Subgroups — #34030069

Order 18688.

706 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	C₂×#17989646	C₂	1	2	yes	maximal
2	#22626268	C₂	1	2	yes	maximal, complemented
3	#22626267	C₂	1	4	yes	maximal, complemented
4	#22626264	C₂	1	2	yes	maximal, complemented
5	#22626263	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626261	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14556599	C₂	1	1	yes	maximal, characteristic
9	#17989646	—	2	8	no	—
10	C₂×#14556666	C₂²	1	4	yes	—
11	C₂×#14556665	C₂²	1	2	yes	—
12	C₂×#14556664	C₂²	1	4	yes	—
13	C₂²×#11924031	C₂²	1	2	yes	—
14	#17989508	C₂²	1	4	yes	—
15	#17989508	C₂²	1	4	yes	—
16	C₄⋊#11924031	C₂²	1	1	yes	characteristic
17	C₄⋊#11924031	C₂²	1	1	yes	characteristic
18	C₄⋊#11924031	C₂²	1	2	yes	—
19	C₄⋊#11924031	C₂²	1	2	yes	—
20	C₄⋊#11924031	C₂²	1	2	yes	—
21	C₂×#14556599	C₄	1	1	yes	characteristic
22	C₂×#14556599	C₂²	1	1	yes	characteristic
23	C₄×#11924028	C₄	1	1	yes	characteristic
24	#17988545	C₄	1	1	yes	characteristic
25	#17988545	C₂²	1	1	yes	characteristic
26	#17988544	C₄	1	2	yes	—
27	#17988544	C₂²	1	2	yes	—
28	C₄×#11924011	C₄	1	2	yes	—
29	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
32	#14556666	—	2	8	no	—
33	#14556665	—	2	4	no	—
34	#14556664	—	2	8	no	—
35	C₂×#11924031	—	2	4	no	—
36	C₂×#11924031	C₂³	1	2	yes	—
37	C₂×#11924031	C₂³	1	2	yes	—
38	C₂×#11924031	C₂³	1	4	yes	—
39	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
40	#14556599	—	2	4	no	—
41	#14556599	D₄	1	4	yes	—
42	#14556599	Q₈	1	4	yes	—
43	C₂×#11924011	C₂×C₄	1	2	yes	—
44	C₂×#11924011	C₂×C₄	1	2	yes	—
45	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
46	C₂²×#10596831	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₃	C₂³	1	2	yes	—
49	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
50	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
51	C₄×D₂₉₂	C₈	1	1	yes	characteristic
52	C₄×D₂₉₂	C₈	1	1	yes	characteristic
53	C₄×C₂²⋊D₇₃	C₈	1	2	yes	—
54	C₄×C₂²⋊D₇₃	C₈	1	2	yes	—
55	#14556550	C₂×C₄	1	1	yes	characteristic
56	#14556548	C₂×C₄	1	2	yes	—
57	C₄²×D₇₃	C₂×C₄	1	2	yes	—
58	#14556501	C₂×C₄	1	2	yes	—
59	#14556501	C₂×C₄	1	2	yes	—
60	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
61	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
62	C₄×#10596831	C₂³	1	1	yes	characteristic
63	C₄×#10596831	C₂³	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
65	C₈×Dic₇₃	C₂³	1	2	yes	—
66	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
69	#97×C₇₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
71	#11924031	—	2	2	no	—
72	#11924031	—	2	2	no	—
73	#11924031	—	2	4	no	—
74	#11924028	C₄⋊C₄	1	4	yes	—
75	#11924028	M₁₆	1	4	yes	—
76	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
77	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
82	#11924011	—	2	4	no	—
83	#11924011	C₄⋊C₄	1	8	yes	—
84	C₂×#10596831	—	2	2	no	—
85	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
88	C₂×#10596831	C₂×D₄	1	2	yes	—
89	C₂×#10596831	C₂×Q₈	1	2	yes	—
90	C₂×#10596831	C₂⁴	1	1	yes	characteristic
91	C₈×D₇₃	—	2	4	no	—
92	C₈×D₇₃	C₂×D₄	1	4	yes	—
93	C₈×D₇₃	C₂×Q₈	1	4	yes	—
94	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
96	#11924007	C₂×C₈	1	2	yes	—
97	#11924007	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
99	#11924005	C₂×C₈	1	2	yes	—
100	#11924005	C₂×C₈	1	2	yes	—
101	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
102	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₇₃	—	2	2	no	—
108	M₁₆×C₇₃	C₄⋊C₄	1	4	yes	—
109	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	4	1	no	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
127	C₂²⋊D₇₃	C₄⋊C₈	1	8	yes	—
128	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
129	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
130	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
131	D₂₉₂	C₄⋊C₈	1	4	yes	—
132	C₄×D₇₃	—	2	4	no	—
133	C₄×D₇₃	—	4	2	no	—
134	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
136	Dic₁₄₆	C₄⋊C₈	1	4	yes	—
137	C₈×C₇₃	—	2	2	no	—
138	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
139	#10596831	—	2	1	no	—
140	#10596831	—	2	1	no	—
141	#10596831	C₂×C₄⋊C₄	1	2	yes	—
142	#10596831	C₂²×D₄	1	1	yes	characteristic
143	#10596831	C₂²×Q₈	1	1	yes	characteristic
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
147	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
148	C₂×D₇₃	—	2	2	no	—
149	C₂×D₇₃	—	2	2	no	—
150	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
155	C₄×C₇₃	#504	1	2	yes	complemented
156	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
158	Dic₇₃	—	2	2	no	—
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
161	#33398	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	—	2	2	no	—
163	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
165	C₂×C₇₃	#3407	1	1	yes	characteristic
166	D₇₃	—	2	4	no	—
167	C₂×#504	—	73	2	no	—
168	C₄×#131	—	73	1	no	—
169	#2151	—	73	4	no	—
170	#2151	—	73	2	no	—
171	#2150	—	73	1	no	—
172	#2150	—	73	2	no	—
173	C₂×C₄⋊C₁₆	—	73	1	no	—
174	C₂×C₄⋊C₁₆	—	73	2	no	—
175	C₇₃	#33398	1	1	yes	characteristic, complemented, Sylow
176	C₂×#131	—	73	1	no	—
177	C₂×#131	—	73	1	no	—
178	C₄×#41	—	73	1	no	—
179	#504	—	146	8	no	—
180	C₂×M₃₂	—	73	2	no	—
181	C₂×M₃₂	—	73	4	no	—
182	C₂²×C₁₆	—	73	2	no	—
183	C₂²×C₁₆	—	73	4	no	—
184	#405	—	73	1	no	—
185	#405	—	73	1	no	—
186	#405	—	73	2	no	—
187	#405	—	73	2	no	—
188	C₄×M₁₆	—	73	2	no	—
189	C₄×M₁₆	#64408	1	1	yes	characteristic
190	C₂×C₄×C₈	—	73	1	no	—
191	C₂×C₄×C₈	—	73	2	no	—
192	C₄⋊C₁₆	—	73	1	no	—
193	C₄⋊C₁₆	—	73	1	no	—
194	C₄⋊C₁₆	—	73	2	no	—
195	C₄⋊C₁₆	—	73	2	no	—
196	C₄⋊C₁₆	—	73	4	no	—
197	C₄⋊C₁₆	—	73	2	no	—
198	C₄⋊C₁₆	—	73	4	no	—
199	C₂×#41	—	73	1	no	—
200	#131	—	73	4	no	—
201	#131	—	73	4	no	—
202	#131	—	146	4	no	—
203	C₂×M₁₆	—	73	2	no	—
204	C₂×M₁₆	—	73	2	no	—
205	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
206	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₄×Q₈	—	73	1	no	—
212	C₄×Q₈	—	73	1	no	—
213	C₄×D₄	—	73	2	no	—
214	C₄×D₄	—	73	2	no	—
215	C₄×D₄	—	73	1	no	—
216	C₄×D₄	—	73	1	no	—
217	#117	—	73	1	no	—
218	#117	—	73	2	no	—
219	C₂×C₄²	—	73	2	no	—
220	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
221	M₃₂	—	146	4	no	—
222	M₃₂	—	146	8	no	—
223	C₂×C₁₆	—	73	2	no	—
224	C₂×C₁₆	—	73	2	no	—
225	C₂×C₁₆	—	73	4	no	—
226	C₂×C₁₆	—	146	4	no	—
227	C₂×C₁₆	—	146	8	no	—
228	#97	—	73	2	no	—
229	#97	—	73	2	no	—
230	#97	C₂×#64408	1	2	yes	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	2	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	1	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	C₂×#64408	1	2	yes	—
238	#41	—	73	4	no	—
239	#41	—	73	4	no	—
240	C₂×Q₈	—	73	1	no	—
241	C₂×D₄	—	73	2	no	—
242	C₂×D₄	—	73	1	no	—
243	C₂²×C₄	—	73	2	no	—
244	C₂²×C₄	—	73	2	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	#11924036	1	1	yes	characteristic
247	C₂²×C₄	#11924036	1	1	yes	characteristic
248	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
249	M₁₆	—	2	2	no	—
250	M₁₆	—	73	8	no	—
251	M₁₆	—	146	4	no	—
252	M₁₆	C₄⋊#64408	1	4	yes	—
253	C₂×C₈	—	73	1	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	4	no	—
258	C₂×C₈	—	73	4	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	146	4	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	C₂²×#64408	1	2	yes	—
264	C₂×C₈	C₂²×#64408	1	2	yes	—
265	C₄⋊C₄	—	73	2	no	—
266	C₄⋊C₄	—	73	2	no	—
267	C₄⋊C₄	—	73	1	no	—
268	C₄⋊C₄	—	73	1	no	—
269	C₂²⋊C₄	—	73	2	no	—
270	C₂²⋊C₄	—	73	2	no	—
271	C₂²⋊C₄	—	73	2	no	—
272	C₄²	—	73	2	no	—
273	C₄²	—	73	2	no	—
274	C₄²	—	73	2	no	—
275	C₄²	#11924036	1	1	yes	characteristic
276	C₄²	#11924036	1	1	yes	characteristic
277	C₄²	C₂²×#64408	1	1	yes	characteristic
278	C₄²	C₂²×#64408	1	1	yes	characteristic
279	C₁₆	—	146	2	no	—
280	C₁₆	—	146	2	no	—
281	C₁₆	—	146	4	no	—
282	C₂³	—	73	2	no	—
283	C₂³	C₂×#11924036	1	1	yes	characteristic
284	Q₈	—	73	4	no	—
285	D₄	—	73	8	no	—
286	D₄	—	73	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	292	2	no	—
290	C₂×C₄	—	73	2	no	—
291	C₂×C₄	—	73	4	no	—
292	C₂×C₄	—	73	4	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	146	4	no	—
296	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
297	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
298	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
299	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
300	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
301	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
302	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
303	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	73	1	no	—
306	C₈	—	73	1	no	—
307	C₈	—	73	2	no	—
308	C₈	—	146	1	no	—
309	C₈	—	146	1	no	—
310	C₈	C₂×C₄⋊#64408	1	4	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	73	4	no	—
314	C₂²	—	146	2	no	—
315	C₂²	—	146	2	no	—
316	C₂²	#17989588	1	2	yes	—
317	C₂²	C₂²×#11924036	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	73	4	no	—
322	C₄	—	146	2	no	—
323	C₄	—	146	2	no	—
324	C₄	#17989588	1	2	yes	—
325	C₄	#17989646	1	2	yes	complemented
326	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
327	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	146	4	no	—
330	C₂	C₂×#17989588	1	1	yes	characteristic
331	C₂	C₂×#17989646	1	1	yes	characteristic
332	C₂	#22626900	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×M₁₆×C₇₃	C₄
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#33398
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22626268	C₂	semidirect
3. #22626267	C₂	semidirect
4. #22626264	C₂	semidirect
5. #22626263	C₂	semidirect
6. #22626261	C₂	semidirect
7. C₂×C₄⋊#11924031	C₂	semidirect
155. C₄×C₇₃	#504	semidirect
175. C₇₃	#33398	semidirect
325. C₄	#17989646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989646	2
#22626268	2
#22626267	2
#22626264	2
#22626263	2
#22626261	2
C₂×C₄⋊#11924031	2
C₄×#14556599	2
#33398	73