Subgroups — #30763766

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568773	C₂	1	1	yes	maximal, characteristic
2	#22559942	C₂	1	1	yes	maximal, characteristic
3	#22559918	C₂	1	2	yes	maximal
4	C₂×#16863156	C₂	1	2	yes	maximal
5	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863423	—	2	4	no	—
9	#16863420	—	2	2	no	—
10	#16863163	—	2	2	no	—
11	#16863156	—	2	4	no	—
12	#16863155	—	2	4	no	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₄	1	2	yes	complemented
17	#382×C₆₇	C₄	1	2	yes	complemented
18	#380×C₆₇	—	2	2	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	D₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	#117×C₆₇	—	2	2	no	—
38	#117×C₆₇	—	2	2	no	—
39	#117×C₆₇	—	2	2	no	—
40	#117×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
55	#95×C₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	#11761612	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	C₂²⋊Dic₆₇	—	2	4	no	—
67	C₂²⋊Dic₆₇	—	4	2	no	—
68	C₄×Dic₆₇	—	2	4	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	4	8	no	—
91	C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	—	4	1	no	—
101	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
103	C₂³×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₆₇	#121	1	2	yes	—
106	C₂³×C₆₇	#121	1	1	yes	characteristic
107	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#123	1	2	yes	—
131	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#125	1	2	yes	—
133	C₂×C₄×C₆₇	#125	1	2	yes	—
134	C₂×C₄×C₆₇	#126	1	2	yes	—
135	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
137	C₂×Dic₆₇	—	4	4	no	—
138	C₂×Dic₆₇	—	8	4	no	—
139	C₆₇⋊C₈	—	4	4	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	4	1	no	—
143	C₂²×C₆₇	—	4	1	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	#380	1	1	yes	characteristic
146	C₂²×C₆₇	#386	1	1	yes	characteristic
147	C₂²×C₆₇	#391	1	1	yes	characteristic
148	C₂²×C₆₇	#396	1	2	yes	—
149	C₂²×C₆₇	#398	1	2	yes	—
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	4	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#387	1	2	yes	—
155	C₄×C₆₇	#389	1	2	yes	—
156	C₄×C₆₇	#394	1	1	yes	characteristic
157	C₄×C₆₇	#395	1	1	yes	characteristic
158	C₄×C₆₇	#398	1	1	yes	characteristic
159	C₄×C₆₇	#400	1	1	yes	characteristic
160	Dic₆₇	—	8	4	no	—
161	#11212	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	2	2	no	—
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	#1441	1	1	yes	characteristic
167	C₂×C₆₇	#1884	1	2	yes	—
168	#2905	—	67	1	no	—
169	#2285	D₆₇	1	1	yes	characteristic
170	#1740	—	67	1	no	—
171	#1733	—	67	2	no	—
172	C₂×#339	—	67	2	no	—
173	C₆₇	#11212	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	67	2	no	—
175	C₂×#117	—	67	2	no	—
176	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
177	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
178	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
179	#424	—	134	4	no	—
180	#423	—	134	2	no	—
181	#384	Dic₆₇	1	2	yes	complemented
182	#382	Dic₆₇	1	2	yes	complemented
183	#380	—	2	2	no	—
184	C₄×C₄⋊C₄	—	2	2	no	—
185	#342	—	134	2	no	—
186	#339	—	134	4	no	—
187	#339	—	134	4	no	—
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
190	C₂×M₁₆	—	67	4	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	2	no	—
196	C₂²×C₈	—	134	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	67	4	no	—
202	#117	—	134	2	no	—
203	#117	—	134	2	no	—
204	#117	—	134	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	134	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	134	2	no	—
211	C₂×C₂²⋊C₄	C₂²⋊D₆₇	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	134	2	no	—
217	C₂×C₄²	Dic₁₃₄	1	2	yes	—
218	C₂×C₄²	D₂₆₈	1	2	yes	—
219	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
222	C₄⋊C₈	—	134	8	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	268	2	no	—
232	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₆₇	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
237	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
238	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
239	C₂²×C₄	#11761627	1	2	yes	—
240	C₂²×C₄	#11761628	1	1	yes	characteristic
241	C₂²×C₄	#11761628	1	1	yes	characteristic
242	C₂²×C₄	#11761629	1	2	yes	—
243	M₁₆	—	268	2	no	—
244	M₁₆	—	268	2	no	—
245	M₁₆	—	134	4	no	—
246	M₁₆	—	134	4	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	—	268	2	no	—
249	C₂×C₈	—	134	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄⋊C₄	—	268	2	no	—
254	C₄⋊C₄	—	134	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	268	2	no	—
258	C₂²⋊C₄	—	134	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	268	2	no	—
268	C₄²	—	134	4	no	—
269	C₄²	C₄⋊Dic₆₇	1	2	yes	—
270	C₄²	C₄⋊Dic₆₇	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
273	C₂³	#14524015	1	2	yes	—
274	C₂³	#14524017	1	1	yes	characteristic
275	C₂³	#14524019	1	2	yes	—
276	C₂³	#14524022	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	268	4	no	—
288	C₂×C₄	—	536	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
290	C₂×C₄	#14524002	1	1	yes	characteristic
291	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
292	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
293	C₂×C₄	#14524011	1	1	yes	characteristic
294	C₂×C₄	#14524017	1	1	yes	characteristic
295	C₂×C₄	#14524018	1	2	yes	—
296	C₂×C₄	#14524021	1	2	yes	—
297	C₂×C₄	#14524022	1	1	yes	characteristic
298	C₂×C₄	#14524023	1	2	yes	—
299	C₂×C₄	#14524024	1	1	yes	characteristic
300	C₂×C₄	#14524028	1	2	yes	—
301	C₂×C₄	#14524032	1	1	yes	characteristic
302	C₂×C₄	#14524033	1	2	yes	—
303	C₂×C₄	#14524034	1	2	yes	—
304	C₂×C₄	#14524038	1	2	yes	—
305	C₂×C₄	#14524041	1	1	yes	characteristic
306	C₂×C₄	#14524042	1	2	yes	—
307	C₈	—	268	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#16863256	1	1	yes	characteristic
314	C₂²	#16863276	1	1	yes	characteristic
315	C₂²	#16863303	1	1	yes	characteristic
316	C₂²	#16863318	1	2	yes	—
317	C₂²	#16863329	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	536	4	no	—
323	C₄	#16863282	1	2	yes	—
324	C₄	#16863295	1	2	yes	—
325	C₄	#16863313	1	1	yes	characteristic
326	C₄	#16863316	1	1	yes	characteristic
327	C₄	#16863326	1	1	yes	characteristic
328	C₄	#16863335	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#22558462	1	1	yes	characteristic
334	C₂	#22560593	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524011
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#2285	C₂
socle	C₂²×C₆₇	#386
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#2285	D₆₇
hypocenter	C₆₇	#11212
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
16. #384×C₆₇	C₄	semidirect
17. #382×C₆₇	C₄	semidirect
173. C₆₇	#11212	semidirect
181. #384	Dic₆₇	semidirect
182. #382	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568773	2
#22559942	2
#22559918	2
C₂×#16863156	2
C₆₇×#2285	2
#11212	67