Subgroups — C₂×C₆₇×#3338 #30740480

Order 17152.

1942 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#3338	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#484×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#477×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#467×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#107×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	#554×C₆₇	—	2	8	no	—
13	#554×C₆₇	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#125×C₆₇	—	2	2	no	—
16	C₂×C₂²⋊Q₈×C₆₇	—	2	4	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	—	2	2	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	#500×C₆₇	C₂²	1	8	yes	complemented
23	#484×C₆₇	C₂²	1	32	yes	complemented
24	#477×C₆₇	C₂²	1	16	yes	complemented
25	#467×C₆₇	C₂²	1	16	yes	complemented
26	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
27	C₈×D₄×C₆₇	C₂²	1	8	yes	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×#107×C₆₇	C₂²	1	4	yes	—
30	C₂×#107×C₆₇	C₂²	1	2	yes	—
31	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#107×C₆₇	C₂²	1	16	yes	complemented
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	4	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	4	no	—
42	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₆₇	—	2	4	no	—
44	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₆₇	—	2	8	no	—
46	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	—	2	4	no	—
51	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
52	#125×C₆₇	—	2	16	no	—
53	C₂²⋊Q₈×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	32	no	—
55	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
58	C₄×D₄×C₆₇	—	2	16	no	—
59	C₄×D₄×C₆₇	D₄	1	4	yes	—
60	C₄×D₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	C₂³	1	8	yes	—
75	#107×C₆₇	C₂³	1	16	yes	—
76	#107×C₆₇	C₂³	1	8	yes	—
77	#107×C₆₇	C₂³	1	4	yes	—
78	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
79	#103×C₆₇	C₂³	1	16	yes	—
80	#103×C₆₇	C₂³	1	8	yes	—
81	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
82	C₄×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	8	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	8	4	no	—
89	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
90	C₂×D₄×C₆₇	Q₁₆	1	8	yes	complemented
91	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	16	no	—
98	C₂²×C₄×C₆₇	—	8	8	no	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	Q₁₆×C₆₇	—	2	8	no	—
106	C₂×C₈×C₆₇	—	2	4	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	16	no	—
109	C₂×C₈×C₆₇	#41	1	4	yes	—
110	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
111	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
112	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	16	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	8	no	—
118	C₄⋊C₄×C₆₇	—	2	16	no	—
119	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
120	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
121	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
122	C₂²⋊C₄×C₆₇	—	2	16	no	—
123	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
124	C₄²×C₆₇	—	2	8	no	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₄²×C₆₇	C₂⁴	1	2	yes	—
127	C₂³×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	C₂×Q₁₆	1	8	yes	—
130	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
131	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₆₇	—	2	4	no	—
133	Q₈×C₆₇	—	2	4	no	—
134	Q₈×C₆₇	—	4	4	no	—
135	Q₈×C₆₇	—	8	8	no	—
136	D₄×C₆₇	C₂×Q₁₆	1	16	yes	complemented
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	8	16	no	—
149	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
156	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
157	C₈×C₆₇	—	2	4	no	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	C₂×#41	1	4	yes	—
160	C₂²×C₆₇	—	2	16	no	—
161	C₂²×C₆₇	—	2	4	no	—
162	C₂²×C₆₇	—	2	4	no	—
163	C₂²×C₆₇	#547	1	2	yes	—
164	C₂²×C₆₇	C₂²×Q₁₆	1	8	yes	—
165	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
166	C₂²×C₆₇	#576	1	2	yes	—
167	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	4	8	no	—
173	C₄×C₆₇	#547	1	2	yes	—
174	C₄×C₆₇	C₂²×Q₁₆	1	4	yes	—
175	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
176	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
177	C₂×#3338	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	2	8	no	—
179	C₂×C₆₇	#3338	1	4	yes	direct product factor
180	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
181	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
182	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
183	C₂×#554	C₂×C₆₇	1	2	yes	—
184	#3338	C₂×C₆₇	1	16	yes	direct product factor
185	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₂×#484	C₂×C₆₇	1	4	yes	complemented
187	C₂×#477	C₂×C₆₇	1	2	yes	complemented
188	C₂×#467	C₂×C₆₇	1	2	yes	complemented
189	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
190	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
191	C₂²×#107	C₂×C₆₇	1	2	yes	complemented
192	C₆₇	C₂×#3338	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
194	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
195	#554	—	2	8	no	—
196	#554	C₂²×C₆₇	1	16	yes	—
197	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
198	C₂×#125	—	2	2	no	—
199	C₂×C₂²⋊Q₈	—	2	4	no	—
200	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
201	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
202	C₂×C₄×D₄	—	2	2	no	—
203	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
204	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
205	#500	C₂²×C₆₇	1	8	yes	complemented
206	#484	C₂²×C₆₇	1	32	yes	complemented
207	#477	C₂²×C₆₇	1	16	yes	complemented
208	#467	C₂²×C₆₇	1	16	yes	complemented
209	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
210	C₈×D₄	C₂²×C₆₇	1	8	yes	—
211	C₂×#107	C₂²×C₆₇	1	4	yes	—
212	C₂×#107	C₂²×C₆₇	1	2	yes	—
213	C₂×#107	C₂²×C₆₇	1	2	yes	—
214	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×#107	C₂²×C₆₇	1	16	yes	complemented
216	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×#103	C₂²×C₆₇	1	4	yes	—
218	C₂×#103	C₂²×C₆₇	1	2	yes	—
219	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂²×Q₈	—	2	2	no	—
222	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
223	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
224	C₂³×C₄	—	2	4	no	—
225	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
226	C₂×Q₁₆	—	2	4	no	—
227	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
228	C₂²×C₈	—	2	8	no	—
229	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
230	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
231	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	2	4	no	—
234	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
235	#125	—	2	16	no	—
236	C₂²⋊Q₈	—	2	8	no	—
237	C₂²⋊Q₈	—	2	32	no	—
238	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
239	C₄×Q₈	—	2	4	no	—
240	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
241	C₄×D₄	—	2	16	no	—
242	C₄×D₄	D₄×C₆₇	1	4	yes	—
243	C₄×D₄	C₂³×C₆₇	1	4	yes	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	8	no	—
246	C₂×C₄⋊C₄	—	2	2	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	4	no	—
249	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
252	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
253	C₂×C₂²⋊C₄	—	2	4	no	—
254	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
257	#107	C₂³×C₆₇	1	4	yes	—
258	#107	C₂³×C₆₇	1	8	yes	—
259	#107	C₂³×C₆₇	1	16	yes	—
260	#107	C₂³×C₆₇	1	8	yes	—
261	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
262	#103	C₂³×C₆₇	1	8	yes	—
263	#103	C₂³×C₆₇	1	16	yes	—
264	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
265	C₄×C₈	C₂³×C₆₇	1	4	yes	—
266	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	8	no	—
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	2	4	no	—
271	C₂×Q₈	—	8	4	no	—
272	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
273	C₂×D₄	Q₁₆×C₆₇	1	8	yes	complemented
274	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	16	no	—
281	C₂²×C₄	—	8	8	no	—
282	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
285	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
287	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
288	Q₁₆	—	2	8	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	16	no	—
292	C₂×C₈	#41×C₆₇	1	4	yes	—
293	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
294	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
295	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	16	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	8	no	—
300	C₄⋊C₄	—	2	8	no	—
301	C₄⋊C₄	—	2	16	no	—
302	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
303	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
304	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
305	C₂²⋊C₄	—	2	16	no	—
306	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
307	C₄²	—	2	8	no	—
308	C₄²	C₂×D₄×C₆₇	1	2	yes	—
309	C₄²	C₂⁴×C₆₇	1	2	yes	—
310	C₂³	—	2	4	no	—
311	C₂³	—	2	4	no	—
312	C₂³	C₂×Q₁₆×C₆₇	1	8	yes	—
313	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
314	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
315	Q₈	—	2	4	no	—
316	Q₈	—	2	4	no	—
317	Q₈	—	4	4	no	—
318	Q₈	—	8	8	no	—
319	D₄	C₂×Q₁₆×C₆₇	1	16	yes	complemented
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	8	no	—
331	C₂×C₄	—	8	16	no	—
332	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
333	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
334	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
335	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
336	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
337	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
338	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
339	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
340	C₈	—	2	4	no	—
341	C₈	—	2	2	no	—
342	C₈	C₂×#41×C₆₇	1	4	yes	—
343	C₂²	—	2	4	no	—
344	C₂²	—	2	16	no	—
345	C₂²	—	2	4	no	—
346	C₂²	#547×C₆₇	1	2	yes	—
347	C₂²	C₂²×Q₁₆×C₆₇	1	8	yes	—
348	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
349	C₂²	#576×C₆₇	1	2	yes	—
350	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	4	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	8	no	—
355	C₄	—	4	4	no	—
356	C₄	#547×C₆₇	1	2	yes	—
357	C₄	C₂²×Q₁₆×C₆₇	1	4	yes	—
358	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
359	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
360	C₂	—	2	8	no	—
361	C₂	C₆₇×#3338	1	4	yes	direct product factor
362	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
363	C₂	C₂³×Q₁₆×C₆₇	1	1	yes	characteristic
364	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3338	C₂	direct (N × H)
3. C₂×#500×C₆₇	C₂	semidirect
4. C₂×#484×C₆₇	C₂	semidirect
5. C₂×#477×C₆₇	C₂	semidirect
6. C₂×#467×C₆₇	C₂	semidirect
7. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
9. C₂²×#107×C₆₇	C₂	semidirect
22. #500×C₆₇	C₂²	semidirect
23. #484×C₆₇	C₂²	semidirect
24. #477×C₆₇	C₂²	semidirect
25. #467×C₆₇	C₂²	semidirect
26. C₄×Q₁₆×C₆₇	C₂²	semidirect
32. C₂×#107×C₆₇	C₂²	semidirect
90. C₂×D₄×C₆₇	Q₁₆	semidirect
136. D₄×C₆₇	C₂×Q₁₆	semidirect
177. C₂×#3338	C₆₇	direct (N × H)
179. C₂×C₆₇	#3338	direct (N × H)
184. #3338	C₂×C₆₇	direct (N × H)
185. C₂×#500	C₂×C₆₇	semidirect
186. C₂×#484	C₂×C₆₇	semidirect
187. C₂×#477	C₂×C₆₇	semidirect
188. C₂×#467	C₂×C₆₇	semidirect
189. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
191. C₂²×#107	C₂×C₆₇	semidirect
192. C₆₇	C₂×#3338	direct (N × H)
205. #500	C₂²×C₆₇	semidirect
206. #484	C₂²×C₆₇	semidirect
207. #477	C₂²×C₆₇	semidirect
208. #467	C₂²×C₆₇	semidirect
209. C₄×Q₁₆	C₂²×C₆₇	semidirect
215. C₂×#107	C₂²×C₆₇	semidirect
273. C₂×D₄	Q₁₆×C₆₇	semidirect
319. D₄	C₂×Q₁₆×C₆₇	semidirect
361. C₂	C₆₇×#3338	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×#554×C₆₇	2
C₆₇×#3338	2
C₂×#500×C₆₇	2
C₂×#484×C₆₇	2
C₂×#477×C₆₇	2
C₂×#467×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#3338	67

Subgroups — C2×C67×#3338 #30740480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3338 #30740480