Subgroups — #30761566

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863160	C₂	1	2	yes	maximal
3	C₂×#16863160	C₂	1	2	yes	maximal
4	#22559903	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	—	2	4	no	—
10	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
11	#16863160	—	2	4	no	—
12	#16863160	C₄	1	8	yes	complemented
13	#16863151	—	2	2	no	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	2	no	—
17	C₂×#11761612	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
34	C₄×C₆₇⋊C₈	—	2	8	no	—
35	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	#11761612	—	2	4	no	—
49	#11761612	—	2	4	no	—
50	#11761612	—	4	2	no	—
51	#11761612	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
64	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	—
65	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	#41	1	2	yes	—
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	M₁₆×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	4	2	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₄²×C₆₇	—	2	2	no	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	4	no	—
89	C₂³×C₆₇	—	2	4	no	—
90	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
92	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₆₇	#125	1	1	yes	characteristic
94	C₂³×C₆₇	#125	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	16	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	#108	1	8	yes	—
99	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#117	1	2	yes	—
103	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	#121	1	2	yes	—
111	C₂×C₄×C₆₇	#123	1	2	yes	—
112	C₂×C₄×C₆₇	#123	1	2	yes	—
113	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#126	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₈×C₆₇	—	4	4	no	—
118	C₆₇⋊C₈	—	2	4	no	—
119	C₆₇⋊C₈	—	2	4	no	—
120	C₆₇⋊C₈	—	4	4	no	—
121	C₂²×C₆₇	—	2	4	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	4	no	—
125	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₇	#380	1	1	yes	characteristic
127	C₂²×C₆₇	#382	1	1	yes	characteristic
128	C₂²×C₆₇	#387	1	2	yes	—
129	C₂²×C₆₇	#387	1	2	yes	—
130	C₂²×C₆₇	C₂×#108	1	4	yes	—
131	C₄×C₆₇	—	2	8	no	—
132	C₄×C₆₇	#376	1	1	yes	characteristic
133	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₆₇	#381	1	1	yes	characteristic
135	C₄×C₆₇	#383	1	1	yes	characteristic
136	C₄×C₆₇	#388	1	2	yes	—
137	C₄×C₆₇	#388	1	2	yes	—
138	#10822	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	2	no	—
140	C₂×C₆₇	—	2	2	no	—
141	C₂×C₆₇	#1434	1	1	yes	characteristic
142	C₂×C₆₇	C₄×#108	1	2	yes	—
143	C₂×C₆₇	#1810	1	2	yes	—
144	C₂×C₆₇	#1837	1	2	yes	—
145	C₂²×C₄×C₈	—	67	1	no	—
146	C₂×#341	—	67	2	no	—
147	C₂×#341	—	67	2	no	—
148	#1728	—	67	1	no	—
149	#1728	D₆₇	1	1	yes	characteristic
150	C₆₇	#10822	1	1	yes	characteristic, complemented, Sylow
151	C₂²×M₁₆	—	67	2	no	—
152	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
153	C₂³×C₈	—	67	1	no	—
154	C₂³×C₈	—	67	1	no	—
155	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
156	C₂×C₄×C₈	—	67	4	no	—
157	C₂×C₄×C₈	—	134	4	no	—
158	#341	—	67	8	no	—
159	#341	—	134	4	no	—
160	#336	—	2	2	no	—
161	#336	—	134	2	no	—
162	C₂³×C₄	Dic₁₃₄	1	2	yes	—
163	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	67	4	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂×M₁₆	—	134	2	no	—
172	C₂×M₁₆	—	134	2	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	67	4	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	—	67	2	no	—
180	C₂²×C₈	—	67	4	no	—
181	C₂²×C₈	—	134	2	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	—	134	1	no	—
184	C₂²×C₈	—	134	1	no	—
185	C₂²×C₈	—	134	2	no	—
186	C₂²×C₈	—	134	1	no	—
187	C₂²×C₈	—	134	1	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₄×D₆₇	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
192	C₄×C₈	—	134	8	no	—
193	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
197	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
199	C₂²×C₄	D₄×D₆₇	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂²×C₄	#11761627	1	2	yes	—
202	C₂²×C₄	#11761627	1	1	yes	characteristic
203	C₂²×C₄	#11761627	1	1	yes	characteristic
204	C₂²×C₄	#11761628	1	2	yes	—
205	C₂²×C₄	#11761629	1	1	yes	characteristic
206	C₂²×C₄	#11761629	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	268	2	no	—
212	M₁₆	—	268	2	no	—
213	M₁₆	—	134	4	no	—
214	M₁₆	—	134	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	268	2	no	—
220	C₂×C₈	—	67	4	no	—
221	C₂×C₈	—	67	4	no	—
222	C₂×C₈	—	134	4	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	—	134	2	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	—	134	4	no	—
228	C₂×C₈	—	134	4	no	—
229	C₂×C₈	—	134	4	no	—
230	C₂×C₈	—	134	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
236	C₂³	C₄×Dic₁₃₄	1	2	yes	—
237	C₂³	#14524020	1	2	yes	—
238	C₂³	#14524034	1	1	yes	characteristic
239	C₂³	#14524034	1	1	yes	characteristic
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	#14523978	1	8	yes	—
244	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₄	#14523999	1	1	yes	characteristic
247	C₂×C₄	#14523999	1	2	yes	—
248	C₂×C₄	#14524000	1	1	yes	characteristic
249	C₂×C₄	#14524001	1	1	yes	characteristic
250	C₂×C₄	#14524001	1	2	yes	—
251	C₂×C₄	#14524001	1	1	yes	characteristic
252	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
254	C₂×C₄	#14524005	1	2	yes	—
255	C₂×C₄	#14524015	1	2	yes	—
256	C₂×C₄	#14524025	1	2	yes	—
257	C₂×C₄	#14524028	1	2	yes	—
258	C₂×C₄	#14524030	1	1	yes	characteristic
259	C₂×C₄	#14524032	1	1	yes	characteristic
260	C₂×C₄	#14524036	1	2	yes	—
261	C₂×C₄	#14524038	1	2	yes	—
262	C₈	—	4	4	no	—
263	C₈	—	268	4	no	—
264	C₈	—	134	4	no	—
265	C₈	—	134	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₂²	#16863254	1	1	yes	characteristic
272	C₂²	#16863261	1	1	yes	characteristic
273	C₂²	#16863279	1	2	yes	—
274	C₂²	#16863283	1	2	yes	—
275	C₂²	C₂×#14523978	1	4	yes	—
276	C₄	—	2	8	no	—
277	C₄	#16863241	1	1	yes	characteristic
278	C₄	#16863243	1	1	yes	characteristic
279	C₄	#16863258	1	1	yes	characteristic
280	C₄	#16863263	1	1	yes	characteristic
281	C₄	#16863286	1	2	yes	—
282	C₄	#16863290	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	#22558435	1	1	yes	characteristic
286	C₂	C₄×#14523978	1	2	yes	—
287	C₂	#22560245	1	2	yes	—
288	C₂	#22560346	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1728	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1728	D₆₇
hypocenter	C₆₇	#10822
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
12. #16863160	C₄	semidirect
150. C₆₇	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₆₇⋊C₈	2
C₂×#16863160	2
#22559903	2
C₆₇×#1728	2
#10822	67