Subgroups — #30761539

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559948	C₂	1	1	yes	maximal, characteristic
2	#22559942	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863156	C₂	1	2	yes	maximal
4	C₂×#16863152	C₂	1	2	yes	maximal
5	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863165	—	2	2	no	—
9	#16863163	—	2	2	no	—
10	#16863156	—	2	4	no	—
11	#16863152	C₄	1	8	yes	complemented
12	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
17	#376×C₆₇	C₄	1	2	yes	complemented
18	C₄³×C₆₇	C₄	1	2	yes	complemented
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	—	2	2	no	—
24	C₂×#11761612	C₂×C₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄×Dic₆₇	—	2	2	no	—
33	C₂³×C₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	#117×C₆₇	—	2	2	no	—
36	#117×C₆₇	—	2	2	no	—
37	#117×C₆₇	—	2	2	no	—
38	#117×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	—	2	1	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	—	2	4	no	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
53	#95×C₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	#11761612	—	2	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₄⋊Dic₆₇	—	2	4	no	—
63	C₄⋊Dic₆₇	—	4	2	no	—
64	C₂²⋊Dic₆₇	—	2	4	no	—
65	C₂²⋊Dic₆₇	—	4	2	no	—
66	C₄×Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	4	2	no	—
68	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₂²⋊C₄×C₆₇	—	2	4	no	—
89	C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	4	8	no	—
98	C₄²×C₆₇	C₄²	1	4	yes	—
99	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
100	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₇	C₄×D₄	1	2	yes	—
104	C₂³×C₆₇	#121	1	1	yes	characteristic
105	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	#123	1	2	yes	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	4	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	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#117	1	2	yes	—
120	C₂×C₄×C₆₇	#117	1	2	yes	—
121	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
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	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#123	1	2	yes	—
131	C₂×C₄×C₆₇	#124	1	2	yes	—
132	C₂×C₄×C₆₇	#125	1	2	yes	—
133	C₂×C₄×C₆₇	#126	1	2	yes	—
134	C₂×C₄×C₆₇	#126	1	2	yes	—
135	C₂×Dic₆₇	—	4	4	no	—
136	C₂×Dic₆₇	—	8	4	no	—
137	C₆₇⋊C₈	—	4	4	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	4	1	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₆₇	#380	1	1	yes	characteristic
145	C₂²×C₆₇	#381	1	1	yes	characteristic
146	C₂²×C₆₇	#387	1	2	yes	—
147	C₂²×C₆₇	#388	1	2	yes	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	4	4	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	4	4	no	—
152	C₄×C₆₇	#376	1	1	yes	characteristic
153	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₆₇	#382	1	1	yes	characteristic
155	C₄×C₆₇	#383	1	1	yes	characteristic
156	C₄×C₆₇	#387	1	2	yes	—
157	C₄×C₆₇	#388	1	2	yes	—
158	Dic₆₇	—	8	4	no	—
159	#10816	—	67	1	no	maximal, Sylow
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	#1434	1	1	yes	characteristic
165	C₂×C₆₇	#1827	1	2	yes	—
166	C₄×#117	D₆₇	1	1	yes	characteristic
167	#1742	—	67	1	no	—
168	#1740	—	67	1	no	—
169	C₂×#339	—	67	2	no	—
170	C₂×H(3,ℤ/4)	—	67	2	no	—
171	C₆₇	#10816	1	1	yes	characteristic, complemented, Sylow
172	C₂²×M₁₆	—	67	2	no	—
173	C₂×#117	—	67	2	no	—
174	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
175	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
176	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₄×C₂²⋊C₄	—	2	2	no	—
179	#376	Dic₆₇	1	2	yes	complemented
180	C₄³	Dic₆₇	1	2	yes	complemented
181	#343	—	134	2	no	—
182	#342	—	134	2	no	—
183	#339	—	134	4	no	—
184	H(3,ℤ/4)	—	67	8	no	—
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
187	C₂×M₁₆	—	67	4	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂×M₁₆	—	134	2	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂²×C₈	—	134	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	67	4	no	—
198	#117	—	134	2	no	—
199	#117	—	134	2	no	—
200	#117	—	134	2	no	—
201	#117	C₄×D₆₇	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	134	2	no	—
207	C₂×C₂²⋊C₄	C₂²⋊D₆₇	1	2	yes	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	—	134	2	no	—
213	C₂×C₄²	Dic₁₃₄	1	2	yes	—
214	C₂×C₄²	D₂₆₈	1	2	yes	—
215	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
216	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
217	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
219	#95	—	2	4	no	—
220	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	268	2	no	—
228	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
229	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
234	C₂²×C₄	#11761628	1	1	yes	characteristic
235	C₂²×C₄	#11761628	1	1	yes	characteristic
236	C₂²×C₄	#11761628	1	2	yes	—
237	C₂²×C₄	#11761628	1	2	yes	—
238	C₂²×C₄	#11761628	1	2	yes	—
239	M₁₆	—	268	2	no	—
240	M₁₆	—	268	2	no	—
241	M₁₆	—	134	4	no	—
242	M₁₆	—	134	4	no	—
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	268	2	no	—
245	C₂×C₈	—	134	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	268	2	no	—
249	C₄⋊C₄	—	134	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	268	2	no	—
253	C₂²⋊C₄	—	134	4	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	8	no	—
262	C₄²	—	268	2	no	—
263	C₄²	—	134	4	no	—
264	C₄²	C₄×Dic₆₇	1	4	yes	—
265	C₄²	C₄⋊Dic₆₇	1	2	yes	—
266	C₄²	C₄⋊Dic₆₇	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
269	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
270	C₂³	#14524017	1	1	yes	characteristic
271	C₂³	#14524022	1	1	yes	characteristic
272	C₂³	#14524026	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	268	4	no	—
284	C₂×C₄	—	536	4	no	—
285	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	#14524000	1	2	yes	—
288	C₂×C₄	#14524000	1	2	yes	—
289	C₂×C₄	#14524002	1	1	yes	characteristic
290	C₂×C₄	#14524002	1	1	yes	characteristic
291	C₂×C₄	#14524002	1	1	yes	characteristic
292	C₂×C₄	#14524002	1	1	yes	characteristic
293	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
294	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
295	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
296	C₂×C₄	#14524017	1	1	yes	characteristic
297	C₂×C₄	#14524022	1	1	yes	characteristic
298	C₂×C₄	#14524026	1	2	yes	—
299	C₂×C₄	#14524031	1	2	yes	—
300	C₂×C₄	#14524035	1	2	yes	—
301	C₂×C₄	#14524037	1	2	yes	—
302	C₂×C₄	#14524037	1	2	yes	—
303	C₈	—	268	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
310	C₂²	#16863256	1	1	yes	characteristic
311	C₂²	#16863259	1	1	yes	characteristic
312	C₂²	#16863280	1	2	yes	—
313	C₂²	#16863287	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	536	4	no	—
319	C₄	#16863242	1	1	yes	characteristic
320	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
321	C₄	#16863262	1	1	yes	characteristic
322	C₄	#16863264	1	1	yes	characteristic
323	C₄	#16863280	1	2	yes	—
324	C₄	#16863287	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	#22558436	1	1	yes	characteristic
330	C₂	#22560313	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₄×#117×C₆₇	C₂
socle	C₂²×C₆₇	#381
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#10816
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
11. #16863152	C₄	semidirect
17. #376×C₆₇	C₄	semidirect
18. C₄³×C₆₇	C₄	semidirect
171. C₆₇	#10816	semidirect
179. #376	Dic₆₇	semidirect
180. C₄³	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559948	2
#22559942	2
C₂×#16863156	2
C₂×#16863152	2
C₄×#117×C₆₇	2
#10816	67