Subgroups — C₆₇×#25905 #30705156

Order 17152.

1218 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#486×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2025	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1875	C₂	1	4	yes	maximal, complemented
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₆₇	—	2	8	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
13	C₂×#121×C₆₇	—	2	2	no	—
14	C₂×#121×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#486×C₆₇	—	2	8	no	—
19	C₂×#102×C₆₇	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#390×C₆₇	—	2	2	no	—
23	#390×C₆₇	C₂²	1	2	yes	—
24	#384×C₆₇	C₂²	1	1	yes	characteristic
25	#340×C₆₇	C₂²	1	8	yes	—
26	#336×C₆₇	C₂²	1	2	yes	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	2	no	—
29	C₂²×D₄×C₆₇	—	2	2	no	—
30	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₆₇	—	2	4	no	—
35	C₄⋊Q₈×C₆₇	D₄	1	8	yes	complemented
36	#127×C₆₇	—	2	4	no	—
37	#125×C₆₇	—	2	8	no	—
38	C₂²⋊Q₈×C₆₇	—	2	16	no	—
39	#121×C₆₇	—	2	8	no	—
40	#121×C₆₇	—	4	8	no	—
41	C₄×Q₈×C₆₇	—	2	16	no	—
42	#117×C₆₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	—	2	4	no	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#102×C₆₇	—	2	16	no	—
57	C₂²⋊C₈×C₆₇	—	2	4	no	—
58	C₄×C₈×C₆₇	—	2	4	no	—
59	C₂⁴×C₆₇	—	2	2	no	—
60	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₆₇	—	2	8	no	—
62	C₂×Q₈×C₆₇	—	4	4	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂×D₄×C₆₇	—	2	4	no	—
65	C₂×D₄×C₆₇	—	2	4	no	—
66	C₂×D₄×C₆₇	—	2	4	no	—
67	C₂×D₄×C₆₇	—	4	8	no	—
68	C₂×D₄×C₆₇	—	4	4	no	—
69	C₂×D₄×C₆₇	—	4	4	no	—
70	C₂×D₄×C₆₇	—	4	4	no	—
71	C₂×D₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
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	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	16	no	—
88	C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
93	C₂²⋊C₄×C₆₇	—	2	8	no	—
94	C₂²⋊C₄×C₆₇	—	4	8	no	—
95	C₂²⋊C₄×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	—	2	8	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	C₂×D₄	1	4	yes	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	2	1	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	—	4	2	no	—
105	C₂³×C₆₇	—	4	2	no	—
106	C₂³×C₆₇	—	4	2	no	—
107	C₂³×C₆₇	—	4	4	no	—
108	C₂³×C₆₇	—	4	4	no	—
109	C₂³×C₆₇	#124	1	2	yes	—
110	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	16	no	—
112	D₄×C₆₇	—	4	8	no	—
113	D₄×C₆₇	—	8	8	no	—
114	D₄×C₆₇	—	8	8	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	8	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	8	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	8	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	#120	1	4	yes	—
128	C₂×C₄×C₆₇	#124	1	2	yes	—
129	C₂×C₄×C₆₇	#135	1	4	yes	—
130	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
134	C₈×C₆₇	—	4	4	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	4	2	no	—
139	C₂²×C₆₇	—	4	2	no	—
140	C₂²×C₆₇	—	8	2	no	—
141	C₂²×C₆₇	—	8	2	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	8	4	no	—
144	C₂²×C₆₇	—	8	4	no	—
145	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
147	C₂²×C₆₇	#536	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#135	1	2	yes	—
149	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	8	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#449	1	8	yes	—
155	C₄×C₆₇	#548	1	4	yes	—
156	#25905	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	4	2	no	—
158	C₂×C₆₇	—	16	2	no	—
159	C₂×C₆₇	#2436	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#449	1	2	yes	—
161	C₂×C₆₇	#3131	1	1	yes	characteristic
162	C₂×C₆₇	#3138	1	1	yes	characteristic
163	C₂×C₆₇	#3228	1	2	yes	—
164	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
165	C₂×#486	C₂×C₆₇	1	2	yes	complemented
166	#2856	C₂×C₆₇	1	1	yes	characteristic
167	#2025	C₂×C₆₇	1	4	yes	complemented
168	#1990	C₂×C₆₇	1	2	yes	complemented
169	#1875	C₂×C₆₇	1	4	yes	complemented
170	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
171	C₆₇	#25905	1	1	yes	characteristic, direct product factor, Sylow
172	#533	—	2	8	no	—
173	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
174	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#125	C₂²×C₆₇	1	2	yes	—
176	C₂×C₂²⋊Q₈	—	2	2	no	—
177	C₂×#121	—	2	2	no	—
178	C₂×#121	C₂²×C₆₇	1	2	yes	—
179	C₂×C₄×Q₈	—	2	2	no	—
180	C₂×#117	C₂²×C₆₇	1	2	yes	—
181	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
182	#486	—	2	8	no	—
183	C₂×#102	C₂²×C₆₇	1	8	yes	—
184	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
185	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
186	#390	—	2	2	no	—
187	#390	C₂²×C₆₇	1	2	yes	—
188	#384	C₂²×C₆₇	1	1	yes	characteristic
189	#340	C₂²×C₆₇	1	8	yes	—
190	#336	C₂²×C₆₇	1	2	yes	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×D₄	—	2	2	no	—
193	C₂²×D₄	—	2	2	no	—
194	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
195	C₂³×C₄	D₄×C₆₇	1	2	yes	—
196	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	D₄×C₆₇	1	8	yes	complemented
200	#127	—	2	4	no	—
201	#125	—	2	8	no	—
202	C₂²⋊Q₈	—	2	16	no	—
203	#121	—	2	8	no	—
204	#121	—	4	8	no	—
205	C₄×Q₈	—	2	16	no	—
206	#117	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	D₄×C₆₇	1	2	yes	—
218	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
220	#102	—	2	16	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₄×C₈	—	2	4	no	—
223	C₂⁴	—	2	2	no	—
224	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
246	C₂²×C₄	#41×C₆₇	1	4	yes	—
247	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₄⋊C₄	—	2	16	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	4	8	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₄²	—	2	8	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂×D₄×C₆₇	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	4	no	—
272	C₂³	—	4	4	no	—
273	C₂³	#124×C₆₇	1	2	yes	—
274	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	Q₈	—	4	16	no	—
276	D₄	—	4	8	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	8	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	#120×C₆₇	1	4	yes	—
292	C₂×C₄	#124×C₆₇	1	2	yes	—
293	C₂×C₄	#135×C₆₇	1	4	yes	—
294	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
296	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
297	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	4	no	—
308	C₂²	—	8	4	no	—
309	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
310	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
311	C₂²	#536×C₆₇	1	1	yes	characteristic
312	C₂²	C₂×#135×C₆₇	1	2	yes	—
313	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	#449×C₆₇	1	8	yes	—
319	C₄	#548×C₆₇	1	4	yes	—
320	C₂	—	4	2	no	—
321	C₂	—	16	2	no	—
322	C₂	C₆₇×#2436	1	1	yes	characteristic
323	C₂	C₂×#449×C₆₇	1	2	yes	—
324	C₂	C₆₇×#3131	1	1	yes	characteristic
325	C₂	C₆₇×#3138	1	1	yes	characteristic
326	C₂	C₆₇×#3228	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#533×C₆₇	C₂	semidirect
2. C₂×#486×C₆₇	C₂	semidirect
4. C₆₇×#2025	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
6. C₆₇×#1875	C₂	semidirect
7. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
35. C₄⋊Q₈×C₆₇	D₄	semidirect
156. #25905	C₆₇	direct (N × H)
164. C₂×#533	C₂×C₆₇	semidirect
165. C₂×#486	C₂×C₆₇	semidirect
167. #2025	C₂×C₆₇	semidirect
168. #1990	C₂×C₆₇	semidirect
169. #1875	C₂×C₆₇	semidirect
170. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
171. C₆₇	#25905	direct (N × H)
199. C₄⋊Q₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2856	2
C₆₇×#2025	2
C₆₇×#1990	2
C₆₇×#1875	2
C₄×C₂²⋊C₈×C₆₇	2
#25905	67

Subgroups — C67×#25905 #30705156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25905 #30705156