Subgroups — C₂×C₆₇×#3293 #30740355

Order 17152.

2598 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₆₇×#3293	C₂	1	16	yes	maximal, direct product factor
3	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#473×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#456×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#443×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
8	#578×C₆₇	—	2	16	no	—
9	#578×C₆₇	C₂²	1	32	yes	complemented
10	C₂×#137×C₆₇	—	2	8	no	—
11	C₂×#137×C₆₇	C₂²	1	8	yes	—
12	C₂×#135×C₆₇	—	2	4	no	—
13	C₂×#135×C₆₇	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₆₇	—	2	4	no	—
15	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
16	C₂×#131×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	#494×C₆₇	C₂²	1	8	yes	complemented
19	#473×C₆₇	C₂²	1	32	yes	complemented
20	#456×C₆₇	C₂²	1	32	yes	complemented
21	#443×C₆₇	C₂²	1	16	yes	complemented
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	C₂×#104×C₆₇	C₂²	1	4	yes	—
24	C₂×#101×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	2₋¹⁺⁴×C₆₇	—	2	16	no	—
27	2₋¹⁺⁴×C₆₇	—	4	4	no	—
28	2₋¹⁺⁴×C₆₇	C₂³	1	16	yes	—
29	C₂×#41×C₆₇	—	2	8	no	—
30	C₂×#41×C₆₇	—	4	4	no	—
31	C₂×#41×C₆₇	C₂³	1	4	yes	—
32	C₂²×Q₈×C₆₇	—	2	8	no	—
33	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
34	#137×C₆₇	—	2	16	no	—
35	#137×C₆₇	—	2	64	no	—
36	#137×C₆₇	C₂³	1	32	yes	—
37	#135×C₆₇	—	2	32	no	—
38	#135×C₆₇	D₄	1	8	yes	—
39	#135×C₆₇	C₂³	1	8	yes	—
40	C₂×Q₁₆×C₆₇	—	2	8	no	—
41	C₂×Q₁₆×C₆₇	—	2	32	no	—
42	C₂×Q₁₆×C₆₇	—	2	4	no	—
43	C₂×Q₁₆×C₆₇	—	2	4	no	—
44	C₂×Q₁₆×C₆₇	—	2	8	no	—
45	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
46	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	—
47	C₂×SD₁₆×C₆₇	—	2	8	no	—
48	C₂×SD₁₆×C₆₇	D₄	1	4	yes	—
49	C₂×D₈×C₆₇	D₄	1	2	yes	—
50	#131×C₆₇	—	2	8	no	—
51	#131×C₆₇	C₂³	1	16	yes	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	—	4	2	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#108×C₆₇	C₂³	1	8	yes	—
61	#104×C₆₇	C₂³	1	16	yes	—
62	#101×C₆₇	C₂³	1	16	yes	—
63	C₄×C₈×C₆₇	C₂³	1	4	yes	—
64	#41×C₆₇	—	2	4	no	—
65	#41×C₆₇	—	2	4	no	—
66	#41×C₆₇	—	2	32	no	—
67	#41×C₆₇	—	4	16	no	—
68	#41×C₆₇	—	4	16	no	—
69	#41×C₆₇	—	4	16	no	—
70	#41×C₆₇	C₂×D₄	1	8	yes	—
71	#41×C₆₇	C₂⁴	1	8	yes	—
72	C₂×Q₈×C₆₇	—	2	4	no	—
73	C₂×Q₈×C₆₇	—	2	32	no	—
74	C₂×Q₈×C₆₇	—	2	8	no	—
75	C₂×Q₈×C₆₇	—	2	8	no	—
76	C₂×Q₈×C₆₇	—	4	4	no	—
77	C₂×Q₈×C₆₇	—	4	16	no	—
78	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
80	C₂×D₄×C₆₇	—	2	4	no	—
81	C₂×D₄×C₆₇	—	4	4	no	—
82	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	4	8	no	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₆₇	—	2	32	no	—
89	Q₁₆×C₆₇	—	2	16	no	—
90	Q₁₆×C₆₇	—	2	16	no	—
91	Q₁₆×C₆₇	C₂×D₄	1	8	yes	—
92	SD₁₆×C₆₇	—	2	32	no	—
93	SD₁₆×C₆₇	C₂×D₄	1	16	yes	—
94	D₈×C₆₇	C₂×D₄	1	8	yes	—
95	M₁₆×C₆₇	—	2	4	no	—
96	M₁₆×C₆₇	—	2	16	no	—
97	M₁₆×C₆₇	C₂⁴	1	8	yes	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	16	no	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
104	C₄⋊C₄×C₆₇	—	4	8	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	C₂⁴	1	2	yes	—
107	C₂³×C₆₇	—	2	4	no	—
108	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	4	no	—
110	Q₈×C₆₇	—	2	16	no	—
111	Q₈×C₆₇	—	4	16	no	—
112	Q₈×C₆₇	—	4	8	no	—
113	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
114	D₄×C₆₇	—	2	8	no	—
115	D₄×C₆₇	—	2	8	no	—
116	D₄×C₆₇	—	2	4	no	—
117	D₄×C₆₇	—	4	16	no	—
118	D₄×C₆₇	C₂²×D₄	1	8	yes	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	2	8	no	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	4	32	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
132	C₈×C₆₇	—	2	8	no	—
133	C₈×C₆₇	C₂²×D₄	1	4	yes	—
134	C₈×C₆₇	C₂²×D₄	1	4	yes	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	8	no	—
137	C₂²×C₆₇	—	4	8	no	—
138	C₂²×C₆₇	D₄²	1	1	yes	characteristic
139	C₂²×C₆₇	D₄²	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	C₄×C₆₇	—	2	4	no	—
142	C₄×C₆₇	—	2	8	no	—
143	C₄×C₆₇	—	4	8	no	—
144	C₄×C₆₇	D₄²	1	2	yes	—
145	C₄×C₆₇	C₂³×D₄	1	2	yes	—
146	C₂×#3293	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	—	4	8	no	—
150	C₂×C₆₇	#3293	1	2	yes	direct product factor
151	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
152	C₂×#578	C₂×C₆₇	1	4	yes	complemented
153	#3293	C₂×C₆₇	1	16	yes	direct product factor
154	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
155	C₂×#473	C₂×C₆₇	1	4	yes	complemented
156	C₂×#456	C₂×C₆₇	1	4	yes	complemented
157	C₂×#443	C₂×C₆₇	1	2	yes	complemented
158	C₆₇	C₂×#3293	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
160	#578	—	2	16	no	—
161	#578	C₂²×C₆₇	1	32	yes	complemented
162	C₂×#137	—	2	8	no	—
163	C₂×#137	C₂²×C₆₇	1	8	yes	—
164	C₂×#135	—	2	4	no	—
165	C₂×#135	C₂²×C₆₇	1	2	yes	—
166	C₂²×Q₁₆	—	2	4	no	—
167	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
168	C₂×#131	C₂²×C₆₇	1	4	yes	—
169	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
170	#494	C₂²×C₆₇	1	8	yes	complemented
171	#473	C₂²×C₆₇	1	32	yes	complemented
172	#456	C₂²×C₆₇	1	32	yes	complemented
173	#443	C₂²×C₆₇	1	16	yes	complemented
174	C₂×#108	C₂²×C₆₇	1	2	yes	—
175	C₂×#104	C₂²×C₆₇	1	4	yes	—
176	C₂×#101	C₂²×C₆₇	1	4	yes	—
177	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
178	2₋¹⁺⁴	—	2	16	no	—
179	2₋¹⁺⁴	—	4	4	no	—
180	2₋¹⁺⁴	C₂³×C₆₇	1	16	yes	—
181	C₂×#41	—	2	8	no	—
182	C₂×#41	—	4	4	no	—
183	C₂×#41	C₂³×C₆₇	1	4	yes	—
184	C₂²×Q₈	—	2	8	no	—
185	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
186	#137	—	2	16	no	—
187	#137	—	2	64	no	—
188	#137	C₂³×C₆₇	1	32	yes	—
189	#135	—	2	32	no	—
190	#135	D₄×C₆₇	1	8	yes	—
191	#135	C₂³×C₆₇	1	8	yes	—
192	C₂×Q₁₆	—	2	8	no	—
193	C₂×Q₁₆	—	2	32	no	—
194	C₂×Q₁₆	—	2	4	no	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×Q₁₆	—	2	8	no	—
197	C₂×Q₁₆	D₄×C₆₇	1	2	yes	—
198	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	—
199	C₂×SD₁₆	—	2	8	no	—
200	C₂×SD₁₆	D₄×C₆₇	1	4	yes	—
201	C₂×D₈	D₄×C₆₇	1	2	yes	—
202	#131	—	2	8	no	—
203	#131	C₂³×C₆₇	1	16	yes	—
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
208	C₄⋊Q₈	—	4	2	no	—
209	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
210	C₂×C₄⋊C₄	—	4	2	no	—
211	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
212	#108	C₂³×C₆₇	1	8	yes	—
213	#104	C₂³×C₆₇	1	16	yes	—
214	#101	C₂³×C₆₇	1	16	yes	—
215	C₄×C₈	C₂³×C₆₇	1	4	yes	—
216	#41	—	2	4	no	—
217	#41	—	2	4	no	—
218	#41	—	2	32	no	—
219	#41	—	4	16	no	—
220	#41	—	4	16	no	—
221	#41	—	4	16	no	—
222	#41	C₂×D₄×C₆₇	1	8	yes	—
223	#41	C₂⁴×C₆₇	1	8	yes	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	32	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	16	no	—
230	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
231	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	8	no	—
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	Q₁₆	—	2	32	no	—
241	Q₁₆	—	2	16	no	—
242	Q₁₆	—	2	16	no	—
243	Q₁₆	C₂×D₄×C₆₇	1	8	yes	—
244	SD₁₆	—	2	32	no	—
245	SD₁₆	C₂×D₄×C₆₇	1	16	yes	—
246	D₈	C₂×D₄×C₆₇	1	8	yes	—
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	16	no	—
249	M₁₆	C₂⁴×C₆₇	1	8	yes	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	16	no	—
252	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
253	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
255	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂⁴×C₆₇	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
261	Q₈	—	2	4	no	—
262	Q₈	—	2	16	no	—
263	Q₈	—	4	16	no	—
264	Q₈	—	4	8	no	—
265	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	8	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	16	no	—
270	D₄	C₂²×D₄×C₆₇	1	8	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	32	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
282	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
284	C₈	—	2	8	no	—
285	C₈	C₂²×D₄×C₆₇	1	4	yes	—
286	C₈	C₂²×D₄×C₆₇	1	4	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	8	no	—
290	C₂²	D₄²×C₆₇	1	1	yes	characteristic
291	C₂²	D₄²×C₆₇	1	1	yes	characteristic
292	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	D₄²×C₆₇	1	2	yes	—
297	C₄	C₂³×D₄×C₆₇	1	2	yes	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	8	no	—
301	C₂	C₆₇×#3293	1	2	yes	direct product factor
302	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#578×C₆₇	C₂	semidirect
2. C₆₇×#3293	C₂	direct (N × H)
3. C₂×#494×C₆₇	C₂	semidirect
4. C₂×#473×C₆₇	C₂	semidirect
5. C₂×#456×C₆₇	C₂	semidirect
6. C₂×#443×C₆₇	C₂	semidirect
9. #578×C₆₇	C₂²	semidirect
18. #494×C₆₇	C₂²	semidirect
19. #473×C₆₇	C₂²	semidirect
20. #456×C₆₇	C₂²	semidirect
21. #443×C₆₇	C₂²	semidirect
146. C₂×#3293	C₆₇	direct (N × H)
150. C₂×C₆₇	#3293	direct (N × H)
152. C₂×#578	C₂×C₆₇	semidirect
153. #3293	C₂×C₆₇	direct (N × H)
154. C₂×#494	C₂×C₆₇	semidirect
155. C₂×#473	C₂×C₆₇	semidirect
156. C₂×#456	C₂×C₆₇	semidirect
157. C₂×#443	C₂×C₆₇	semidirect
158. C₆₇	C₂×#3293	direct (N × H)
161. #578	C₂²×C₆₇	semidirect
170. #494	C₂²×C₆₇	semidirect
171. #473	C₂²×C₆₇	semidirect
172. #456	C₂²×C₆₇	semidirect
173. #443	C₂²×C₆₇	semidirect
301. C₂	C₆₇×#3293	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×#578×C₆₇	2
C₆₇×#3293	2
C₂×#494×C₆₇	2
C₂×#473×C₆₇	2
C₂×#456×C₆₇	2
C₂×#443×C₆₇	2
C₂×#3293	67

Subgroups — C2×C67×#3293 #30740355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3293 #30740355