Subgroups — #34030327

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626347	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626347	C₂	1	2	yes	maximal, complemented
4	#22626346	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626345	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626267	C₂	1	2	yes	maximal
8	#22626261	C₂	1	1	yes	maximal, characteristic
9	#22626141	C₂	1	2	yes	maximal
10	C₄×#14556666	C₂	1	1	yes	maximal, characteristic
11	#22618862	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556666	C₂²	1	1	yes	characteristic
13	C₂×#14556666	C₂²	1	1	yes	characteristic
14	C₂×#14556664	C₂²	1	2	yes	—
15	#17989508	C₂²	1	2	yes	—
16	#17989508	C₂²	1	2	yes	—
17	#17989508	C₂²	1	2	yes	—
18	C₄⋊#11924031	C₂²	1	1	yes	characteristic
19	C₄⋊#11924031	C₂²	1	2	yes	—
20	C₄⋊#11924031	C₂²	1	1	yes	characteristic
21	C₄⋊#11924031	C₂²	1	2	yes	—
22	#17989476	C₂²	1	1	yes	characteristic
23	#17989476	C₂²	1	2	yes	—
24	#17989476	C₂²	1	1	yes	characteristic
25	#17989475	C₂²	1	2	yes	—
26	#17989473	C₂²	1	2	yes	—
27	C₄×#11924031	C₂²	1	1	yes	characteristic
28	C₄×#11924031	C₂²	1	2	yes	—
29	C₄×#11924031	C₂²	1	1	yes	characteristic
30	#17989141	C₄	1	1	yes	characteristic
31	#17988717	C₄	1	1	yes	characteristic
32	#17988715	C₄	1	2	yes	—
33	Q₈×#10596831	C₂²	1	1	yes	characteristic
34	#17988711	C₂²	1	1	yes	characteristic
35	#17988711	C₂²	1	2	yes	—
36	#17988612	C₄	1	2	yes	—
37	#17988609	C₄	1	1	yes	characteristic
38	#17988544	C₂²	1	2	yes	—
39	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
40	#446×C₇₃	C₄	1	1	yes	characteristic
41	#14556666	—	2	2	no	—
42	#14556666	Q₈	1	4	yes	complemented
43	#14556664	—	2	4	no	—
44	C₂×#11924031	C₂³	1	2	yes	—
45	C₂×#11924031	C₂³	1	2	yes	—
46	C₂×#11924031	C₂³	1	1	yes	characteristic
47	C₂×#11924031	C₂³	1	1	yes	characteristic
48	C₂×#11924031	C₂³	1	1	yes	characteristic
49	C₂×#11924031	C₂³	1	1	yes	characteristic
50	C₂×#11924011	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	#14556590	C₈	1	1	yes	characteristic
54	#14556589	C₈	1	1	yes	characteristic
55	#14556581	C₈	1	2	yes	—
56	#14556571	C₈	1	1	yes	characteristic
57	#14556569	C₈	1	2	yes	—
58	#14556567	C₈	1	1	yes	characteristic
59	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
61	#14556548	C₂×C₄	1	2	yes	—
62	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
64	C₄⋊#10596831	C₂×C₄	1	2	yes	—
65	#14556520	C₂×C₄	1	2	yes	—
66	#14556520	C₂×C₄	1	2	yes	—
67	#14556520	C₂×C₄	1	2	yes	—
68	#14556501	C₂×C₄	1	2	yes	—
69	C₄×#10596831	C₂³	1	1	yes	characteristic
70	C₄×#10596831	C₂³	1	2	yes	—
71	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₇₃	C₂³	1	2	yes	—
73	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
77	#97×C₇₃	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
79	#11924031	—	2	2	no	—
80	#11924031	—	2	2	no	—
81	#11924031	—	2	1	no	—
82	#11924031	—	2	1	no	—
83	#11924031	C₂×Q₈	1	2	yes	—
84	#11924031	C₂×Q₈	1	2	yes	—
85	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
86	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	#11924011	—	2	4	no	—
91	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596831	C₂²×C₄	1	2	yes	—
93	C₂×#10596831	C₂⁴	1	1	yes	characteristic
94	C₈×D₇₃	—	2	2	no	—
95	C₈×D₇₃	C₂×Q₈	1	2	yes	—
96	C₈×D₇₃	#41	1	2	yes	—
97	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
98	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
99	#11924007	C₂×C₈	1	2	yes	—
100	#11924007	C₂×C₈	1	2	yes	—
101	#11924007	C₂×C₈	1	2	yes	—
102	#11924007	C₂×C₈	1	2	yes	—
103	#11924007	C₂×C₈	1	2	yes	—
104	#11924005	C₂×C₈	1	2	yes	—
105	#11924005	C₂×C₈	1	2	yes	—
106	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
110	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
115	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
116	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
118	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
129	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	4	1	no	—
132	C₄×D₇₃	C₄×Q₈	1	2	yes	—
133	C₄×D₇₃	#131	1	2	yes	—
134	Dic₁₄₆	—	2	2	no	—
135	Dic₁₄₆	—	4	2	no	—
136	C₈×C₇₃	—	2	1	no	—
137	C₈×C₇₃	—	2	2	no	—
138	C₈×C₇₃	C₄×Q₈	1	2	yes	—
139	#10596831	—	2	1	no	—
140	#10596831	—	2	2	no	—
141	#10596831	C₂²×Q₈	1	1	yes	characteristic
142	#10596831	C₂×#41	1	1	yes	characteristic
143	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	C₈×Q₈	1	2	yes	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	#504	1	2	yes	—
151	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
153	Dic₇₃	—	2	2	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	2	no	—
157	Dic₇₃	C₈×Q₈	1	2	yes	—
158	#33476	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
161	C₂×C₇₃	#3407	1	1	yes	characteristic
162	D₇₃	—	2	2	no	—
163	#2960	—	73	1	no	—
164	#2183	—	73	1	no	—
165	#2183	—	73	1	no	—
166	#2183	—	73	2	no	—
167	#2183	—	73	2	no	—
168	Q₈×C₁₆	—	73	1	no	—
169	Q₈×C₁₆	—	73	1	no	—
170	#2151	—	73	2	no	—
171	C₂×C₄⋊C₁₆	—	73	1	no	—
172	#2108	—	73	2	no	—
173	C₄×M₃₂	—	73	1	no	—
174	C₇₃	#33476	1	1	yes	characteristic, complemented, Sylow
175	#533	—	73	1	no	—
176	C₂×M₃₂	—	73	1	no	—
177	C₂×M₃₂	—	73	1	no	—
178	C₂²×C₁₆	—	73	2	no	—
179	#446	—	73	2	no	—
180	#446	—	73	1	no	—
181	#446	#64408	1	1	yes	characteristic
182	C₈×Q₈	—	73	1	no	—
183	C₈×Q₈	—	73	1	no	—
184	C₈×Q₈	—	73	2	no	—
185	#424	—	73	2	no	—
186	#423	—	73	1	no	—
187	#405	—	73	2	no	—
188	C₂×C₄×C₈	—	73	1	no	—
189	C₄⋊C₁₆	—	73	1	no	—
190	C₄⋊C₁₆	—	73	1	no	—
191	C₄⋊C₁₆	—	73	2	no	—
192	C₄⋊C₁₆	—	73	2	no	—
193	C₄⋊C₁₆	—	73	2	no	—
194	C₄⋊C₁₆	—	73	2	no	—
195	C₄⋊C₁₆	—	73	2	no	—
196	#346	—	73	1	no	—
197	#346	—	73	1	no	—
198	#346	—	73	2	no	—
199	#346	—	73	2	no	—
200	C₄×C₁₆	—	73	1	no	—
201	C₄×C₁₆	—	73	1	no	—
202	C₄×C₁₆	—	73	2	no	—
203	C₄×C₁₆	—	73	2	no	—
204	C₂×M₁₆	—	73	2	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₄⋊Q₈	—	73	1	no	—
208	C₄⋊Q₈	—	73	1	no	—
209	#125	—	73	2	no	—
210	C₂²⋊Q₈	—	73	1	no	—
211	C₂²⋊Q₈	—	73	2	no	—
212	C₂²⋊Q₈	—	73	1	no	—
213	C₄×Q₈	—	73	2	no	—
214	C₄×Q₈	—	73	1	no	—
215	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
216	#117	—	73	2	no	—
217	C₂×C₄²	—	73	1	no	—
218	M₃₂	—	73	4	no	—
219	M₃₂	—	146	2	no	—
220	C₂×C₁₆	—	73	1	no	—
221	C₂×C₁₆	—	73	1	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₂×C₁₆	—	73	1	no	—
224	C₂×C₁₆	—	73	1	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	146	4	no	—
227	C₄⋊C₈	—	73	2	no	—
228	C₄⋊C₈	—	73	2	no	—
229	C₄⋊C₈	—	73	2	no	—
230	C₄⋊C₈	—	73	1	no	—
231	C₄⋊C₈	—	73	2	no	—
232	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
233	C₄⋊C₈	C₂×#64408	1	2	yes	—
234	#97	—	73	2	no	—
235	#97	C₂×#64408	1	2	yes	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	1	no	—
239	C₄×C₈	—	73	1	no	—
240	C₄×C₈	—	73	2	no	—
241	C₄×C₈	C₂×#64408	1	1	yes	characteristic
242	C₂×Q₈	—	73	2	no	—
243	C₂×Q₈	—	73	1	no	—
244	C₂×Q₈	#11924036	1	1	yes	characteristic
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	M₁₆	—	146	4	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
256	C₂×C₈	C₂²×#64408	1	2	yes	—
257	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
258	C₄⋊C₄	—	73	2	no	—
259	C₄⋊C₄	—	73	2	no	—
260	C₄⋊C₄	—	73	2	no	—
261	C₄⋊C₄	—	73	2	no	—
262	C₄⋊C₄	—	73	2	no	—
263	C₄⋊C₄	—	73	2	no	—
264	C₄⋊C₄	—	73	1	no	—
265	C₄⋊C₄	#11924036	1	2	yes	—
266	C₄⋊C₄	#11924036	1	1	yes	characteristic
267	C₂²⋊C₄	—	73	2	no	—
268	C₂²⋊C₄	—	73	2	no	—
269	C₄²	—	73	1	no	—
270	C₄²	—	73	1	no	—
271	C₄²	—	73	1	no	—
272	C₄²	—	73	2	no	—
273	C₄²	C₂²×#64408	1	1	yes	characteristic
274	C₄²	C₂²×#64408	1	2	yes	—
275	C₁₆	—	73	2	no	—
276	C₁₆	—	73	2	no	—
277	C₁₆	—	146	2	no	—
278	C₁₆	—	146	2	no	—
279	C₁₆	—	146	1	no	—
280	C₁₆	—	146	1	no	—
281	C₂³	—	73	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	292	2	no	—
284	Q₈	—	146	2	no	—
285	C₂×C₄	—	292	1	no	—
286	C₂×C₄	—	73	1	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	2	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	146	2	no	—
294	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
295	C₂×C₄	C₂×#11924036	1	2	yes	—
296	C₂×C₄	C₂×#11924036	1	2	yes	—
297	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
298	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	73	1	no	—
302	C₈	—	73	1	no	—
303	C₈	—	146	1	no	—
304	C₈	—	146	2	no	—
305	C₈	Q₈×#64408	1	2	yes	—
306	C₂²	—	73	2	no	—
307	C₂²	—	146	1	no	—
308	C₂²	—	146	1	no	—
309	C₂²	C₂²×#11924036	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	73	2	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	2	no	—
316	C₄	—	146	1	no	—
317	C₄	—	146	2	no	—
318	C₄	#17989646	1	2	yes	—
319	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
320	C₄	D₇₃⋊#131	1	1	yes	characteristic
321	C₂	—	146	2	no	—
322	C₂	Q₈×#11924036	1	1	yes	characteristic
323	C₂	C₂×#17989646	1	1	yes	characteristic
324	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#446×C₇₃	C₄
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#446	#64408
hypocenter	C₇₃	#33476
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #22626348	C₂	semidirect
2. #22626347	C₂	semidirect
3. #22626347	C₂	semidirect
4. #22626346	C₂	semidirect
5. #22626345	C₂	semidirect
6. Q₈×#11924031	C₂	semidirect
42. #14556666	Q₈	semidirect
174. C₇₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#22626348	2
#22626347	2
#22626346	2
#22626345	2
Q₈×#11924031	2
#22626267	2
#22626261	2
#22626141	2
C₄×#14556666	2
#22618862	2
#33476	73