	73	24	no	—
240	M₃₂	—	73	24	no	—
241	C₂×C₁₆	—	73	8	no	—
242	C₂×C₁₆	—	73	24	no	—
243	C₂×C₁₆	—	73	24	no	—
244	C₂⁴	—	73	3	no	—
245	#41	—	73	4	no	—
246	#41	—	73	4	no	—
247	#41	—	73	12	no	—
248	#41	—	73	12	no	—
249	#41	—	73	12	no	—
250	#41	—	73	12	no	—
251	#41	#11924036	1	4	yes	—
252	#41	C₂²×#64408	1	4	yes	—
253	C₂×Q₈	—	73	4	no	—
254	C₂×Q₈	—	73	4	no	—
255	C₂×Q₈	—	73	3	no	—
256	C₂×Q₈	#11924036	1	1	yes	characteristic
257	C₂×D₄	—	73	6	no	—
258	C₂×D₄	—	73	12	no	—
259	C₂×D₄	—	73	12	no	—
260	C₂×D₄	—	73	3	no	—
261	C₂×D₄	#11924036	1	3	yes	—
262	C₂²×C₄	—	73	3	no	—
263	C₂²×C₄	—	73	3	no	—
264	C₂²×C₄	—	73	3	no	—
265	C₂²×C₄	—	73	6	no	—
266	C₂²×C₄	—	73	6	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	3	no	—
269	C₂²×C₄	—	73	6	no	—
270	C₂²×C₄	—	73	6	no	—
271	C₂²×C₄	C₂²×#64408	1	3	yes	—
272	M₁₆	—	73	12	no	—
273	M₁₆	—	73	24	no	—
274	M₁₆	C₂²×#64408	1	12	yes	—
275	C₂×C₈	—	73	3	no	—
276	C₂×C₈	—	73	6	no	—
277	C₂×C₈	—	73	6	no	—
278	C₂×C₈	—	73	1	no	—
279	C₂×C₈	—	73	1	no	—
280	C₂×C₈	—	73	6	no	—
281	C₂×C₈	—	73	6	no	—
282	C₂×C₈	—	73	6	no	—
283	C₂×C₈	—	73	6	no	—
284	C₂×C₈	—	73	6	no	—
285	C₂×C₈	—	73	1	no	—
286	C₂×C₈	—	73	1	no	—
287	C₂×C₈	—	73	6	no	—
288	C₂×C₈	—	73	1	no	—
289	C₂×C₈	C₂²×#64408	1	3	yes	—
290	C₂×C₈	C₂²×#64408	1	6	yes	—
291	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
292	C₂×C₈	C₂²×#64408	1	6	yes	—
293	C₁₆	—	73	16	no	—
294	C₂³	—	292	6	no	—
295	C₂³	—	73	6	no	—
296	C₂³	—	73	3	no	—
297	C₂³	—	73	1	no	—
298	C₂³	—	73	6	no	—
299	C₂³	C₂×#11924036	1	3	yes	—
300	Q₈	—	73	12	no	—
301	Q₈	C₂×#11924036	1	4	yes	—
302	D₄	—	73	24	no	—
303	D₄	—	73	12	no	—
304	D₄	C₂×#11924036	1	12	yes	—
305	C₂×C₄	—	73	1	no	—
306	C₂×C₄	—	73	3	no	—
307	C₂×C₄	—	73	6	no	—
308	C₂×C₄	—	73	6	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	—	73	1	no	—
311	C₂×C₄	—	73	6	no	—
312	C₂×C₄	—	73	6	no	—
313	C₂×C₄	—	73	6	no	—
314	C₂×C₄	—	73	6	no	—
315	C₂×C₄	—	73	6	no	—
316	C₂×C₄	—	73	1	no	—
317	C₂×C₄	—	73	1	no	—
318	C₂×C₄	—	73	6	no	—
319	C₂×C₄	C₂×#11924036	1	3	yes	—
320	C₂×C₄	C₂×#11924036	1	6	yes	—
321	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
322	C₂×C₄	C₂³×#64408	1	6	yes	—
323	C₈	—	73	1	no	—
324	C₈	—	73	1	no	—
325	C₈	—	73	6	no	—
326	C₈	C₂³×#64408	1	1	yes	characteristic
327	C₈	C₂³×#64408	1	6	yes	—
328	C₈	C₂³×#64408	1	1	yes	characteristic
329	C₂²	—	4	3	no	—
330	C₂²	—	292	3	no	—
331	C₂²	—	292	6	no	—
332	C₂²	—	73	1	no	—
333	C₂²	—	73	1	no	—
334	C₂²	—	73	6	no	—
335	C₂²	—	146	2	no	—
336	C₂²	—	146	12	no	—
337	C₂²	C₂²×#11924036	1	1	yes	characteristic
338	C₂²	C₂²×#11924036	1	6	yes	—
339	C₄	—	73	1	no	—
340	C₄	—	73	1	no	—
341	C₄	—	73	6	no	—
342	C₄	C₂²×#11924036	1	1	yes	characteristic
343	C₄	C₂²×#11924036	1	6	yes	—
344	C₄	C₂⁴×#64408	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	6	no	—
347	C₂	—	73	2	no	—
348	C₂	—	146	1	no	—
349	C₂	—	146	6	no	—
350	C₂	C₂³×#11924036	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂³×#64408
commutator, socle	C₂×C₇₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#64408
Fitting	C₂×#131×C₇₃	C₄
prime radical	C₂×#41×D₇₃	C₄
cosocle	#10596831	C₂⁵
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	C₂×#3407
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #22626900	C₂	semidirect
2. #22626899	C₂	semidirect
3. #22626898	C₂	semidirect
4. #22626896	C₂	semidirect
5. C₂×#17989643	C₂	semidirect
7. #17989646	C₂²	semidirect
8. #17989645	C₂²	semidirect
9. C₂×#14556666	C₂²	semidirect
11. #17989643	C₂²	semidirect
12. #17989643	C₂²	semidirect
13. #17989643	C₂²	semidirect
14. C₂×#14556665	C₂²	semidirect
15. C₂×#14556664	C₂²	semidirect
28. #14556666	C₂³	semidirect
30. #14556665	C₂³	semidirect
32. #14556664	C₂³	semidirect
33. C₂×#11924031	C₂³	semidirect
175. D₇₃	#3407	semidirect
182. C₇₃	C₂×#3407	semidirect

Maximal subgroups

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

subgroup	index
#22626900	2
#22626899	2
#22626898	2
#22626896	2
C₂×#17989643	2
C₂×#131×D₇₃	2
C₂×#3407	73