Subgroups — #30764099

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863160	C₂	1	2	yes	maximal
3	C₂×#16863160	C₂	1	2	yes	maximal
4	C₂×#16863151	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	2	yes	—
8	#16863420	—	2	4	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
10	#16863160	—	2	4	no	—
11	#16863160	—	2	4	no	—
12	#16863151	C₄	1	2	yes	—
13	#16863151	C₄	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	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	—	2	2	no	—
23	C₂×#11761612	D₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
33	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	2	yes	—
50	#11761612	—	2	4	no	—
51	#11761612	—	2	4	no	—
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	4	2	no	—
83	M₁₆×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	—	4	1	no	—
88	C₂×C₈×C₆₇	—	4	1	no	—
89	C₂×C₈×C₆₇	—	4	1	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	—	2	2	no	—
96	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₆₇	#125	1	2	yes	—
99	C₂³×C₆₇	#125	1	1	yes	characteristic
100	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	4	8	no	—
107	C₂×C₄×C₆₇	#108	1	4	yes	—
108	C₂×C₄×C₆₇	#117	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#121	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#123	1	2	yes	—
119	C₂×C₄×C₆₇	#123	1	2	yes	—
120	C₂×C₄×C₆₇	#124	1	2	yes	—
121	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#125	1	2	yes	—
124	C₂×C₄×C₆₇	#126	1	2	yes	—
125	C₂×C₄×C₆₇	#126	1	2	yes	—
126	C₈×C₆₇	—	4	2	no	—
127	C₈×C₆₇	—	4	2	no	—
128	C₆₇⋊C₈	—	4	4	no	—
129	C₆₇⋊C₈	—	4	4	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	4	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	#382	1	1	yes	characteristic
136	C₂²×C₆₇	#385	1	1	yes	characteristic
137	C₂²×C₆₇	#389	1	1	yes	characteristic
138	C₂²×C₆₇	#398	1	2	yes	—
139	C₂²×C₆₇	#400	1	2	yes	—
140	C₂²×C₆₇	C₂×#108	1	2	yes	—
141	C₂²×C₆₇	#430	1	2	yes	—
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	#387	1	2	yes	—
145	C₄×C₆₇	#388	1	2	yes	—
146	C₄×C₆₇	#393	1	1	yes	characteristic
147	C₄×C₆₇	#394	1	1	yes	characteristic
148	C₄×C₆₇	#397	1	1	yes	characteristic
149	C₄×C₆₇	#400	1	1	yes	characteristic
150	#11269	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	#1442	1	1	yes	characteristic
154	C₂×C₆₇	#1837	1	1	yes	characteristic
155	C₂×C₆₇	#1838	1	1	yes	characteristic
156	C₂×C₆₇	#1855	1	2	yes	—
157	C₂×C₆₇	#1950	1	2	yes	—
158	C₂×#423	—	67	1	no	—
159	C₂×#341	—	67	2	no	—
160	C₂×#341	—	67	2	no	—
161	#1728	D₆₇	1	1	yes	characteristic
162	C₂×#336	—	67	1	no	—
163	C₆₇	#11269	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	67	2	no	—
165	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
166	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
167	C₂³×C₈	—	67	2	no	—
168	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
169	#423	—	134	4	no	—
170	C₂×C₄⋊C₈	—	134	2	no	—
171	#341	—	134	4	no	—
172	#341	—	134	4	no	—
173	#336	—	2	2	no	—
174	#336	—	67	2	no	—
175	#336	—	67	2	no	—
176	C₂³×C₄	Dic₁₃₄	1	2	yes	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	67	4	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	—	134	2	no	—
187	C₂×M₁₆	—	134	2	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
191	C₂×M₁₆	D₂₆₈	1	2	yes	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	67	4	no	—
195	C₂²×C₈	—	134	2	no	—
196	C₂²×C₈	—	134	2	no	—
197	C₂²×C₈	—	134	2	no	—
198	C₂²×C₈	—	134	2	no	—
199	C₂²×C₈	—	134	2	no	—
200	C₂²×C₈	—	134	2	no	—
201	C₂²×C₈	—	134	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
207	C₄⋊C₈	—	134	8	no	—
208	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
213	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₆₇	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
219	C₂²×C₄	#11761627	1	2	yes	—
220	C₂²×C₄	#11761627	1	2	yes	—
221	C₂²×C₄	#11761628	1	1	yes	characteristic
222	C₂²×C₄	#11761628	1	1	yes	characteristic
223	C₂²×C₄	#11761629	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	268	2	no	—
229	M₁₆	—	268	2	no	—
230	M₁₆	—	134	4	no	—
231	M₁₆	—	134	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	268	2	no	—
239	C₂×C₈	—	268	2	no	—
240	C₂×C₈	—	268	2	no	—
241	C₂×C₈	—	268	2	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	134	4	no	—
245	C₂×C₈	—	134	4	no	—
246	C₂×C₈	—	134	4	no	—
247	C₂×C₈	—	134	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
255	C₂³	#14524020	1	2	yes	—
256	C₂³	#14524034	1	2	yes	—
257	C₂³	#14524035	1	1	yes	characteristic
258	C₂³	#14524043	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#14523978	1	4	yes	—
266	C₂×C₄	#14524001	1	2	yes	—
267	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
268	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
269	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
270	C₂×C₄	#14524011	1	1	yes	characteristic
271	C₂×C₄	#14524017	1	1	yes	characteristic
272	C₂×C₄	#14524018	1	2	yes	—
273	C₂×C₄	#14524020	1	2	yes	—
274	C₂×C₄	#14524022	1	1	yes	characteristic
275	C₂×C₄	#14524024	1	1	yes	characteristic
276	C₂×C₄	#14524027	1	2	yes	—
277	C₂×C₄	#14524028	1	2	yes	—
278	C₂×C₄	#14524029	1	2	yes	—
279	C₂×C₄	#14524030	1	1	yes	characteristic
280	C₂×C₄	#14524032	1	1	yes	characteristic
281	C₂×C₄	#14524034	1	2	yes	—
282	C₂×C₄	#14524036	1	2	yes	—
283	C₂×C₄	#14524038	1	2	yes	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	268	4	no	—
287	C₈	—	268	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#16863261	1	1	yes	characteristic
294	C₂²	#16863268	1	1	yes	characteristic
295	C₂²	#16863294	1	1	yes	characteristic
296	C₂²	#16863328	1	2	yes	—
297	C₂²	#16863338	1	2	yes	—
298	C₂²	C₂×#14523978	1	2	yes	—
299	C₂²	#16863444	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#16863283	1	2	yes	—
303	C₄	#16863288	1	2	yes	—
304	C₄	#16863308	1	1	yes	characteristic
305	C₄	#16863313	1	1	yes	characteristic
306	C₄	#16863321	1	1	yes	characteristic
307	C₄	#16863335	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#22558465	1	1	yes	characteristic
311	C₂	#22560346	1	1	yes	characteristic
312	C₂	#22560349	1	1	yes	characteristic
313	C₂	#22560413	1	2	yes	—
314	C₂	#22560984	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
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	#11269
LCS	G ▷ C₂²×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
163. C₆₇	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
C₂×#16863160	2
C₂×#16863151	2
C₆₇×#1728	2
#11269	67