Subgroups — #30761642

Order 17152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 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×C₄³×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	—	2	2	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄³×C₆₇	—	2	2	no	—
19	C₄³×C₆₇	C₄	1	4	yes	complemented
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	16	no	—
37	C₂×C₄²×C₆₇	—	2	8	no	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	—	2	2	no	—
40	C₂×C₄²×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	4	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
51	C₂²×Dic₆₇	—	2	4	no	—
52	#11761612	—	2	8	no	—
53	#11761612	—	4	4	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₄⋊Dic₆₇	—	2	8	no	—
58	C₄⋊Dic₆₇	—	4	8	no	—
59	C₂²⋊Dic₆₇	—	4	4	no	—
60	C₄×Dic₆₇	—	4	2	no	—
61	C₄×Dic₆₇	—	4	2	no	—
62	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	16	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₄²×C₆₇	—	2	32	no	—
83	C₄²×C₆₇	—	2	4	no	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	C₄²	1	2	yes	—
87	C₄²×C₆₇	C₄²	1	2	yes	—
88	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
92	C₄²×C₆₇	C₄⋊C₄	1	8	yes	complemented
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₇	#121	1	2	yes	—
98	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₆₇	—	2	16	no	—
100	C₂×C₄×C₆₇	—	2	16	no	—
101	C₂×C₄×C₆₇	—	2	16	no	—
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	2	8	no	—
106	C₂×C₄×C₆₇	#95	1	4	yes	—
107	C₂×C₄×C₆₇	#95	1	4	yes	—
108	C₂×C₄×C₆₇	#104	1	8	yes	—
109	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	#121	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²⋊Q₈	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	#127	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	4	8	no	—
129	C₆₇⋊C₈	—	4	4	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
135	C₂²×C₆₇	#380	1	2	yes	—
136	C₂²×C₆₇	#389	1	2	yes	—
137	C₂²×C₆₇	#390	1	1	yes	characteristic
138	C₂²×C₆₇	#391	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×#104	1	4	yes	—
140	C₄×C₆₇	—	2	16	no	—
141	C₄×C₆₇	—	2	4	no	—
142	C₄×C₆₇	#339	1	8	yes	complemented
143	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₆₇	#382	1	1	yes	characteristic
145	C₄×C₆₇	#384	1	1	yes	characteristic
146	C₄×C₆₇	#389	1	2	yes	—
147	C₄×C₆₇	#390	1	1	yes	characteristic
148	C₄×C₆₇	#391	1	1	yes	characteristic
149	Dic₆₇	—	8	4	no	—
150	#10839	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#339	1	2	yes	—
155	C₂×C₆₇	#1962	1	2	yes	—
156	C₂×C₆₇	#1994	1	2	yes	—
157	C₂×#423	—	67	1	no	—
158	#2285	—	67	1	no	—
159	C₂×C₄³	D₆₇	1	1	yes	characteristic
160	C₂×#339	—	67	4	no	—
161	C₆₇	#10839	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	67	2	no	—
163	C₂×#117	—	67	2	no	—
164	C₂²×C₄²	—	2	2	no	—
165	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
166	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
167	#423	—	67	4	no	—
168	#423	—	134	2	no	—
169	C₂×C₄⋊C₈	—	134	2	no	—
170	#384	—	67	2	no	—
171	#382	—	67	2	no	—
172	#380	—	134	2	no	—
173	C₄×C₄⋊C₄	—	134	2	no	—
174	C₄³	—	2	2	no	—
175	C₄³	Dic₆₇	1	4	yes	complemented
176	#339	—	134	8	no	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
179	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×M₁₆	—	67	4	no	—
181	C₂×M₁₆	—	134	4	no	—
182	C₂×M₁₆	—	134	4	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂²×C₈	—	134	2	no	—
185	#117	—	134	4	no	—
186	#117	—	134	4	no	—
187	C₂×C₄⋊C₄	—	67	4	no	—
188	C₂×C₄⋊C₄	—	134	4	no	—
189	C₂×C₂²⋊C₄	—	134	2	no	—
190	C₂×C₄²	—	2	16	no	—
191	C₂×C₄²	—	2	8	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	134	2	no	—
196	C₂×C₄²	Dic₁₃₄	1	4	yes	—
197	C₂×C₄²	Dic₁₃₄	1	2	yes	—
198	C₂×C₄²	C₄×D₆₇	1	2	yes	—
199	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
201	C₂×C₄²	D₂₆₈	1	4	yes	—
202	C₂×C₄²	D₂₆₈	1	2	yes	—
203	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
204	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
206	C₄⋊C₈	—	134	8	no	—
207	#95	—	134	4	no	—
208	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	134	4	no	—
215	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
216	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
218	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
219	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
224	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
227	C₂²×C₄	#11761628	1	2	yes	—
228	C₂²×C₄	#11761628	1	2	yes	—
229	M₁₆	—	268	4	no	—
230	M₁₆	—	134	8	no	—
231	C₂×C₈	—	268	2	no	—
232	C₂×C₈	—	268	2	no	—
233	C₂×C₈	—	134	4	no	—
234	C₄⋊C₄	—	268	8	no	—
235	C₄⋊C₄	—	134	8	no	—
236	C₂²⋊C₄	—	268	4	no	—
237	C₄²	—	2	32	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	268	2	no	—
242	C₄²	—	268	2	no	—
243	C₄²	C₄×Dic₆₇	1	2	yes	—
244	C₄²	C₄×Dic₆₇	1	2	yes	—
245	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
246	C₄²	Dic₆₇⋊C₄	1	2	yes	—
247	C₄²	Dic₆₇⋊C₄	1	2	yes	—
248	C₄²	C₄⋊Dic₆₇	1	4	yes	—
249	C₄²	C₄⋊Dic₆₇	1	8	yes	complemented
250	C₂³	—	2	4	no	—
251	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
253	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
254	C₂³	#14524017	1	2	yes	—
255	C₂³	#14524022	1	2	yes	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	268	4	no	—
264	C₂×C₄	—	268	8	no	—
265	C₂×C₄	C₆₇⋊#95	1	4	yes	—
266	C₂×C₄	C₆₇⋊#95	1	4	yes	—
267	C₂×C₄	#14523971	1	8	yes	—
268	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
273	C₂×C₄	#14524002	1	2	yes	—
274	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
275	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
276	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
277	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
278	C₂×C₄	#14524017	1	2	yes	—
279	C₂×C₄	#14524022	1	2	yes	—
280	C₂×C₄	#14524031	1	1	yes	characteristic
281	C₂×C₄	#14524031	1	1	yes	characteristic
282	C₂×C₄	#14524035	1	2	yes	—
283	C₂×C₄	#14524040	1	1	yes	characteristic
284	C₂×C₄	#14524043	1	2	yes	—
285	C₂×C₄	#14524043	1	1	yes	characteristic
286	C₈	—	268	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
292	C₂²	#16863256	1	2	yes	—
293	C₂²	#16863294	1	2	yes	—
294	C₂²	#16863298	1	1	yes	characteristic
295	C₂²	#16863302	1	1	yes	characteristic
296	C₂²	C₂×#14523971	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	536	4	no	—
300	C₄	#16863156	1	8	yes	complemented
301	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
302	C₄	#16863262	1	1	yes	characteristic
303	C₄	#16863266	1	1	yes	characteristic
304	C₄	#16863294	1	2	yes	—
305	C₄	#16863298	1	1	yes	characteristic
306	C₄	#16863302	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#22558439	1	1	yes	characteristic
310	C₂	C₂×#16863156	1	2	yes	—
311	C₂	#22561037	1	2	yes	—
312	C₂	#22561186	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₄³×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	#10839
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄³

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
19. C₄³×C₆₇	C₄	semidirect
92. C₄²×C₆₇	C₄⋊C₄	semidirect
142. C₄×C₆₇	#339	semidirect
161. C₆₇	#10839	semidirect
175. C₄³	Dic₆₇	semidirect
249. C₄²	C₄⋊Dic₆₇	semidirect
300. C₄	#16863156	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×C₄³×C₆₇	2
#10839	67