Subgroups — C₆₇×#17746 #30696997

Order 17152.

1178 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	2	yes	maximal
3	C₆₇×#1558	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1548	C₂	1	8	yes	maximal, complemented
5	C₂³×Q₈×C₆₇	C₄	1	2	yes	—
6	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#432×C₆₇	—	2	2	no	—
16	#432×C₆₇	C₂²	1	4	yes	—
17	#431×C₆₇	—	2	2	no	—
18	#431×C₆₇	C₂²	1	4	yes	—
19	#423×C₆₇	C₂²	1	4	yes	—
20	#407×C₆₇	—	2	2	no	—
21	C₄×M₁₆×C₆₇	—	2	2	no	—
22	#333×C₆₇	C₂²	1	16	yes	complemented
23	C₂²×Q₈×C₆₇	—	2	4	no	—
24	C₂²×Q₈×C₆₇	—	2	2	no	—
25	C₂²×Q₈×C₆₇	—	4	8	no	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₆₇	—	2	2	no	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	D₄	1	2	yes	—
31	C₂×M₁₆×C₆₇	—	2	4	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	—	2	4	no	—
35	C₄⋊Q₈×C₆₇	—	2	2	no	—
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	C₄⋊Q₈×C₆₇	—	2	4	no	—
38	C₄⋊Q₈×C₆₇	—	2	4	no	—
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	—	2	4	no	—
56	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
57	C₂²⋊C₈×C₆₇	—	2	4	no	—
58	C₂²⋊C₈×C₆₇	—	2	4	no	—
59	#97×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂×Q₈×C₆₇	—	2	8	no	—
64	C₂×Q₈×C₆₇	—	4	16	no	—
65	C₂×Q₈×C₆₇	—	4	16	no	—
66	C₂×Q₈×C₆₇	—	4	8	no	—
67	C₂×Q₈×C₆₇	—	4	8	no	—
68	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	4	8	no	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²⋊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	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	M₁₆×C₆₇	—	4	8	no	—
85	C₂×C₈×C₆₇	—	2	8	no	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂×D₄	1	2	yes	—
99	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	2	4	no	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
104	Q₈×C₆₇	—	4	8	no	—
105	Q₈×C₆₇	—	4	8	no	—
106	Q₈×C₆₇	—	4	16	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
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	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	4	16	no	—
117	C₂×C₄×C₆₇	—	4	8	no	—
118	C₂×C₄×C₆₇	#100	1	4	yes	—
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₆₇	#137	1	8	yes	—
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
127	C₈×C₆₇	—	4	8	no	—
128	C₂²×C₆₇	—	2	8	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	C₂×#100	1	2	yes	—
133	C₂²×C₆₇	#419	1	4	yes	—
134	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	4	8	no	—
140	C₄×C₆₇	C₂×#100	1	4	yes	—
141	C₄×C₆₇	#419	1	4	yes	—
142	C₄×C₆₇	C₂×#137	1	4	yes	—
143	#17746	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	—	2	2	no	—
145	C₂×C₆₇	—	2	4	no	—
146	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#419	1	2	yes	—
148	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
149	#2920	C₂×C₆₇	1	2	yes	—
150	#1558	C₂×C₆₇	1	4	yes	complemented
151	#1548	C₂×C₆₇	1	8	yes	complemented
152	C₆₇	#17746	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×Q₈	C₄×C₆₇	1	2	yes	—
154	C₂×C₄⋊Q₈	—	2	2	no	—
155	C₂×C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊Q₈	—	2	2	no	—
157	C₂×C₄⋊Q₈	C₄×C₆₇	1	2	yes	—
158	C₂×C₄⋊Q₈	C₄×C₆₇	1	4	yes	—
159	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
161	C₂²×C₄⋊C₄	—	2	2	no	—
162	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
163	#432	—	2	2	no	—
164	#432	C₂²×C₆₇	1	4	yes	—
165	#431	—	2	2	no	—
166	#431	C₂²×C₆₇	1	4	yes	—
167	#423	C₂²×C₆₇	1	4	yes	—
168	#407	—	2	2	no	—
169	C₄×M₁₆	—	2	2	no	—
170	#333	C₂²×C₆₇	1	16	yes	complemented
171	C₂²×Q₈	—	2	4	no	—
172	C₂²×Q₈	—	2	2	no	—
173	C₂²×Q₈	—	4	8	no	—
174	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
175	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
178	C₂³×C₄	D₄×C₆₇	1	2	yes	—
179	C₂×M₁₆	—	2	4	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	—	2	4	no	—
187	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
188	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
189	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
198	C₂×C₄²	D₄×C₆₇	1	2	yes	—
199	C₂×C₄²	D₄×C₆₇	1	2	yes	—
200	C₂×C₄²	D₄×C₆₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
202	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	4	no	—
208	C₄×C₈	—	2	4	no	—
209	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	16	no	—
213	C₂×Q₈	—	4	16	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	8	no	—
222	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
232	M₁₆	—	4	8	no	—
233	C₂×C₈	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
239	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
240	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
241	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
242	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
243	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
244	C₄²	C₂×D₄×C₆₇	1	2	yes	—
245	C₄²	C₂×D₄×C₆₇	1	2	yes	—
246	C₄²	C₂×D₄×C₆₇	1	2	yes	—
247	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
251	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
252	Q₈	—	4	8	no	—
253	Q₈	—	4	8	no	—
254	Q₈	—	4	16	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	#100×C₆₇	1	4	yes	—
267	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
272	C₂×C₄	#137×C₆₇	1	8	yes	—
273	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
275	C₈	—	4	8	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	C₂×#100×C₆₇	1	2	yes	—
281	C₂²	#419×C₆₇	1	4	yes	—
282	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	8	no	—
288	C₄	C₂×#100×C₆₇	1	4	yes	—
289	C₄	#419×C₆₇	1	4	yes	—
290	C₄	C₂×#137×C₆₇	1	4	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	2	4	no	—
293	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
294	C₂	C₂×#419×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1558	C₂	semidirect
4. C₆₇×#1548	C₂	semidirect
22. #333×C₆₇	C₂²	semidirect
143. #17746	C₆₇	direct (N × H)
150. #1558	C₂×C₆₇	semidirect
151. #1548	C₂×C₆₇	semidirect
152. C₆₇	#17746	direct (N × H)
170. #333	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₆₇×#2920	2
C₆₇×#1558	2
C₆₇×#1548	2
#17746	67

Subgroups — C67×#17746 #30696997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17746 #30696997