Subgroups — C₂×C₆₇×#1763 #30699661

Order 17152.

1686 conjugacy classes of subgroups.

The subgroup lattice has 362 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	C₂²×#103×C₆₇	C₂	1	2	yes	maximal
3	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1763	C₂	1	8	yes	maximal, direct product factor
6	C₂×#340×C₆₇	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₆₇	—	2	4	no	—
10	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
11	C₂×C₄×Q₈×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	—	2	1	no	—
17	C₂²×C₄²×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₆₇	C₄	1	16	yes	complemented
20	C₂×#103×C₆₇	C₂²	1	8	yes	—
21	C₂×#97×C₆₇	—	2	4	no	—
22	C₂×#97×C₆₇	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂×#95×C₆₇	—	2	1	no	—
26	#340×C₆₇	C₂²	1	8	yes	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂²×Q₈×C₆₇	—	2	4	no	—
30	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
32	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
33	C₂³×C₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	—	2	1	no	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	2	4	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₄×Q₈×C₆₇	—	2	16	no	—
44	C₄×Q₈×C₆₇	—	2	8	no	—
45	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
46	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₆₇	—	2	8	no	—
57	C₂×C₄²×C₆₇	—	2	4	no	—
58	C₂×C₄²×C₆₇	—	2	4	no	—
59	C₂×C₄²×C₆₇	—	4	4	no	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₆₇	D₄	1	2	yes	—
62	C₂×C₄²×C₆₇	D₄	1	4	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	#103×C₆₇	C₂×C₄	1	32	yes	complemented
65	#97×C₆₇	—	2	8	no	—
66	#95×C₆₇	—	2	4	no	—
67	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	—	2	8	no	—
70	C₂×Q₈×C₆₇	—	2	8	no	—
71	C₂×Q₈×C₆₇	—	2	16	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	4	no	—
74	C₂×Q₈×C₆₇	C₄²	1	16	yes	complemented
75	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
76	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	8	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	8	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	4	4	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	4	no	—
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	#41	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₆₇	—	2	16	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
105	C₄⋊C₄×C₆₇	—	2	16	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
111	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₄²×C₆₇	—	2	16	no	—
113	C₄²×C₆₇	—	4	8	no	—
114	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
115	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
116	C₄²×C₆₇	C₂×D₄	1	4	yes	—
117	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂³×C₆₇	C₄×D₄	1	4	yes	—
119	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
120	C₂³×C₆₇	#137	1	2	yes	—
121	C₂³×C₆₇	#137	1	4	yes	—
122	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₆₇	—	2	4	no	—
124	Q₈×C₆₇	—	2	8	no	—
125	Q₈×C₆₇	—	2	16	no	—
126	Q₈×C₆₇	C₂×C₄²	1	8	yes	complemented
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	8	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	16	no	—
133	C₂×C₄×C₆₇	—	2	8	no	—
134	C₂×C₄×C₆₇	—	2	8	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	8	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	4	8	no	—
142	C₂×C₄×C₆₇	—	4	16	no	—
143	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
144	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
145	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
146	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
147	C₂×C₄×C₆₇	#117	1	4	yes	—
148	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
149	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
150	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
154	C₈×C₆₇	—	2	8	no	—
155	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
156	C₂²×C₆₇	#417	1	4	yes	—
157	C₂²×C₆₇	#419	1	4	yes	—
158	C₂²×C₆₇	#440	1	8	yes	—
159	C₂²×C₆₇	#441	1	8	yes	—
160	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
162	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂²×C₆₇	C₂×#137	1	2	yes	—
164	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
165	C₄×C₆₇	—	2	4	no	—
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	4	4	no	—
169	C₄×C₆₇	—	4	8	no	—
170	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
171	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
172	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
173	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
174	C₂×#1763	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₆₇	#1763	1	8	yes	direct product factor
176	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#440	1	2	yes	—
180	C₂×C₆₇	C₂×#441	1	2	yes	—
181	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
182	C₂²×#103	C₂×C₆₇	1	2	yes	—
183	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
184	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
185	#1763	C₂×C₆₇	1	8	yes	direct product factor
186	C₂×#340	C₂×C₆₇	1	2	yes	—
187	C₆₇	C₂×#1763	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
189	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
190	C₂×C₄×Q₈	—	2	4	no	—
191	C₂×C₄×Q₈	—	2	2	no	—
192	C₂×C₄×Q₈	C₄×C₆₇	1	8	yes	complemented
193	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
194	C₂²×C₄⋊C₄	—	2	1	no	—
195	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
196	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
197	C₂²×C₄²	—	2	1	no	—
198	C₂²×C₄²	—	2	1	no	—
199	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×#103	C₄×C₆₇	1	16	yes	complemented
201	C₂×#103	C₂²×C₆₇	1	8	yes	—
202	C₂×#97	—	2	4	no	—
203	C₂×#97	C₂²×C₆₇	1	4	yes	—
204	C₄×C₄⋊C₄	—	2	2	no	—
205	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
206	C₂×#95	—	2	1	no	—
207	#340	C₂²×C₆₇	1	8	yes	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×Q₈	—	2	4	no	—
211	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
212	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
213	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
214	C₂³×C₄	—	2	1	no	—
215	C₂³×C₄	—	2	1	no	—
216	C₂³×C₄	—	2	2	no	—
217	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
223	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
224	C₄×Q₈	—	2	16	no	—
225	C₄×Q₈	—	2	8	no	—
226	C₄×Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
227	C₂×C₄⋊C₄	—	2	8	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
234	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
235	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
237	C₂×C₄²	—	2	8	no	—
238	C₂×C₄²	—	2	4	no	—
239	C₂×C₄²	—	2	4	no	—
240	C₂×C₄²	—	4	4	no	—
241	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
242	C₂×C₄²	D₄×C₆₇	1	2	yes	—
243	C₂×C₄²	D₄×C₆₇	1	4	yes	—
244	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
245	#103	C₂×C₄×C₆₇	1	32	yes	complemented
246	#97	—	2	8	no	—
247	#95	—	2	4	no	—
248	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	8	no	—
251	C₂×Q₈	—	2	8	no	—
252	C₂×Q₈	—	2	16	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	C₄²×C₆₇	1	16	yes	complemented
256	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
257	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	8	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	4	4	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
273	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
280	C₂²×C₄	#41×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
282	C₂×C₈	—	2	16	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
286	C₄⋊C₄	—	2	16	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	8	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
292	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
293	C₄²	—	2	16	no	—
294	C₄²	—	4	8	no	—
295	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
296	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
297	C₄²	C₂×D₄×C₆₇	1	4	yes	—
298	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
299	C₂³	C₄×D₄×C₆₇	1	4	yes	—
300	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
301	C₂³	#137×C₆₇	1	2	yes	—
302	C₂³	#137×C₆₇	1	4	yes	—
303	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
304	Q₈	—	2	4	no	—
305	Q₈	—	2	8	no	—
306	Q₈	—	2	16	no	—
307	Q₈	C₂×C₄²×C₆₇	1	8	yes	complemented
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	16	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	16	no	—
324	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
325	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
326	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
327	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
328	C₂×C₄	#117×C₆₇	1	4	yes	—
329	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
330	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
331	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
333	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
335	C₈	—	2	8	no	—
336	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
337	C₂²	#417×C₆₇	1	4	yes	—
338	C₂²	#419×C₆₇	1	4	yes	—
339	C₂²	#440×C₆₇	1	8	yes	—
340	C₂²	#441×C₆₇	1	8	yes	—
341	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
343	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
344	C₂²	C₂×#137×C₆₇	1	2	yes	—
345	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
346	C₄	—	2	8	no	—
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	8	no	—
351	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
352	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
353	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
354	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
355	C₂	C₆₇×#1763	1	8	yes	direct product factor
356	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
357	C₂	C₂×#419×C₆₇	1	1	yes	characteristic
358	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
359	C₂	C₂×#440×C₆₇	1	2	yes	—
360	C₂	C₂×#441×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1763	C₂	direct (N × H)
11. C₂×C₄×Q₈×C₆₇	C₄	semidirect
19. C₂×#103×C₆₇	C₄	semidirect
45. C₄×Q₈×C₆₇	C₂×C₄	semidirect
64. #103×C₆₇	C₂×C₄	semidirect
74. C₂×Q₈×C₆₇	C₄²	semidirect
126. Q₈×C₆₇	C₂×C₄²	semidirect
174. C₂×#1763	C₆₇	direct (N × H)
175. C₂×C₆₇	#1763	direct (N × H)
185. #1763	C₂×C₆₇	direct (N × H)
187. C₆₇	C₂×#1763	direct (N × H)
192. C₂×C₄×Q₈	C₄×C₆₇	semidirect
200. C₂×#103	C₄×C₆₇	semidirect
226. C₄×Q₈	C₂×C₄×C₆₇	semidirect
245. #103	C₂×C₄×C₆₇	semidirect
255. C₂×Q₈	C₄²×C₆₇	semidirect
307. Q₈	C₂×C₄²×C₆₇	semidirect
355. C₂	C₆₇×#1763	direct (N × H)

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
C₂²×#103×C₆₇	2
C₂²×#97×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1763	2
C₂×#340×C₆₇	2
C₂×#1763	67

Subgroups — C2×C67×#1763 #30699661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1763 #30699661