Subgroups — #30761633

Order 17152.

850 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562616	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524002	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863153	C₂	1	4	yes	maximal
4	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	C₂²	1	2	yes	—
6	C₂×#14524002	C₂²	1	2	yes	—
7	#16863266	—	2	1	no	—
8	#16863262	—	2	1	no	—
9	#16863256	—	2	2	no	—
10	C₄×C₄⋊Dic₆₇	—	2	2	no	—
11	C₄×C₄⋊Dic₆₇	C₄	1	4	yes	complemented
12	C₄×C₂²⋊Dic₆₇	—	2	2	no	—
13	#16863242	C₄	1	2	yes	complemented
14	C₄²×Dic₆₇	C₄	1	2	yes	complemented
15	#16863153	—	2	8	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#423×C₆₇	—	2	2	no	—
19	#423×C₆₇	—	2	1	no	—
20	#423×C₆₇	—	2	1	no	—
21	C₂×C₄⋊C₈×C₆₇	C₄	1	4	yes	complemented
22	#14524002	—	2	4	no	—
23	#14524002	—	2	2	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	4	no	—
26	#14524002	D₄	1	4	yes	—
27	#14524002	Q₈	1	4	yes	—
28	C₂×C₄⋊Dic₆₇	—	2	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	—	2	4	no	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
36	C₆₇⋊#95	—	2	4	no	—
37	C₆₇⋊#95	—	2	4	no	—
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	—	2	4	no	—
55	C₄⋊C₈×C₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	2	4	no	—
57	C₂²×Dic₆₇	—	2	4	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	8	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²⋊Dic₆₇	—	2	4	no	—
64	C₂²⋊Dic₆₇	—	2	4	no	—
65	C₂²⋊Dic₆₇	—	4	4	no	—
66	C₄×Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	2	8	no	—
68	C₄×Dic₆₇	—	4	8	no	—
69	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	complemented
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	M₁₆×C₆₇	—	4	4	no	—
89	M₁₆×C₆₇	—	4	2	no	—
90	M₁₆×C₆₇	—	4	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	8	no	—
93	C₄²×C₆₇	C₄²	1	2	yes	—
94	C₄²×C₆₇	C₄²	1	2	yes	—
95	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₆₇	#121	1	2	yes	—
108	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	#95	1	2	yes	—
117	C₂×C₄×C₆₇	#95	1	2	yes	—
118	C₂×C₄×C₆₇	#95	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	#121	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₆₇	#125	1	2	yes	—
134	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
137	C₂×Dic₆₇	—	2	8	no	—
138	C₂×Dic₆₇	—	4	8	no	—
139	C₂×Dic₆₇	—	4	8	no	—
140	C₂×Dic₆₇	—	4	4	no	—
141	C₈×C₆₇	—	4	4	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₆₇	#380	1	2	yes	—
149	C₂²×C₆₇	#389	1	2	yes	—
150	C₂²×C₆₇	#390	1	1	yes	characteristic
151	C₂²×C₆₇	#391	1	1	yes	characteristic
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	H(3,ℤ/4)	1	4	yes	complemented
156	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
157	C₄×C₆₇	#382	1	1	yes	characteristic
158	C₄×C₆₇	#384	1	1	yes	characteristic
159	C₄×C₆₇	#389	1	2	yes	—
160	C₄×C₆₇	#390	1	1	yes	characteristic
161	C₄×C₆₇	#391	1	1	yes	characteristic
162	Dic₆₇	—	4	8	no	—
163	Dic₆₇	—	8	4	no	—
164	#10836	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	—	2	1	no	—
167	C₂×C₆₇	—	2	2	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₆₇	C₂×H(3,ℤ/4)	1	2	yes	—
171	C₂×#423	D₆₇	1	1	yes	characteristic
172	#2285	—	67	1	no	—
173	C₄×#117	—	67	1	no	—
174	C₂×H(3,ℤ/4)	—	67	4	no	—
175	C₆₇	#10836	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
177	C₂×#117	—	67	2	no	—
178	C₂×#117	—	67	2	no	—
179	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
180	#423	—	2	2	no	—
181	#423	—	2	1	no	—
182	#423	—	2	1	no	—
183	C₂×C₄⋊C₈	Dic₆₇	1	4	yes	complemented
184	#384	—	134	1	no	—
185	#382	—	134	1	no	—
186	#380	—	134	2	no	—
187	C₄×C₄⋊C₄	—	67	4	no	—
188	C₄×C₄⋊C₄	—	134	2	no	—
189	C₄×C₂²⋊C₄	—	134	2	no	—
190	#376	—	67	2	no	—
191	C₄³	—	67	2	no	—
192	H(3,ℤ/4)	—	134	8	no	—
193	C₂³×C₄	D₂₆₈	1	2	yes	—
194	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
200	#117	—	67	4	no	—
201	#117	—	67	4	no	—
202	#117	—	134	4	no	—
203	#117	—	134	4	no	—
204	#117	—	134	2	no	—
205	#117	—	134	2	no	—
206	C₂×C₄⋊C₄	—	134	4	no	—
207	C₂×C₄⋊C₄	—	134	2	no	—
208	C₂×C₄⋊C₄	—	134	2	no	—
209	C₂×C₂²⋊C₄	—	134	2	no	—
210	C₂×C₂²⋊C₄	—	134	2	no	—
211	C₂×C₄²	—	67	4	no	—
212	C₂×C₄²	—	67	4	no	—
213	C₂×C₄²	—	134	4	no	—
214	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
215	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₆₇	1	2	yes	—
219	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	#95	—	134	4	no	—
226	#95	—	134	4	no	—
227	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	134	4	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
234	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₆₇	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
244	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
245	C₂²×C₄	#11761627	1	2	yes	—
246	C₂²×C₄	#11761629	1	2	yes	—
247	M₁₆	—	4	4	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	8	no	—
252	C₄⋊C₄	—	268	8	no	—
253	C₄⋊C₄	—	268	2	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	134	4	no	—
256	C₄⋊C₄	—	134	4	no	—
257	C₂²⋊C₄	—	268	4	no	—
258	C₂²⋊C₄	—	134	4	no	—
259	C₂²⋊C₄	—	134	4	no	—
260	C₄²	—	268	8	no	—
261	C₄²	—	67	8	no	—
262	C₄²	—	134	8	no	—
263	C₄²	—	134	4	no	—
264	C₄²	C₄×Dic₆₇	1	2	yes	—
265	C₄²	C₄×Dic₆₇	1	2	yes	—
266	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
267	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
268	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
269	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
270	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₄²	Dic₆₇⋊C₄	1	2	yes	—
272	C₄²	Dic₆₇⋊C₄	1	2	yes	—
273	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
274	C₂³	—	4	2	no	—
275	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
276	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
277	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₂³	#14524018	1	2	yes	—
279	C₂³	#14524023	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	268	8	no	—
287	C₂×C₄	—	268	8	no	—
288	C₂×C₄	—	268	4	no	—
289	C₂×C₄	—	134	8	no	—
290	C₂×C₄	C₆₇⋊#95	1	2	yes	—
291	C₂×C₄	C₆₇⋊#95	1	2	yes	—
292	C₂×C₄	C₆₇⋊#95	1	2	yes	—
293	C₂×C₄	C₆₇⋊#95	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
296	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
297	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#14524001	1	2	yes	—
300	C₂×C₄	#14524005	1	2	yes	—
301	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
302	C₂×C₄	#14524008	1	2	yes	—
303	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
304	C₂×C₄	#14524014	1	2	yes	—
305	C₂×C₄	#14524019	1	2	yes	—
306	C₂×C₄	#14524030	1	1	yes	characteristic
307	C₂×C₄	#14524032	1	1	yes	characteristic
308	C₂×C₄	#14524034	1	2	yes	—
309	C₂×C₄	#14524039	1	1	yes	characteristic
310	C₂×C₄	#14524041	1	1	yes	characteristic
311	C₂×C₄	#14524042	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
319	C₂²	#16863253	1	2	yes	—
320	C₂²	#16863295	1	2	yes	—
321	C₂²	#16863297	1	1	yes	characteristic
322	C₂²	#16863301	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	268	8	no	—
327	C₄	—	536	4	no	—
328	C₄	#16863153	1	4	yes	complemented
329	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
330	C₄	#16863261	1	1	yes	characteristic
331	C₄	#16863265	1	1	yes	characteristic
332	C₄	#16863292	1	2	yes	—
333	C₄	#16863299	1	1	yes	characteristic
334	C₄	#16863303	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	#22558438	1	1	yes	characteristic
340	C₂	C₂×#16863153	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	#10836
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
11. C₄×C₄⋊Dic₆₇	C₄	semidirect
13. #16863242	C₄	semidirect
14. C₄²×Dic₆₇	C₄	semidirect
21. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
69. C₄×Dic₆₇	C₄⋊C₄	semidirect
155. C₄×C₆₇	H(3,ℤ/4)	semidirect
175. C₆₇	#10836	semidirect
183. C₂×C₄⋊C₈	Dic₆₇	semidirect
328. C₄	#16863153	semidirect

Maximal subgroups

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

subgroup	index
#22562616	2
C₄×#14524002	2
C₂×#16863153	2
C₂×#423×C₆₇	2
#10836	67