Subgroups — #30761580

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863167	C₂	1	2	yes	maximal
2	C₂×#16863156	C₂	1	2	yes	maximal
3	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863167	—	2	4	no	—
9	#16863156	C₄	1	8	yes	complemented
10	C₆₇⋊#338	—	2	2	no	—
11	#16863153	—	2	2	no	—
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₆₇	—	2	4	no	—
16	C₂×#97×C₆₇	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	C₂×C₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	2	no	—
32	C₂³×C₄×C₆₇	—	2	1	no	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
40	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
41	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#97×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	2	4	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²⋊Dic₆₇	—	2	4	no	—
62	C₂²⋊Dic₆₇	—	4	2	no	—
63	C₄×Dic₆₇	—	2	4	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₂⁴×C₆₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	C₄²	1	2	yes	—
94	C₄²×C₆₇	C₄²	1	2	yes	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂³×C₆₇	C₄×D₄	1	2	yes	—
98	C₂³×C₆₇	#124	1	1	yes	characteristic
99	C₂³×C₆₇	#125	1	1	yes	characteristic
100	C₂³×C₆₇	#126	1	2	yes	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	2	yes	—
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	#121	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	#123	1	2	yes	—
120	C₂×C₄×C₆₇	#123	1	2	yes	—
121	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#126	1	2	yes	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	—	8	4	no	—
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	—	2	4	no	—
128	C₆₇⋊C₈	—	4	4	no	—
129	C₂²×C₆₇	—	2	4	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	1	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₆₇	#382	1	1	yes	characteristic
136	C₂²×C₆₇	#383	1	1	yes	characteristic
137	C₂²×C₆₇	#387	1	2	yes	—
138	C₂²×C₆₇	#388	1	2	yes	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	4	2	no	—
141	C₄×C₆₇	#376	1	1	yes	characteristic
142	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₆₇	#380	1	1	yes	characteristic
144	C₄×C₆₇	#381	1	1	yes	characteristic
145	C₄×C₆₇	#387	1	2	yes	—
146	C₄×C₆₇	#388	1	2	yes	—
147	Dic₆₇	—	8	4	no	—
148	#10825	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	#1434	1	1	yes	characteristic
154	C₂×C₆₇	#1841	1	2	yes	—
155	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
156	C₂×#344	—	67	2	no	—
157	C₂×#339	—	67	2	no	—
158	C₂×#338	—	67	1	no	—
159	C₂×H(3,ℤ/4)	—	67	1	no	—
160	C₆₇	#10825	1	1	yes	characteristic, complemented, Sylow
161	C₂²×M₁₆	—	67	2	no	—
162	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
163	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	—	67	2	no	—
165	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
166	C₄×M₁₆	—	2	4	no	—
167	C₂×#97	Dic₆₇	1	2	yes	complemented
168	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
169	#344	—	134	4	no	—
170	#339	—	67	8	no	—
171	#338	—	134	2	no	—
172	H(3,ℤ/4)	—	134	2	no	—
173	C₂³×C₄	—	2	1	no	—
174	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	67	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
187	C₂×M₁₆	D₂₆₈	1	2	yes	—
188	C₂×M₁₆	C₂²⋊D₆₇	1	4	yes	—
189	C₂²×C₈	—	134	2	no	—
190	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
191	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
192	#117	—	67	4	no	—
193	#117	—	134	2	no	—
194	#117	—	134	2	no	—
195	#117	—	134	2	no	—
196	C₂×C₄⋊C₄	—	134	2	no	—
197	C₂×C₂²⋊C₄	—	134	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	134	2	no	—
201	C₂×C₄²	C₄×D₆₇	1	4	yes	—
202	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
204	#97	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	C₂⁴	#11761628	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	268	2	no	—
212	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
213	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
215	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
217	C₂²×C₄	#11761628	1	2	yes	—
218	C₂²×C₄	#11761628	1	1	yes	characteristic
219	C₂²×C₄	#11761628	1	1	yes	characteristic
220	C₂²×C₄	#11761628	1	2	yes	—
221	C₂²×C₄	#11761628	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	268	2	no	—
227	M₁₆	—	268	2	no	—
228	M₁₆	—	134	4	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	268	2	no	—
235	C₂×C₈	—	268	2	no	—
236	C₂×C₈	—	134	4	no	—
237	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
238	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
239	C₄⋊C₄	—	268	2	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₂²⋊C₄	—	268	2	no	—
242	C₂²⋊C₄	—	134	4	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	268	2	no	—
247	C₄²	—	134	4	no	—
248	C₄²	C₄×Dic₆₇	1	2	yes	—
249	C₄²	C₄×Dic₆₇	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	#14524002	1	1	yes	characteristic
252	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
253	C₂³	#14524031	1	1	yes	characteristic
254	C₂³	#14524035	1	1	yes	characteristic
255	C₂³	#14524037	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	268	4	no	—
262	C₂×C₄	—	536	4	no	—
263	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	#14524000	1	2	yes	—
267	C₂×C₄	#14524000	1	2	yes	—
268	C₂×C₄	#14524002	1	1	yes	characteristic
269	C₂×C₄	#14524002	1	1	yes	characteristic
270	C₂×C₄	#14524002	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
272	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
273	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
274	C₂×C₄	#14524017	1	2	yes	—
275	C₂×C₄	#14524022	1	2	yes	—
276	C₂×C₄	#14524026	1	2	yes	—
277	C₂×C₄	#14524026	1	2	yes	—
278	C₂×C₄	#14524031	1	1	yes	characteristic
279	C₂×C₄	#14524035	1	1	yes	characteristic
280	C₂×C₄	#14524037	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	268	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
290	C₂²	#16863262	1	1	yes	characteristic
291	C₂²	#16863264	1	1	yes	characteristic
292	C₂²	#16863280	1	2	yes	—
293	C₂²	#16863287	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	536	4	no	—
297	C₄	#16863242	1	1	yes	characteristic
298	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
299	C₄	#16863256	1	1	yes	characteristic
300	C₄	#16863259	1	1	yes	characteristic
301	C₄	#16863280	1	2	yes	—
302	C₄	#16863287	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	#22558436	1	1	yes	characteristic
308	C₂	#22560358	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#383
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#10825
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. #16863156	C₄	semidirect
16. C₂×#97×C₆₇	C₄	semidirect
17. C₂×C₄×C₈×C₆₇	C₄	semidirect
160. C₆₇	#10825	semidirect
167. C₂×#97	Dic₆₇	semidirect
168. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863167	2
C₂×#16863156	2
C₂×C₆₇⋊#338	2
C₂×#16863153	2
C₂×C₄×M₁₆×C₆₇	2
#10825	67