Subgroups — C₆₇×#33362 #30712613

Order 17152.

1220 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3500	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2884	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2128	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2126	C₂	1	6	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
6	C₂×H(5,𝔽₂)×C₆₇	C₄	1	1	yes	characteristic, complemented
7	#548×C₆₇	—	2	2	no	—
8	#546×C₆₇	—	2	6	no	—
9	#536×C₆₇	C₄	1	3	yes	complemented
10	#535×C₆₇	C₂²	1	3	yes	—
11	#532×C₆₇	C₂²	1	3	yes	—
12	C₂×#124×C₆₇	C₄	1	3	yes	complemented
13	#519×C₆₇	—	2	2	no	—
14	C₄×#41×C₆₇	C₄	1	1	yes	characteristic, complemented
15	#410×C₆₇	—	2	6	no	—
16	#410×C₆₇	C₂²	1	12	yes	complemented
17	#354×C₆₇	C₂²	1	4	yes	complemented
18	#353×C₆₇	C₂²	1	12	yes	complemented
19	2₋¹⁺⁴×C₆₇	—	4	4	no	—
20	H(5,𝔽₂)×C₆₇	—	4	4	no	—
21	C₂×#41×C₆₇	—	2	1	no	—
22	C₂×#41×C₆₇	C₂×C₄	1	3	yes	—
23	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₆₇	D₄	1	6	yes	—
25	C₂×#41×C₆₇	C₂³	1	3	yes	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	3	yes	—
27	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
28	C₂²×D₄×C₆₇	—	2	3	no	—
29	C₂²×D₄×C₆₇	C₂×C₄	1	3	yes	—
30	C₄⋊Q₈×C₆₇	C₂×C₄	1	3	yes	—
31	#127×C₆₇	C₂³	1	3	yes	—
32	#124×C₆₇	—	2	6	no	—
33	#124×C₆₇	C₂×C₄	1	3	yes	—
34	#124×C₆₇	C₂×C₄	1	6	yes	—
35	#124×C₆₇	C₂³	1	1	yes	characteristic
36	#123×C₆₇	—	2	6	no	—
37	C₂²⋊Q₈×C₆₇	—	2	6	no	—
38	#121×C₆₇	—	2	6	no	—
39	#121×C₆₇	—	2	6	no	—
40	#120×C₆₇	—	2	6	no	—
41	C₄×Q₈×C₆₇	—	2	1	no	—
42	C₄×Q₈×C₆₇	—	4	8	no	—
43	C₄×D₄×C₆₇	—	2	3	no	—
44	C₄×D₄×C₆₇	—	2	6	no	—
45	#117×C₆₇	—	2	12	no	—
46	#117×C₆₇	C₂×C₄	1	3	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
49	#99×C₆₇	—	2	12	no	—
50	#99×C₆₇	C₂³	1	8	yes	—
51	C₂⁴×C₆₇	—	2	3	no	—
52	#41×C₆₇	—	2	4	no	—
53	#41×C₆₇	—	4	6	no	—
54	#41×C₆₇	—	4	12	no	—
55	#41×C₆₇	—	4	6	no	—
56	#41×C₆₇	—	4	4	no	—
57	C₂×Q₈×C₆₇	—	2	8	no	—
58	C₂×Q₈×C₆₇	—	4	6	no	—
59	C₂×Q₈×C₆₇	C₂²⋊C₄	1	6	yes	—
60	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₆₇	C₂²×C₄	1	3	yes	—
63	C₂×D₄×C₆₇	—	2	3	no	—
64	C₂×D₄×C₆₇	—	2	6	no	—
65	C₂×D₄×C₆₇	—	2	3	no	—
66	C₂×D₄×C₆₇	—	4	6	no	—
67	C₂×D₄×C₆₇	—	4	12	no	—
68	C₂×D₄×C₆₇	C₂²⋊C₄	1	3	yes	—
69	C₂×D₄×C₆₇	C₂²×C₄	1	3	yes	—
70	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
71	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	3	no	—
73	C₂²×C₄×C₆₇	—	2	3	no	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
77	C₄⋊C₄×C₆₇	—	2	3	no	—
78	C₄⋊C₄×C₆₇	—	2	6	no	—
79	C₄⋊C₄×C₆₇	—	4	12	no	—
80	C₂²⋊C₄×C₆₇	—	2	3	no	—
81	C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂²⋊C₄×C₆₇	—	2	6	no	—
83	C₂²⋊C₄×C₆₇	—	2	12	no	—
84	C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₄²×C₆₇	—	2	3	no	—
86	C₄²×C₆₇	—	4	12	no	—
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
89	C₂³×C₆₇	—	2	3	no	—
90	C₂³×C₆₇	—	4	6	no	—
91	C₂³×C₆₇	—	8	6	no	—
92	C₂³×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
94	Q₈×C₆₇	—	2	12	no	—
95	Q₈×C₆₇	—	4	3	no	—
96	Q₈×C₆₇	—	4	1	no	—
97	Q₈×C₆₇	#99	1	4	yes	—
98	D₄×C₆₇	—	2	6	no	—
99	D₄×C₆₇	—	4	6	no	—
100	D₄×C₆₇	—	4	1	no	—
101	D₄×C₆₇	—	4	3	no	—
102	D₄×C₆₇	—	4	6	no	—
103	D₄×C₆₇	—	4	6	no	—
104	D₄×C₆₇	—	8	6	no	—
105	C₂×C₄×C₆₇	—	2	12	no	—
106	C₂×C₄×C₆₇	—	2	6	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	3	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	4	6	no	—
111	C₂×C₄×C₆₇	—	4	6	no	—
112	C₂×C₄×C₆₇	—	4	8	no	—
113	C₂×C₄×C₆₇	—	8	3	no	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	3	no	—
119	C₂²×C₆₇	—	4	3	no	—
120	C₂²×C₆₇	—	4	2	no	—
121	C₂²×C₆₇	—	4	6	no	—
122	C₂²×C₆₇	—	4	3	no	—
123	C₂²×C₆₇	—	16	6	no	—
124	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	6	no	—
127	C₄×C₆₇	—	4	1	no	—
128	C₄×C₆₇	—	4	3	no	—
129	C₄×C₆₇	—	4	1	no	—
130	C₄×C₆₇	—	8	8	no	—
131	C₄×C₆₇	C₂×#99	1	6	yes	—
132	#33362	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₆₇	—	2	1	no	—
134	C₂×C₆₇	—	4	2	no	—
135	C₂×C₆₇	—	4	3	no	—
136	C₂×C₆₇	—	8	3	no	—
137	C₂×C₆₇	C₂²×#99	1	1	yes	characteristic
138	#3500	C₂×C₆₇	1	1	yes	characteristic
139	#2884	C₂×C₆₇	1	2	yes	complemented
140	#2128	C₂×C₆₇	1	6	yes	complemented
141	#2126	C₂×C₆₇	1	6	yes	complemented
142	C₆₇	#33362	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
144	C₂×H(5,𝔽₂)	C₄×C₆₇	1	1	yes	characteristic, complemented
145	#548	—	2	2	no	—
146	#546	—	2	6	no	—
147	#536	C₄×C₆₇	1	3	yes	complemented
148	#535	C₂²×C₆₇	1	3	yes	—
149	#532	C₂²×C₆₇	1	3	yes	—
150	C₂×#124	C₄×C₆₇	1	3	yes	complemented
151	#519	—	2	2	no	—
152	C₄×#41	C₄×C₆₇	1	1	yes	characteristic, complemented
153	#410	—	2	6	no	—
154	#410	C₂²×C₆₇	1	12	yes	complemented
155	#354	C₂²×C₆₇	1	4	yes	complemented
156	#353	C₂²×C₆₇	1	12	yes	complemented
157	2₋¹⁺⁴	—	4	4	no	—
158	H(5,𝔽₂)	—	4	4	no	—
159	C₂×#41	—	2	1	no	—
160	C₂×#41	C₂×C₄×C₆₇	1	3	yes	—
161	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
162	C₂×#41	D₄×C₆₇	1	6	yes	—
163	C₂×#41	C₂³×C₆₇	1	3	yes	—
164	C₂²×Q₈	C₂×C₄×C₆₇	1	3	yes	—
165	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
166	C₂²×D₄	—	2	3	no	—
167	C₂²×D₄	C₂×C₄×C₆₇	1	3	yes	—
168	C₄⋊Q₈	C₂×C₄×C₆₇	1	3	yes	—
169	#127	C₂³×C₆₇	1	3	yes	—
170	#124	—	2	6	no	—
171	#124	C₂×C₄×C₆₇	1	3	yes	—
172	#124	C₂×C₄×C₆₇	1	6	yes	—
173	#124	C₂³×C₆₇	1	1	yes	characteristic
174	#123	—	2	6	no	—
175	C₂²⋊Q₈	—	2	6	no	—
176	#121	—	2	6	no	—
177	#121	—	2	6	no	—
178	#120	—	2	6	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	—	4	8	no	—
181	C₄×D₄	—	2	3	no	—
182	C₄×D₄	—	2	6	no	—
183	#117	—	2	12	no	—
184	#117	C₂×C₄×C₆₇	1	3	yes	—
185	C₂×C₂²⋊C₄	—	2	6	no	—
186	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
187	#99	—	2	12	no	—
188	#99	C₂³×C₆₇	1	8	yes	—
189	C₂⁴	—	2	3	no	—
190	#41	—	2	4	no	—
191	#41	—	4	6	no	—
192	#41	—	4	12	no	—
193	#41	—	4	6	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	4	6	no	—
197	C₂×Q₈	C₂²⋊C₄×C₆₇	1	6	yes	—
198	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
199	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂×Q₈	C₂²×C₄×C₆₇	1	3	yes	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	6	no	—
203	C₂×D₄	—	2	3	no	—
204	C₂×D₄	—	4	6	no	—
205	C₂×D₄	—	4	12	no	—
206	C₂×D₄	C₂²⋊C₄×C₆₇	1	3	yes	—
207	C₂×D₄	C₂²×C₄×C₆₇	1	3	yes	—
208	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
209	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	3	no	—
211	C₂²×C₄	—	2	3	no	—
212	C₂²×C₄	C₂²⋊C₄×C₆₇	1	6	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
215	C₄⋊C₄	—	2	3	no	—
216	C₄⋊C₄	—	2	6	no	—
217	C₄⋊C₄	—	4	12	no	—
218	C₂²⋊C₄	—	2	3	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	6	no	—
221	C₂²⋊C₄	—	2	12	no	—
222	C₂²⋊C₄	—	2	8	no	—
223	C₄²	—	2	3	no	—
224	C₄²	—	4	12	no	—
225	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
227	C₂³	—	2	3	no	—
228	C₂³	—	4	6	no	—
229	C₂³	—	8	6	no	—
230	C₂³	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
231	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
232	Q₈	—	2	12	no	—
233	Q₈	—	4	3	no	—
234	Q₈	—	4	1	no	—
235	Q₈	#99×C₆₇	1	4	yes	—
236	D₄	—	2	6	no	—
237	D₄	—	4	6	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	3	no	—
240	D₄	—	4	6	no	—
241	D₄	—	4	6	no	—
242	D₄	—	8	6	no	—
243	C₂×C₄	—	2	12	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	6	no	—
249	C₂×C₄	—	4	6	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	8	3	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
254	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
255	C₂²	—	2	2	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	4	3	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	6	no	—
260	C₂²	—	4	3	no	—
261	C₂²	—	16	6	no	—
262	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	6	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	3	no	—
267	C₄	—	4	1	no	—
268	C₄	—	8	8	no	—
269	C₄	C₂×#99×C₆₇	1	6	yes	—
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	4	3	no	—
273	C₂	—	8	3	no	—
274	C₂	C₂²×#99×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₇
prime radical	C₆₇×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×Q₈×C₆₇ ◁ C₂×2₋¹⁺⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2884	C₂	semidirect
3. C₆₇×#2128	C₂	semidirect
4. C₆₇×#2126	C₂	semidirect
6. C₂×H(5,𝔽₂)×C₆₇	C₄	semidirect
9. #536×C₆₇	C₄	semidirect
12. C₂×#124×C₆₇	C₄	semidirect
14. C₄×#41×C₆₇	C₄	semidirect
16. #410×C₆₇	C₂²	semidirect
17. #354×C₆₇	C₂²	semidirect
18. #353×C₆₇	C₂²	semidirect
132. #33362	C₆₇	direct (N × H)
139. #2884	C₂×C₆₇	semidirect
140. #2128	C₂×C₆₇	semidirect
141. #2126	C₂×C₆₇	semidirect
142. C₆₇	#33362	direct (N × H)
144. C₂×H(5,𝔽₂)	C₄×C₆₇	semidirect
147. #536	C₄×C₆₇	semidirect
150. C₂×#124	C₄×C₆₇	semidirect
152. C₄×#41	C₄×C₆₇	semidirect
154. #410	C₂²×C₆₇	semidirect
155. #354	C₂²×C₆₇	semidirect
156. #353	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3500	2
C₆₇×#2884	2
C₆₇×#2128	2
C₆₇×#2126	2
#33362	67

Subgroups — C67×#33362 #30712613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33362 #30712613