Subgroups — #30761646

Order 17152.

894 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×#16863420	C₂	1	1	yes	maximal, characteristic
2	#22562616	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863156	C₂	1	4	yes	maximal
4	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	2	yes	—
6	C₂×#14524002	C₂²	1	2	yes	—
7	#16863420	—	2	2	no	—
8	#16863420	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
10	#16863266	C₄	1	2	yes	complemented
11	#16863262	C₄	1	2	yes	complemented
12	#16863256	—	2	2	no	—
13	C₄×C₄⋊Dic₆₇	—	2	2	no	—
14	#16863156	—	2	8	no	—
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#376×C₆₇	—	2	2	no	—
18	#376×C₆₇	C₄	1	4	yes	complemented
19	C₂×#95×C₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	4	no	—
22	C₂×#11761612	—	2	4	no	—
23	C₂×#11761612	C₂×C₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	#14524002	—	2	4	no	—
26	#14524002	—	2	4	no	—
27	C₂×C₄⋊Dic₆₇	—	2	4	no	—
28	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
32	C₆₇⋊#95	—	2	4	no	—
33	C₂³×C₄×C₆₇	—	2	2	no	—
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	D₄	1	4	yes	—
45	C₂×C₄²×C₆₇	D₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
49	#95×C₆₇	—	2	8	no	—
50	C₂²×Dic₆₇	—	2	4	no	—
51	#11761612	—	2	8	no	—
52	#11761612	—	4	4	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₄⋊Dic₆₇	—	2	8	no	—
57	C₄⋊Dic₆₇	—	4	8	no	—
58	C₂²⋊Dic₆₇	—	4	4	no	—
59	C₄×Dic₆₇	—	4	2	no	—
60	C₄×Dic₆₇	—	4	2	no	—
61	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	4	4	no	—
66	C₂²×C₄×C₆₇	—	4	8	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₄²×C₆₇	—	2	4	no	—
82	C₄²×C₆₇	—	2	4	no	—
83	C₄²×C₆₇	—	2	4	no	—
84	C₄²×C₆₇	C₄²	1	2	yes	—
85	C₄²×C₆₇	C₄²	1	2	yes	—
86	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	8	yes	complemented
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₇	#121	1	2	yes	—
96	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	8	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	8	no	—
103	C₂×C₄×C₆₇	—	4	8	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	#95	1	4	yes	—
106	C₂×C₄×C₆₇	#95	1	4	yes	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#117	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×Q₈	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	#124	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#125	1	2	yes	—
122	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×Dic₆₇	—	4	4	no	—
126	C₂×Dic₆₇	—	4	8	no	—
127	C₆₇⋊C₈	—	4	4	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
133	C₂²×C₆₇	#380	1	2	yes	—
134	C₂²×C₆₇	#389	1	2	yes	—
135	C₂²×C₆₇	#390	1	1	yes	characteristic
136	C₂²×C₆₇	#391	1	1	yes	characteristic
137	C₂²×C₆₇	#421	1	4	yes	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	4	8	no	—
141	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
142	C₄×C₆₇	#382	1	1	yes	characteristic
143	C₄×C₆₇	#384	1	1	yes	characteristic
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	#390	1	1	yes	characteristic
146	C₄×C₆₇	#391	1	1	yes	characteristic
147	Dic₆₇	—	8	4	no	—
148	#10840	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
152	C₂×C₆₇	#1733	1	2	yes	—
153	C₂×C₆₇	#1963	1	2	yes	—
154	C₂×C₆₇	#1995	1	2	yes	—
155	C₂×#423	—	67	1	no	—
156	#2285	—	67	1	no	—
157	C₂×#376	D₆₇	1	1	yes	characteristic
158	C₂×#339	—	67	4	no	—
159	C₆₇	#10840	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	2	no	—
161	C₂×#117	—	67	2	no	—
162	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
163	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
164	#423	—	67	4	no	—
165	#423	—	134	2	no	—
166	C₂×C₄⋊C₈	—	134	2	no	—
167	#384	—	67	2	no	—
168	#382	—	67	2	no	—
169	#380	—	134	2	no	—
170	C₄×C₄⋊C₄	—	134	2	no	—
171	#376	—	2	2	no	—
172	#376	Dic₆₇	1	4	yes	complemented
173	C₂×#95	—	2	2	no	—
174	#339	—	134	8	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	134	4	no	—
180	C₂×M₁₆	—	134	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂²×C₈	—	134	2	no	—
183	#117	—	134	4	no	—
184	#117	—	134	4	no	—
185	C₂×C₄⋊C₄	—	67	4	no	—
186	C₂×C₄⋊C₄	—	134	4	no	—
187	C₂×C₂²⋊C₄	—	134	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	134	2	no	—
192	C₂×C₄²	Dic₁₃₄	1	4	yes	—
193	C₂×C₄²	Dic₁₃₄	1	2	yes	—
194	C₂×C₄²	C₄×D₆₇	1	2	yes	—
195	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
197	C₂×C₄²	D₂₆₈	1	4	yes	—
198	C₂×C₄²	D₂₆₈	1	2	yes	—
199	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
200	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
202	C₄⋊C₈	—	134	8	no	—
203	#95	—	2	8	no	—
204	#95	—	134	4	no	—
205	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	134	4	no	—
212	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
213	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
215	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
216	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
221	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
224	C₂²×C₄	#11761628	1	2	yes	—
225	C₂²×C₄	#11761628	1	2	yes	—
226	M₁₆	—	268	4	no	—
227	M₁₆	—	134	8	no	—
228	C₂×C₈	—	268	2	no	—
229	C₂×C₈	—	268	2	no	—
230	C₂×C₈	—	134	4	no	—
231	C₄⋊C₄	—	268	8	no	—
232	C₄⋊C₄	—	134	8	no	—
233	C₂²⋊C₄	—	268	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	268	2	no	—
238	C₄²	—	268	2	no	—
239	C₄²	C₄×Dic₆₇	1	2	yes	—
240	C₄²	C₄×Dic₆₇	1	2	yes	—
241	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
242	C₄²	Dic₆₇⋊C₄	1	2	yes	—
243	C₄²	Dic₆₇⋊C₄	1	2	yes	—
244	C₄²	C₄⋊Dic₆₇	1	4	yes	—
245	C₄²	C₄⋊Dic₆₇	1	8	yes	complemented
246	C₂³	—	2	4	no	—
247	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
250	C₂³	#14524017	1	2	yes	—
251	C₂³	#14524022	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	268	4	no	—
261	C₂×C₄	—	268	8	no	—
262	C₂×C₄	C₆₇⋊#95	1	4	yes	—
263	C₂×C₄	C₆₇⋊#95	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
269	C₂×C₄	#14524002	1	2	yes	—
270	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
271	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
272	C₂×C₄	C₄×Dic₁₃₄	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	#14524031	1	1	yes	characteristic
277	C₂×C₄	#14524031	1	1	yes	characteristic
278	C₂×C₄	#14524035	1	2	yes	—
279	C₂×C₄	#14524040	1	1	yes	characteristic
280	C₂×C₄	#14524043	1	2	yes	—
281	C₂×C₄	#14524043	1	1	yes	characteristic
282	C₈	—	268	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
288	C₂²	#16863256	1	2	yes	—
289	C₂²	#16863294	1	2	yes	—
290	C₂²	#16863298	1	1	yes	characteristic
291	C₂²	#16863302	1	1	yes	characteristic
292	C₂²	#16863414	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	536	4	no	—
297	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
298	C₄	#16863262	1	1	yes	characteristic
299	C₄	#16863266	1	1	yes	characteristic
300	C₄	#16863294	1	2	yes	—
301	C₄	#16863298	1	1	yes	characteristic
302	C₄	#16863302	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#22558439	1	1	yes	characteristic
306	C₂	#22559919	1	2	yes	—
307	C₂	#22561041	1	2	yes	—
308	C₂	#22561190	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#376×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#376	D₆₇
hypocenter	C₆₇	#10840
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
8. #16863420	C₄	semidirect
10. #16863266	C₄	semidirect
11. #16863262	C₄	semidirect
18. #376×C₆₇	C₄	semidirect
90. C₄²×C₆₇	C₄⋊C₄	semidirect
159. C₆₇	#10840	semidirect
172. #376	Dic₆₇	semidirect
245. C₄²	C₄⋊Dic₆₇	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
#22562616	2
C₂×#16863156	2
C₂×#376×C₆₇	2
#10840	67