Subgroups — #30774332

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863780	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863210	C₂	1	1	yes	maximal, characteristic
4	#22561930	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561926	C₂	1	1	yes	maximal, characteristic
6	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
9	C₂²×#11761617	—	2	1	no	—
10	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
11	C₂×#14524031	—	2	1	no	—
12	#16863780	—	2	2	no	—
13	#16863780	C₄	1	4	yes	complemented
14	C₂×#14523979	C₂²	1	1	yes	characteristic
15	#16863434	C₂²	1	1	yes	characteristic
16	#16863351	—	2	1	no	—
17	#16863213	—	2	2	no	—
18	#16863210	—	2	2	no	—
19	#16863201	—	2	2	no	—
20	#16863198	—	2	2	no	—
21	#16863156	—	2	1	no	—
22	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#364×C₆₇	—	2	2	no	—
26	C₂²×Dic₁₃₄	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂×Dic₂₆₈	—	2	1	no	—
29	C₂×Dic₂₆₈	—	2	1	no	—
30	C₂×Dic₂₆₈	—	2	1	no	—
31	C₂×Dic₂₆₈	—	2	1	no	—
32	C₂×Dic₂₆₈	—	2	1	no	—
33	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
34	C₂×#11761617	—	2	4	no	—
35	C₂×#11761617	—	4	1	no	—
36	C₂×#11761617	—	4	1	no	—
37	C₂×#11761617	—	4	2	no	—
38	C₂×D₅₃₆	—	2	1	no	—
39	C₂×D₅₃₆	—	2	1	no	—
40	C₂×D₅₃₆	—	2	1	no	—
41	C₂×D₅₃₆	—	2	1	no	—
42	C₂×D₅₃₆	—	2	1	no	—
43	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
44	C₂×#11761612	—	2	1	no	—
45	#14524031	—	2	4	no	—
46	#14524031	—	4	2	no	—
47	#14524002	—	2	1	no	—
48	C₂×(C₂×D₆₇)⋊C₄	—	4	2	no	—
49	#14523979	—	2	1	no	—
50	#14523979	D₄	1	2	yes	—
51	#14523977	D₄	1	2	yes	—
52	#14523975	—	2	1	no	—
53	#14523953	—	4	1	no	—
54	C₈×Dic₆₇	—	4	1	no	—
55	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	M₃₂×C₆₇	—	2	1	no	—
59	M₃₂×C₆₇	—	2	1	no	—
60	M₃₂×C₆₇	—	2	1	no	—
61	M₃₂×C₆₇	—	2	1	no	—
62	C₂×C₁₆×C₆₇	—	2	1	no	—
63	C₂×C₁₆×C₆₇	—	2	1	no	—
64	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₆₇	D₄	1	2	yes	—
66	C₂³×D₆₇	—	4	1	no	—
67	C₂×Dic₁₃₄	—	2	2	no	—
68	C₂×Dic₁₃₄	—	4	1	no	—
69	C₂×Dic₁₃₄	—	4	1	no	—
70	C₂×Dic₁₃₄	—	4	1	no	—
71	C₂×Dic₁₃₄	—	4	1	no	—
72	C₂×Dic₁₃₄	—	4	1	no	—
73	C₂×D₂₆₈	—	2	2	no	—
74	C₂×D₂₆₈	—	4	1	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	4	1	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂²×Dic₆₇	—	4	1	no	—
80	Dic₂₆₈	—	2	2	no	—
81	Dic₂₆₈	—	2	2	no	—
82	Dic₂₆₈	—	4	1	no	—
83	Dic₂₆₈	—	4	1	no	—
84	#11761617	—	4	4	no	—
85	#11761617	—	4	4	no	—
86	D₅₃₆	—	2	2	no	—
87	D₅₃₆	—	2	2	no	—
88	D₅₃₆	—	4	1	no	—
89	D₅₃₆	—	4	1	no	—
90	#11761612	—	2	2	no	—
91	#11761612	—	4	1	no	—
92	C₂×C₆₇⋊C₈	—	4	1	no	—
93	C₄⋊Dic₆₇	—	2	2	no	—
94	C₂²⋊Dic₆₇	—	4	1	no	—
95	(C₂×D₆₇)⋊C₄	—	4	8	no	—
96	C₄×Dic₆₇	—	4	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
101	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
102	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
108	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
111	C₁₆×C₆₇	—	4	1	no	—
112	C₁₆×C₆₇	—	4	1	no	—
113	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₆₇	—	4	2	no	—
115	C₂²×D₆₇	—	8	4	no	—
116	C₂²×D₆₇	—	8	1	no	—
117	C₂²×D₆₇	—	8	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×Dic₆₇	—	4	2	no	—
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	8	1	no	—
129	C₂×Dic₆₇	—	8	1	no	—
130	D₂₆₈	—	4	2	no	—
131	D₂₆₈	—	4	2	no	—
132	D₂₆₈	—	8	1	no	—
133	D₂₆₈	—	8	1	no	—
134	Dic₁₃₄	—	4	2	no	—
135	Dic₁₃₄	—	4	2	no	—
136	Dic₁₃₄	—	8	1	no	—
137	Dic₁₃₄	—	8	1	no	—
138	C₈×C₆₇	#102	1	2	yes	—
139	C₈×C₆₇	#102	1	2	yes	—
140	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₈×C₆₇	#124	1	1	yes	characteristic
143	C₈×C₆₇	#127	1	1	yes	characteristic
144	C₆₇⋊C₈	—	4	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	#418	1	1	yes	characteristic
147	C₂²×C₆₇	#459	1	1	yes	characteristic
148	C₂²×C₆₇	#460	1	1	yes	characteristic
149	C₂×D₆₇	—	8	4	no	—
150	C₂×D₆₇	—	8	2	no	—
151	C₂×D₆₇	—	16	2	no	—
152	C₂×D₆₇	—	16	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	#390	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
156	C₄×C₆₇	#459	1	1	yes	characteristic
157	C₄×C₆₇	#460	1	1	yes	characteristic
158	Dic₆₇	—	8	2	no	—
159	Dic₆₇	—	8	2	no	—
160	#13260	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	#1982	1	1	yes	characteristic
164	C₂×C₆₇	#2212	1	2	yes	—
165	D₆₇	—	16	2	no	—
166	C₂×#495	—	67	1	no	—
167	C₂×#364	D₆₇	1	1	yes	characteristic, complemented
168	C₂×#362	—	67	1	no	—
169	C₂×#361	—	67	1	no	—
170	#2141	—	67	1	no	—
171	#2140	—	67	1	no	—
172	#1772	—	67	1	no	—
173	C₆₇	#13260	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	67	1	no	—
175	C₂²×SD₁₆	—	134	1	no	—
176	C₂²×D₈	—	67	1	no	—
177	C₂×#124	—	134	1	no	—
178	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
179	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
180	#495	—	67	4	no	—
181	#495	—	134	2	no	—
182	C₂×#108	—	67	1	no	—
183	#427	—	67	1	no	—
184	#405	—	134	1	no	—
185	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
186	#364	—	2	2	no	—
187	#362	—	134	2	no	—
188	#361	—	134	2	no	—
189	#358	—	134	2	no	—
190	#357	—	134	2	no	—
191	#339	—	134	1	no	—
192	C₂²×Q₈	—	134	1	no	—
193	C₂²×D₄	—	134	1	no	—
194	C₂×Q₁₆	—	67	2	no	—
195	C₂×Q₁₆	—	134	1	no	—
196	C₂×Q₁₆	—	134	1	no	—
197	C₂×Q₁₆	—	134	1	no	—
198	C₂×Q₁₆	—	134	1	no	—
199	C₂×Q₁₆	—	134	1	no	—
200	C₂×SD₁₆	—	268	1	no	—
201	C₂×SD₁₆	—	268	1	no	—
202	C₂×SD₁₆	—	268	2	no	—
203	C₂×SD₁₆	—	134	4	no	—
204	C₂×D₈	—	67	2	no	—
205	C₂×D₈	—	134	1	no	—
206	C₂×D₈	—	134	1	no	—
207	C₂×D₈	—	134	1	no	—
208	C₂×D₈	—	134	1	no	—
209	C₂×D₈	—	134	1	no	—
210	C₂×M₁₆	—	134	1	no	—
211	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
213	#124	—	268	2	no	—
214	#124	—	134	4	no	—
215	#117	—	134	1	no	—
216	C₂×C₂²⋊C₄	—	268	2	no	—
217	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	#108	—	67	2	no	—
225	#108	—	134	1	no	—
226	#107	—	67	2	no	—
227	#106	—	134	1	no	—
228	#97	—	268	1	no	—
229	C₄×C₈	—	268	1	no	—
230	C₄×C₈	C₄×D₆₇	1	2	yes	—
231	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
232	C₂⁴	—	268	1	no	—
233	C₂×Q₈	—	268	1	no	—
234	C₂×Q₈	—	268	1	no	—
235	C₂×Q₈	—	268	1	no	—
236	C₂×Q₈	—	268	1	no	—
237	C₂×Q₈	—	268	1	no	—
238	C₂×Q₈	—	134	2	no	—
239	C₂×D₄	—	268	1	no	—
240	C₂×D₄	—	268	1	no	—
241	C₂×D₄	—	268	1	no	—
242	C₂×D₄	—	268	1	no	—
243	C₂×D₄	—	268	1	no	—
244	C₂×D₄	—	134	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	268	1	no	—
247	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
248	Q₁₆	—	268	1	no	—
249	Q₁₆	—	268	1	no	—
250	Q₁₆	—	134	2	no	—
251	Q₁₆	—	134	2	no	—
252	SD₁₆	—	268	4	no	—
253	SD₁₆	—	268	4	no	—
254	D₈	—	268	1	no	—
255	D₈	—	268	1	no	—
256	D₈	—	134	2	no	—
257	D₈	—	134	2	no	—
258	M₁₆	—	268	1	no	—
259	M₁₆	—	134	2	no	—
260	C₂×C₈	—	268	1	no	—
261	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
262	C₂×C₈	D₅₃₆	1	1	yes	characteristic
263	C₂×C₈	D₅₃₆	1	1	yes	characteristic
264	C₂×C₈	#11761617	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
267	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
269	C₂×C₈	#11761629	1	1	yes	characteristic
270	C₂×C₈	#11761629	1	1	yes	characteristic
271	C₄⋊C₄	—	134	2	no	—
272	C₂²⋊C₄	—	268	8	no	—
273	C₂²⋊C₄	—	268	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	268	1	no	—
276	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
277	C₁₆	—	4	1	no	—
278	C₁₆	—	4	1	no	—
279	C₂³	—	268	2	no	—
280	C₂³	—	536	4	no	—
281	C₂³	—	536	1	no	—
282	C₂³	—	536	1	no	—
283	C₂³	#14524071	1	1	yes	characteristic
284	Q₈	—	268	2	no	—
285	Q₈	—	268	2	no	—
286	Q₈	—	536	1	no	—
287	Q₈	—	536	1	no	—
288	D₄	—	268	2	no	—
289	D₄	—	268	2	no	—
290	D₄	—	536	1	no	—
291	D₄	—	536	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	268	2	no	—
295	C₂×C₄	—	268	2	no	—
296	C₂×C₄	—	536	1	no	—
297	C₂×C₄	—	536	1	no	—
298	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#14524018	1	1	yes	characteristic
300	C₂×C₄	#14524018	1	1	yes	characteristic
301	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
303	C₂×C₄	#14524071	1	1	yes	characteristic
304	C₈	—	268	2	no	—
305	C₈	#14523965	1	2	yes	—
306	C₈	#14523965	1	2	yes	—
307	C₈	#14524005	1	1	yes	characteristic
308	C₈	#14524005	1	1	yes	characteristic
309	C₈	#14524032	1	1	yes	characteristic
310	C₈	#14524039	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	536	4	no	—
313	C₂²	—	536	2	no	—
314	C₂²	—	1072	2	no	—
315	C₂²	—	1072	2	no	—
316	C₂²	#16863402	1	1	yes	characteristic
317	C₂²	#16863592	1	1	yes	characteristic
318	C₂²	#16863597	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	536	2	no	—
321	C₄	—	536	2	no	—
322	C₄	#16863297	1	1	yes	characteristic
323	C₄	C₂×#14523965	1	1	yes	characteristic
324	C₄	#16863592	1	1	yes	characteristic
325	C₄	#16863597	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	1072	2	no	—
329	C₂	#22561135	1	1	yes	characteristic
330	C₂	#22562279	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863592
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#459
prime radical	C₂²×D₅₃₆	C₂²
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#13260
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#16863213	C₂	semidirect
4. #22561930	C₂	semidirect
6. #22560088	C₂	semidirect
7. C₂×#364×C₆₇	C₂	semidirect
13. #16863780	C₄	semidirect
167. C₂×#364	D₆₇	semidirect
173. C₆₇	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863780	2
C₂×#16863213	2
C₂×#16863210	2
#22561930	2
#22561926	2
#22560088	2
C₂×#364×C₆₇	2
#13260	67