Subgroups — C₂×C₆₇×#3363 #30740551

Order 17152.

1830 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3363	C₂	1	16	yes	maximal, direct product factor
4	C₂×#487×C₆₇	C₂	1	3	yes	maximal
5	C₂×#477×C₆₇	C₂	1	3	yes	maximal
6	C₂×#462×C₆₇	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	3	yes	maximal
9	C₂²×Q₁₆×C₆₇	C₂²	1	3	yes	—
10	Q₈²×C₆₇	—	2	4	no	—
11	Q₈²×C₆₇	C₂²	1	8	yes	—
12	#557×C₆₇	—	2	4	no	—
13	#557×C₆₇	C₂²	1	8	yes	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	3	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#125×C₆₇	—	2	3	no	—
18	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	#487×C₆₇	C₂²	1	24	yes	—
24	#477×C₆₇	C₂²	1	24	yes	—
25	#462×C₆₇	C₂²	1	24	yes	—
26	C₈×Q₈×C₆₇	C₂²	1	8	yes	—
27	C₄×Q₁₆×C₆₇	C₂²	1	24	yes	—
28	C₂×#107×C₆₇	C₂²	1	3	yes	—
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
30	C₂×#103×C₆₇	C₂²	1	6	yes	—
31	C₂×#103×C₆₇	C₂²	1	3	yes	—
32	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
34	C₂²×Q₈×C₆₇	—	2	3	no	—
35	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×Q₁₆×C₆₇	—	2	12	no	—
39	C₂×Q₁₆×C₆₇	C₂³	1	12	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	—	2	24	no	—
43	C₄⋊Q₈×C₆₇	—	2	6	no	—
44	C₄⋊Q₈×C₆₇	—	2	6	no	—
45	C₄⋊Q₈×C₆₇	C₂³	1	12	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	12	yes	—
47	#125×C₆₇	—	2	24	no	—
48	C₄×Q₈×C₆₇	—	2	8	no	—
49	C₄×Q₈×C₆₇	—	2	8	no	—
50	C₄×Q₈×C₆₇	—	2	6	no	—
51	C₄×Q₈×C₆₇	—	2	6	no	—
52	C₄×Q₈×C₆₇	D₄	1	4	yes	—
53	C₄×Q₈×C₆₇	C₂³	1	12	yes	—
54	C₄×Q₈×C₆₇	C₂³	1	12	yes	—
55	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
61	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
64	C₂×C₄²×C₆₇	—	2	3	no	—
65	C₂×C₄²×C₆₇	—	2	3	no	—
66	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
67	#107×C₆₇	C₂³	1	12	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	12	yes	—
69	#103×C₆₇	C₂³	1	24	yes	—
70	#103×C₆₇	C₂³	1	12	yes	—
71	#103×C₆₇	C₂³	1	4	yes	—
72	C₄×C₈×C₆₇	C₂³	1	12	yes	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	3	no	—
75	C₂×Q₈×C₆₇	—	2	12	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	6	no	—
78	C₂×Q₈×C₆₇	—	2	6	no	—
79	C₂×Q₈×C₆₇	—	8	6	no	—
80	C₂×Q₈×C₆₇	Q₁₆	1	8	yes	complemented
81	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₆₇	#41	1	4	yes	—
83	C₂×Q₈×C₆₇	C₂⁴	1	6	yes	—
84	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	—	2	1	no	—
87	C₂²×C₄×C₆₇	—	2	3	no	—
88	C₂²×C₄×C₆₇	—	2	3	no	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₆₇	—	2	24	no	—
93	C₂×C₈×C₆₇	—	2	6	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
96	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
97	C₄⋊C₄×C₆₇	—	2	12	no	—
98	C₄⋊C₄×C₆₇	—	2	12	no	—
99	C₄⋊C₄×C₆₇	—	2	12	no	—
100	C₄⋊C₄×C₆₇	—	2	12	no	—
101	C₄⋊C₄×C₆₇	—	2	12	no	—
102	C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₄⋊C₄×C₆₇	—	2	3	no	—
104	C₄⋊C₄×C₆₇	C₂×D₄	1	12	yes	—
105	C₄⋊C₄×C₆₇	C₂⁴	1	6	yes	—
106	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
107	C₄²×C₆₇	—	2	12	no	—
108	C₄²×C₆₇	—	2	12	no	—
109	C₄²×C₆₇	C₂×D₄	1	6	yes	—
110	C₄²×C₆₇	C₂⁴	1	6	yes	—
111	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₆₇	—	2	4	no	—
113	Q₈×C₆₇	—	2	2	no	—
114	Q₈×C₆₇	—	2	6	no	—
115	Q₈×C₆₇	—	2	6	no	—
116	Q₈×C₆₇	—	4	6	no	—
117	Q₈×C₆₇	—	8	12	no	—
118	Q₈×C₆₇	C₂×Q₁₆	1	16	yes	complemented
119	Q₈×C₆₇	C₂×#41	1	4	yes	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	6	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	6	no	—
125	C₂×C₄×C₆₇	—	2	6	no	—
126	C₂×C₄×C₆₇	—	4	6	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	6	no	—
129	C₂×C₄×C₆₇	C₂×Q₁₆	1	12	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
133	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
135	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
136	C₈×C₆₇	—	2	6	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	#548	1	2	yes	—
139	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
140	C₂²×C₆₇	#577	1	2	yes	—
141	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	2	6	no	—
144	C₄×C₆₇	—	4	6	no	—
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	6	no	—
147	C₄×C₆₇	#548	1	2	yes	—
148	C₄×C₆₇	C₂²×Q₁₆	1	12	yes	—
149	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
150	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×#3363	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₆₇	#3363	1	4	yes	direct product factor
153	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
154	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
155	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
156	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic
157	C₂×#557	C₂×C₆₇	1	1	yes	characteristic
158	#3363	C₂×C₆₇	1	16	yes	direct product factor
159	C₂×#487	C₂×C₆₇	1	3	yes	—
160	C₂×#477	C₂×C₆₇	1	3	yes	—
161	C₂×#462	C₂×C₆₇	1	3	yes	—
162	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
163	C₂×C₄×Q₁₆	C₂×C₆₇	1	3	yes	—
164	C₆₇	C₂×#3363	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×Q₁₆	C₂²×C₆₇	1	3	yes	—
166	Q₈²	—	2	4	no	—
167	Q₈²	C₂²×C₆₇	1	8	yes	—
168	#557	—	2	4	no	—
169	#557	C₂²×C₆₇	1	8	yes	—
170	C₂×C₄⋊Q₈	—	2	3	no	—
171	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
172	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
173	C₂×#125	—	2	3	no	—
174	C₂×C₄×Q₈	—	2	1	no	—
175	C₂×C₄×Q₈	—	2	1	no	—
176	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
177	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
178	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
179	#487	C₂²×C₆₇	1	24	yes	—
180	#477	C₂²×C₆₇	1	24	yes	—
181	#462	C₂²×C₆₇	1	24	yes	—
182	C₈×Q₈	C₂²×C₆₇	1	8	yes	—
183	C₄×Q₁₆	C₂²×C₆₇	1	24	yes	—
184	C₂×#107	C₂²×C₆₇	1	3	yes	—
185	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
186	C₂×#103	C₂²×C₆₇	1	6	yes	—
187	C₂×#103	C₂²×C₆₇	1	3	yes	—
188	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
190	C₂²×Q₈	—	2	3	no	—
191	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
192	C₂²×Q₈	C₂³×C₆₇	1	3	yes	—
193	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₂×Q₁₆	—	2	12	no	—
195	C₂×Q₁₆	C₂³×C₆₇	1	12	yes	—
196	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
197	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	24	no	—
199	C₄⋊Q₈	—	2	6	no	—
200	C₄⋊Q₈	—	2	6	no	—
201	C₄⋊Q₈	C₂³×C₆₇	1	12	yes	—
202	C₄⋊Q₈	C₂³×C₆₇	1	12	yes	—
203	#125	—	2	24	no	—
204	C₄×Q₈	—	2	8	no	—
205	C₄×Q₈	—	2	8	no	—
206	C₄×Q₈	—	2	6	no	—
207	C₄×Q₈	—	2	6	no	—
208	C₄×Q₈	D₄×C₆₇	1	4	yes	—
209	C₄×Q₈	C₂³×C₆₇	1	12	yes	—
210	C₄×Q₈	C₂³×C₆₇	1	12	yes	—
211	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
212	C₂×C₄⋊C₄	—	2	3	no	—
213	C₂×C₄⋊C₄	—	2	3	no	—
214	C₂×C₄⋊C₄	—	2	3	no	—
215	C₂×C₄⋊C₄	—	2	3	no	—
216	C₂×C₄⋊C₄	—	2	3	no	—
217	C₂×C₄⋊C₄	D₄×C₆₇	1	3	yes	—
218	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
220	C₂×C₄²	—	2	3	no	—
221	C₂×C₄²	—	2	3	no	—
222	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
223	#107	C₂³×C₆₇	1	12	yes	—
224	C₄⋊C₈	C₂³×C₆₇	1	12	yes	—
225	#103	C₂³×C₆₇	1	4	yes	—
226	#103	C₂³×C₆₇	1	12	yes	—
227	#103	C₂³×C₆₇	1	24	yes	—
228	C₄×C₈	C₂³×C₆₇	1	12	yes	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	3	no	—
231	C₂×Q₈	—	2	12	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	6	no	—
234	C₂×Q₈	—	2	6	no	—
235	C₂×Q₈	—	8	6	no	—
236	C₂×Q₈	Q₁₆×C₆₇	1	8	yes	complemented
237	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
238	C₂×Q₈	#41×C₆₇	1	4	yes	—
239	C₂×Q₈	C₂⁴×C₆₇	1	6	yes	—
240	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	3	no	—
244	C₂²×C₄	—	2	3	no	—
245	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
247	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
248	Q₁₆	—	2	24	no	—
249	C₂×C₈	—	2	6	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
252	C₂×C₈	C₂⁴×C₆₇	1	6	yes	—
253	C₄⋊C₄	—	2	12	no	—
254	C₄⋊C₄	—	2	12	no	—
255	C₄⋊C₄	—	2	12	no	—
256	C₄⋊C₄	—	2	12	no	—
257	C₄⋊C₄	—	2	12	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	3	no	—
260	C₄⋊C₄	C₂×D₄×C₆₇	1	12	yes	—
261	C₄⋊C₄	C₂⁴×C₆₇	1	6	yes	—
262	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
263	C₄²	—	2	12	no	—
264	C₄²	—	2	12	no	—
265	C₄²	C₂×D₄×C₆₇	1	6	yes	—
266	C₄²	C₂⁴×C₆₇	1	6	yes	—
267	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
268	Q₈	—	2	4	no	—
269	Q₈	—	2	2	no	—
270	Q₈	—	2	6	no	—
271	Q₈	—	2	6	no	—
272	Q₈	—	4	6	no	—
273	Q₈	—	8	12	no	—
274	Q₈	C₂×Q₁₆×C₆₇	1	16	yes	complemented
275	Q₈	C₂×#41×C₆₇	1	4	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	4	6	no	—
283	C₂×C₄	—	4	6	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₂×Q₁₆×C₆₇	1	12	yes	—
286	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
287	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
289	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
290	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
291	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
292	C₈	—	2	6	no	—
293	C₈	—	2	2	no	—
294	C₂²	#548×C₆₇	1	2	yes	—
295	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
296	C₂²	#577×C₆₇	1	2	yes	—
297	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	6	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	6	no	—
303	C₄	#548×C₆₇	1	2	yes	—
304	C₄	C₂²×Q₁₆×C₆₇	1	12	yes	—
305	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
306	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
307	C₂	C₆₇×#3363	1	4	yes	direct product factor
308	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
309	C₂	C₂³×Q₁₆×C₆₇	1	1	yes	characteristic
310	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#3363	C₂	direct (N × H)
80. C₂×Q₈×C₆₇	Q₁₆	semidirect
118. Q₈×C₆₇	C₂×Q₁₆	semidirect
151. C₂×#3363	C₆₇	direct (N × H)
152. C₂×C₆₇	#3363	direct (N × H)
158. #3363	C₂×C₆₇	direct (N × H)
164. C₆₇	C₂×#3363	direct (N × H)
236. C₂×Q₈	Q₁₆×C₆₇	semidirect
274. Q₈	C₂×Q₁₆×C₆₇	semidirect
307. C₂	C₆₇×#3363	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₆₇	2
C₂×#557×C₆₇	2
C₆₇×#3363	2
C₂×#487×C₆₇	2
C₂×#477×C₆₇	2
C₂×#462×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×#3363	67

Subgroups — C2×C67×#3363 #30740551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3363 #30740551