Subgroups — C₆₇×#21849 #30701100

Order 17152.

1386 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1870	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1797	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	C₄×#41×C₆₇	—	2	4	no	—
11	C₄×#41×C₆₇	—	2	4	no	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#431×C₆₇	—	2	4	no	—
23	#423×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	8	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	8	yes	—
31	C₂×#41×C₆₇	—	2	4	no	—
32	C₂×#41×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	—	4	4	no	—
45	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	C₄×Q₈×C₆₇	—	2	4	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	—	2	8	no	—
51	C₄×D₄×C₆₇	—	2	4	no	—
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	4	no	—
57	#117×C₆₇	—	2	8	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₆₇	—	2	4	no	—
66	C₂×C₄²×C₆₇	—	2	4	no	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	4	yes	—
72	C₂×C₄²×C₆₇	D₄	1	4	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	C₂²⋊C₈×C₆₇	—	2	4	no	—
80	C₂²⋊C₈×C₆₇	—	4	8	no	—
81	C₄×C₈×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
84	#41×C₆₇	—	2	32	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	M₁₆	1	4	yes	complemented
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	4	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	M₁₆	1	4	yes	complemented
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₆₇	—	4	8	no	—
122	C₂×C₈×C₆₇	—	2	4	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₂²⋊C₄×C₆₇	—	2	4	no	—
133	C₂²⋊C₄×C₆₇	—	2	4	no	—
134	C₄²×C₆₇	—	2	4	no	—
135	C₄²×C₆₇	—	2	4	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
139	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
140	C₄²×C₆₇	C₂×D₄	1	2	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₄²×C₆₇	C₂×D₄	1	2	yes	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	1	no	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
154	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
155	Q₈×C₆₇	—	2	8	no	—
156	D₄×C₆₇	—	2	16	no	—
157	D₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	16	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	8	no	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
184	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
186	C₂×C₄×C₆₇	#131	1	4	yes	—
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
190	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
191	C₈×C₆₇	—	4	4	no	—
192	C₈×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	2	4	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	8	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₆₇	#520	1	2	yes	—
205	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
206	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×#131	1	2	yes	—
208	C₄×C₆₇	—	2	2	no	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	4	no	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	4	2	no	—
214	C₄×C₆₇	#407	1	4	yes	—
215	C₄×C₆₇	#408	1	4	yes	—
216	C₄×C₆₇	#535	1	1	yes	characteristic
217	C₄×C₆₇	#536	1	1	yes	characteristic
218	C₄×C₆₇	#537	1	2	yes	—
219	#21849	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	4	2	no	—
221	C₂×C₆₇	—	4	4	no	—
222	C₂×C₆₇	#2311	1	1	yes	characteristic
223	C₂×C₆₇	C₂×#407	1	1	yes	characteristic
224	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
225	C₂×C₆₇	#2990	1	2	yes	—
226	C₂×C₆₇	#2991	1	2	yes	—
227	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
228	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1984	C₂×C₆₇	1	2	yes	complemented
231	#1954	C₂×C₆₇	1	2	yes	complemented
232	#1870	C₂×C₆₇	1	4	yes	complemented
233	#1797	C₂×C₆₇	1	4	yes	complemented
234	C₆₇	#21849	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
236	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
237	C₄×#41	—	2	4	no	—
238	C₄×#41	—	2	4	no	—
239	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
242	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
244	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
246	C₂×#117	C₄×C₆₇	1	2	yes	—
247	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
248	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
249	#431	—	2	4	no	—
250	#423	—	2	4	no	—
251	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
252	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
255	C₂×C₂²⋊C₈	C₂²×C₆₇	1	8	yes	—
256	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
257	#336	C₂²×C₆₇	1	8	yes	—
258	C₂×#41	—	2	4	no	—
259	C₂×#41	—	2	4	no	—
260	C₂×#41	—	2	4	no	—
261	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
263	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
266	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
268	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
269	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	4	4	no	—
272	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
273	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
274	C₄×Q₈	—	2	4	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×D₄	—	2	8	no	—
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	—	2	4	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	—	2	2	no	—
283	#117	—	2	4	no	—
284	#117	—	2	8	no	—
285	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
291	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
292	C₂×C₄²	—	2	4	no	—
293	C₂×C₄²	—	2	4	no	—
294	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
295	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
296	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	D₄×C₆₇	1	4	yes	—
299	C₂×C₄²	D₄×C₆₇	1	4	yes	—
300	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
303	C₄⋊C₈	—	2	4	no	—
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	4	no	—
306	C₂²⋊C₈	—	2	4	no	—
307	C₂²⋊C₈	—	4	8	no	—
308	C₄×C₈	—	2	4	no	—
309	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
311	#41	—	2	32	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	M₁₆×C₆₇	1	4	yes	complemented
315	C₂×D₄	—	2	4	no	—
316	C₂×D₄	—	2	4	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	M₁₆×C₆₇	1	4	yes	complemented
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
337	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
339	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
343	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
348	M₁₆	—	4	8	no	—
349	C₂×C₈	—	2	4	no	—
350	C₂×C₈	—	2	4	no	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	8	no	—
353	C₂×C₈	—	2	4	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	2	4	no	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	4	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
366	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
367	C₄²	C₂×D₄×C₆₇	1	2	yes	—
368	C₄²	C₂×D₄×C₆₇	1	2	yes	—
369	C₄²	C₂×D₄×C₆₇	1	2	yes	—
370	C₄²	C₂×D₄×C₆₇	1	2	yes	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	—	2	4	no	—
379	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
380	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
381	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
382	Q₈	—	2	8	no	—
383	D₄	—	2	16	no	—
384	D₄	—	2	8	no	—
385	C₂×C₄	—	2	16	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	8	no	—
405	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
406	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
407	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
408	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
409	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
411	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
412	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
413	C₂×C₄	#131×C₆₇	1	4	yes	—
414	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
417	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
418	C₈	—	4	4	no	—
419	C₈	—	4	4	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	4	no	—
430	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
431	C₂²	#520×C₆₇	1	2	yes	—
432	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
433	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
434	C₂²	C₂×#131×C₆₇	1	2	yes	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	2	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	2	no	—
441	C₄	#407×C₆₇	1	4	yes	—
442	C₄	#408×C₆₇	1	4	yes	—
443	C₄	#535×C₆₇	1	1	yes	characteristic
444	C₄	#536×C₆₇	1	1	yes	characteristic
445	C₄	#537×C₆₇	1	2	yes	—
446	C₂	—	4	2	no	—
447	C₂	—	4	4	no	—
448	C₂	C₆₇×#2311	1	1	yes	characteristic
449	C₂	C₂×#407×C₆₇	1	1	yes	characteristic
450	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
451	C₂	C₆₇×#2990	1	2	yes	—
452	C₂	C₆₇×#2991	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#431×C₆₇	C₂	semidirect
3. C₂×#423×C₆₇	C₂	semidirect
4. C₆₇×#1984	C₂	semidirect
5. C₆₇×#1954	C₂	semidirect
6. C₆₇×#1870	C₂	semidirect
7. C₆₇×#1797	C₂	semidirect
87. C₂×Q₈×C₆₇	M₁₆	semidirect
93. C₂×D₄×C₆₇	M₁₆	semidirect
219. #21849	C₆₇	direct (N × H)
228. C₂×#431	C₂×C₆₇	semidirect
229. C₂×#423	C₂×C₆₇	semidirect
230. #1984	C₂×C₆₇	semidirect
231. #1954	C₂×C₆₇	semidirect
232. #1870	C₂×C₆₇	semidirect
233. #1797	C₂×C₆₇	semidirect
234. C₆₇	#21849	direct (N × H)
314. C₂×Q₈	M₁₆×C₆₇	semidirect
320. C₂×D₄	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂×#431×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#1984	2
C₆₇×#1954	2
C₆₇×#1870	2
C₆₇×#1797	2
#21849	67

Subgroups — C67×#21849 #30701100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21849 #30701100