Subgroups — #34030143

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626296	C₂	1	4	yes	maximal, complemented
2	#22626292	C₂	1	2	yes	maximal, complemented
3	#22626288	C₂	1	2	yes	maximal, complemented
4	#22626266	C₂	1	2	yes	maximal, complemented
5	#22626158	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊#11924031	C₂	1	1	yes	maximal, characteristic
7	#22626140	C₂	1	2	yes	maximal, complemented
8	C₂×#17988543	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556665	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₂²	1	4	yes	complemented
12	C₄⋊#11924031	C₂²	1	4	yes	complemented
13	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
14	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
15	C₂²⋊#11924031	C₂²	1	2	yes	—
16	C₂²⋊#11924031	C₂²	1	2	yes	—
17	C₂²⋊#11924031	C₂²	1	2	yes	—
18	#17989475	C₂²	1	4	yes	complemented
19	C₄×#11924031	C₂²	1	4	yes	complemented
20	C₂²×#11924012	C₄	1	1	yes	characteristic
21	C₂³×#10596831	C₂²	1	1	yes	characteristic
22	#17988543	C₄	1	2	yes	—
23	#17988543	C₄	1	1	yes	characteristic
24	#17988543	C₄	1	1	yes	characteristic
25	#17988543	C₂²	1	1	yes	characteristic
26	#17988543	C₂²	1	1	yes	characteristic
27	#17988543	C₂²	1	2	yes	—
28	C₂×#14556502	C₄	1	2	yes	—
29	C₂×C₄×#10596831	C₂²	1	2	yes	—
30	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
31	#14556665	—	2	4	no	—
32	C₂×#11924031	—	2	4	no	—
33	C₂×#11924031	C₂³	1	2	yes	—
34	C₂×#11924031	C₂³	1	2	yes	—
35	C₂×#11924031	C₂³	1	4	yes	—
36	C₂×#11924012	—	2	4	no	—
37	C₂×#11924012	C₂×C₄	1	2	yes	—
38	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
40	C₂²×#10596831	—	2	4	no	—
41	C₂²×#10596831	C₂×C₄	1	2	yes	—
42	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
43	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
44	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₂²×#10596831	C₂³	1	1	yes	characteristic
46	C₂²×#10596831	C₂³	1	1	yes	characteristic
47	C₂²×#10596831	C₂³	1	1	yes	characteristic
48	#14556502	C₂×C₄	1	2	yes	—
49	#14556502	C₂×C₄	1	2	yes	—
50	#14556502	C₂×C₄	1	2	yes	—
51	#14556502	C₂×C₄	1	2	yes	—
52	C₄×#10596831	C₂×C₄	1	2	yes	—
53	C₄×#10596831	C₂×C₄	1	2	yes	—
54	C₄×#10596831	C₂³	1	2	yes	—
55	C₄×#10596831	C₂³	1	2	yes	—
56	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
57	#117×C₇₃	C₈	1	1	yes	characteristic
58	#117×C₇₃	C₈	1	1	yes	characteristic
59	#117×C₇₃	C₈	1	2	yes	—
60	#117×C₇₃	C₂×C₄	1	2	yes	—
61	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
62	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₇₃	C₈	1	2	yes	—
66	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
67	#11924031	—	2	2	no	—
68	#11924031	—	2	2	no	—
69	#11924031	—	2	4	no	—
70	#11924012	—	2	2	no	—
71	#11924012	—	2	4	no	—
72	#11924012	M₁₆	1	4	yes	—
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
80	C₂×#10596831	C₂²×C₄	1	2	yes	—
81	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₂×#10596831	#41	1	4	yes	—
85	C₂×#10596831	C₂⁴	1	1	yes	characteristic
86	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	—	2	2	no	—
88	C₂²×C₄×C₇₃	—	2	2	no	—
89	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
98	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
107	C₄²×C₇₃	C₂×C₈	1	2	yes	—
108	C₄²×C₇₃	C₂×C₈	1	2	yes	—
109	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
111	C₂³×C₇₃	—	2	2	no	—
112	C₂³×C₇₃	—	2	2	no	—
113	C₂³×C₇₃	—	4	1	no	—
114	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	4	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	#117	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	#10596831	—	2	1	no	—
137	#10596831	—	2	1	no	—
138	#10596831	—	2	2	no	—
139	#10596831	#117	1	2	yes	—
140	#10596831	C₂×M₁₆	1	4	yes	—
141	#10596831	C₂×#41	1	2	yes	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	—	2	1	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	4	4	no	—
150	C₂²×C₇₃	#431	1	2	yes	—
151	C₂²×C₇₃	#504	1	2	yes	—
152	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₇₃	—	2	2	no	—
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	#431	1	2	yes	—
160	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
161	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
162	#33422	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	—	2	2	no	—
164	C₂×C₇₃	—	2	2	no	—
165	C₂×C₇₃	—	4	1	no	—
166	C₂×C₇₃	C₂×#431	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₃	#3407	1	1	yes	characteristic
169	C₂×#405	—	73	1	no	—
170	#2164	—	73	4	no	—
171	#2163	—	73	2	no	—
172	#2162	—	73	2	no	—
173	#2151	—	73	2	no	—
174	#2112	—	73	1	no	—
175	C₂×C₂²⋊C₁₆	—	73	1	no	—
176	#2108	—	73	2	no	—
177	C₇₃	#33422	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	73	1	no	—
179	C₂³×C₈	—	73	1	no	—
180	C₂×#117	#64408	1	1	yes	characteristic
181	C₂×M₃₂	—	73	2	no	—
182	C₂²×C₁₆	—	73	2	no	—
183	#405	—	73	1	no	—
184	#405	—	73	1	no	—
185	#405	—	73	2	no	—
186	#405	—	73	1	no	—
187	#405	—	73	2	no	—
188	#405	—	73	1	no	—
189	C₂×#97	—	73	2	no	—
190	C₂×C₄×C₈	—	73	2	no	—
191	C₄⋊C₁₆	—	73	4	no	—
192	C₄⋊C₁₆	—	73	4	no	—
193	C₂²⋊C₁₆	—	73	1	no	—
194	C₂²⋊C₁₆	—	73	1	no	—
195	C₂²⋊C₁₆	—	73	2	no	—
196	C₂²⋊C₁₆	—	73	2	no	—
197	C₂²⋊C₁₆	—	73	2	no	—
198	#346	—	73	4	no	—
199	C₄×C₁₆	—	73	4	no	—
200	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
201	C₂×M₁₆	—	73	1	no	—
202	C₂×M₁₆	—	73	2	no	—
203	C₂×M₁₆	—	73	1	no	—
204	C₂×M₁₆	—	146	4	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	—	146	4	no	—
213	#117	#10596833	1	1	yes	characteristic
214	#117	#10596833	1	1	yes	characteristic
215	#117	#10596833	1	2	yes	—
216	#117	C₂×#64408	1	2	yes	—
217	#117	C₂×#64408	1	1	yes	characteristic
218	#117	C₂×#64408	1	1	yes	characteristic
219	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
220	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	#10596833	1	2	yes	—
222	C₂×C₄²	C₂×#64408	1	2	yes	—
223	M₃₂	—	146	4	no	—
224	C₂×C₁₆	—	73	2	no	—
225	C₂×C₁₆	—	73	4	no	—
226	C₂×C₁₆	—	73	2	no	—
227	C₂×C₁₆	—	146	4	no	—
228	#97	—	73	2	no	—
229	#97	—	73	2	no	—
230	#97	—	73	2	no	—
231	#97	—	73	2	no	—
232	C₄×C₈	—	73	2	no	—
233	C₄×C₈	—	73	2	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	2	no	—
236	C₂⁴	C₂×#10596833	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	C₂×#10596833	1	2	yes	—
240	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
241	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
242	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
243	C₂²×C₄	C₂×#10596833	1	2	yes	—
244	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
245	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
246	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
247	M₁₆	—	73	4	no	—
248	M₁₆	—	146	4	no	—
249	M₁₆	—	146	2	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	4	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	—	146	4	no	—
262	C₂×C₈	—	146	2	no	—
263	C₄⋊C₄	C₂×#10596833	1	2	yes	—
264	C₄⋊C₄	C₂×#10596833	1	2	yes	—
265	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
266	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
267	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
269	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
270	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
271	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
272	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
273	C₄²	C₂×#10596833	1	2	yes	—
274	C₄²	C₂×#10596833	1	2	yes	—
275	C₄²	C₂²×#64408	1	2	yes	—
276	C₄²	C₂²×#64408	1	2	yes	—
277	C₁₆	—	146	2	no	—
278	C₁₆	—	146	2	no	—
279	C₁₆	—	146	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₂²×#10596833	1	1	yes	characteristic
284	C₂³	C₂²×#10596833	1	1	yes	characteristic
285	C₂³	C₂²×#10596833	1	1	yes	characteristic
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#14556670	1	2	yes	—
296	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
297	C₂×C₄	C₂²×#10596833	1	2	yes	—
298	C₂×C₄	C₂²×#10596833	1	2	yes	—
299	C₂×C₄	C₂²×#10596833	1	2	yes	—
300	C₂×C₄	C₂²×#10596833	1	2	yes	—
301	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
302	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
303	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
304	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
305	C₈	—	73	2	no	—
306	C₈	—	73	2	no	—
307	C₈	—	73	4	no	—
308	C₈	—	146	1	no	—
309	C₈	—	146	1	no	—
310	C₈	—	146	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	#17989603	1	2	yes	—
320	C₂²	#17989645	1	2	yes	—
321	C₂²	C₂³×#10596833	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	#17989603	1	2	yes	—
329	C₄	C₂×#14556670	1	1	yes	characteristic
330	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#17989603	1	1	yes	characteristic
335	C₂	C₂×#17989645	1	1	yes	characteristic
336	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂×#117×C₇₃	C₄
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#117	#64408
hypocenter	C₇₃	#33422
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22626296	C₂	semidirect
2. #22626292	C₂	semidirect
3. #22626288	C₂	semidirect
4. #22626266	C₂	semidirect
7. #22626140	C₂	semidirect
11. C₄⋊#11924031	C₂²	semidirect
12. C₄⋊#11924031	C₂²	semidirect
18. #17989475	C₂²	semidirect
19. C₄×#11924031	C₂²	semidirect
177. C₇₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
#22626296	2
#22626292	2
#22626288	2
#22626266	2
#22626158	2
C₂×C₂²⋊#11924031	2
#22626140	2
C₂×#17988543	2
#33422	73