Subgroups — C₆₇×#60089 #30739340

Order 17152.

4902 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2898	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2897	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2896	C₂	1	6	yes	maximal, complemented
7	C₂×#417×C₆₇	C₂	1	6	yes	maximal, complemented
8	#585×C₆₇	—	2	8	no	—
9	#585×C₆₇	C₄	1	16	yes	complemented
10	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
11	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
12	C₂×H(5,𝔽₂)×C₆₇	—	2	3	no	—
13	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	—
14	C₂²×#41×C₆₇	—	2	4	no	—
15	C₂²×#41×C₆₇	C₂²	1	6	yes	—
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#568×C₆₇	—	2	8	no	—
18	C₂×#131×C₆₇	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₆₇	C₂²	1	6	yes	complemented
20	#520×C₆₇	—	2	4	no	—
21	#520×C₆₇	C₂²	1	8	yes	complemented
22	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	—	2	3	no	—
24	C₂×#117×C₆₇	C₂²	1	3	yes	complemented
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	complemented
26	#419×C₆₇	C₂²	1	12	yes	complemented
27	#418×C₆₇	C₂²	1	12	yes	complemented
28	#417×C₆₇	C₂²	1	12	yes	complemented
29	#417×C₆₇	C₂²	1	24	yes	complemented
30	#417×C₆₇	C₂²	1	4	yes	complemented
31	#416×C₆₇	C₂²	1	24	yes	complemented
32	C₂×#103×C₆₇	C₂²	1	12	yes	complemented
33	C₂×#102×C₆₇	C₂²	1	12	yes	complemented
34	2₋¹⁺⁴×C₆₇	—	2	16	no	—
35	2₋¹⁺⁴×C₆₇	—	2	16	no	—
36	2₋¹⁺⁴×C₆₇	C₂×C₄	1	32	yes	complemented
37	H(5,𝔽₂)×C₆₇	—	2	16	no	—
38	H(5,𝔽₂)×C₆₇	—	2	48	no	—
39	H(5,𝔽₂)×C₆₇	C₂×C₄	1	32	yes	complemented
40	C₂×#41×C₆₇	—	2	64	no	—
41	C₂×#41×C₆₇	—	2	4	no	—
42	C₂×#41×C₆₇	—	2	12	no	—
43	C₂×#41×C₆₇	—	2	6	no	—
44	C₂×#41×C₆₇	—	2	6	no	—
45	C₂×#41×C₆₇	—	2	24	no	—
46	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
47	C₂×#41×C₆₇	C₂×C₄	1	48	yes	complemented
48	C₂×#41×C₆₇	D₄	1	8	yes	—
49	C₂×#41×C₆₇	D₄	1	4	yes	—
50	C₂×#41×C₆₇	C₂³	1	4	yes	—
51	C₂×#41×C₆₇	C₂³	1	12	yes	—
52	C₂×#41×C₆₇	C₂³	1	4	yes	—
53	C₂²×Q₈×C₆₇	—	2	4	no	—
54	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
56	C₂²×D₄×C₆₇	—	2	6	no	—
57	C₂²×D₄×C₆₇	—	2	12	no	—
58	C₂²×D₄×C₆₇	D₄	1	3	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
60	C₂³×C₄×C₆₇	—	2	6	no	—
61	C₂³×C₄×C₆₇	—	2	3	no	—
62	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
63	#131×C₆₇	—	2	32	no	—
64	C₂×M₁₆×C₆₇	—	2	24	no	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	12	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
68	C₄×Q₈×C₆₇	—	2	8	no	—
69	C₄×D₄×C₆₇	—	2	24	no	—
70	#117×C₆₇	—	2	6	no	—
71	#117×C₆₇	C₂³	1	12	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
78	C₂×C₄²×C₆₇	—	2	3	no	—
79	#103×C₆₇	C₂³	1	24	yes	—
80	#103×C₆₇	C₂³	1	8	yes	—
81	#102×C₆₇	C₂³	1	24	yes	—
82	#102×C₆₇	C₂³	1	8	yes	—
83	C₂⁴×C₆₇	—	2	6	no	—
84	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
85	#41×C₆₇	—	2	32	no	—
86	#41×C₆₇	—	2	96	no	—
87	#41×C₆₇	—	2	48	no	—
88	#41×C₆₇	—	2	48	no	—
89	#41×C₆₇	—	2	8	no	—
90	#41×C₆₇	—	2	24	no	—
91	#41×C₆₇	C₂²⋊C₄	1	32	yes	—
92	#41×C₆₇	C₂²×C₄	1	16	yes	—
93	#41×C₆₇	C₂²×C₄	1	16	yes	—
94	#41×C₆₇	C₂²×C₄	1	48	yes	complemented
95	#41×C₆₇	C₂×D₄	1	16	yes	—
96	C₂×Q₈×C₆₇	—	2	32	no	—
97	C₂×Q₈×C₆₇	—	2	6	no	—
98	C₂×Q₈×C₆₇	—	2	12	no	—
99	C₂×Q₈×C₆₇	C₂²⋊C₄	1	8	yes	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	24	yes	complemented
101	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
103	C₂×D₄×C₆₇	—	2	48	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	12	no	—
106	C₂×D₄×C₆₇	—	2	96	no	—
107	C₂×D₄×C₆₇	—	2	6	no	—
108	C₂×D₄×C₆₇	—	2	12	no	—
109	C₂×D₄×C₆₇	C₂²⋊C₄	1	24	yes	—
110	C₂×D₄×C₆₇	C₂²×C₄	1	24	yes	complemented
111	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	—
112	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₇	—	2	6	no	—
114	C₂²×C₄×C₆₇	—	2	48	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	12	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	3	no	—
119	C₂²×C₄×C₆₇	—	2	12	no	—
120	C₂²×C₄×C₆₇	—	8	6	no	—
121	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
127	M₁₆×C₆₇	—	2	8	no	—
128	M₁₆×C₆₇	—	2	24	no	—
129	C₂×C₈×C₆₇	—	2	24	no	—
130	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
131	C₄⋊C₄×C₆₇	—	2	3	no	—
132	C₄⋊C₄×C₆₇	—	2	12	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	C₂⁴	1	6	yes	—
135	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	2	12	no	—
137	C₄²×C₆₇	—	2	12	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	2	24	no	—
140	C₂³×C₆₇	—	4	3	no	—
141	C₂³×C₆₇	—	4	3	no	—
142	C₂³×C₆₇	—	8	6	no	—
143	C₂³×C₆₇	—	8	6	no	—
144	C₂³×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
145	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
146	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₆₇	—	2	24	no	—
148	Q₈×C₆₇	—	2	4	no	—
149	Q₈×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
150	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
151	D₄×C₆₇	—	2	16	no	—
152	D₄×C₆₇	—	2	48	no	—
153	D₄×C₆₇	—	2	24	no	—
154	D₄×C₆₇	—	2	4	no	—
155	D₄×C₆₇	C₂×C₂²⋊C₄	1	48	yes	—
156	D₄×C₆₇	C₂³×C₄	1	8	yes	—
157	C₂×C₄×C₆₇	—	2	24	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	16	no	—
160	C₂×C₄×C₆₇	—	2	48	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	6	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	6	no	—
165	C₂×C₄×C₆₇	—	8	12	no	—
166	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
168	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
175	C₈×C₆₇	—	2	8	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	6	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	6	no	—
180	C₂²×C₆₇	—	2	8	no	—
181	C₂²×C₆₇	—	4	12	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	8	24	no	—
185	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
186	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
188	C₄×C₆₇	—	2	8	no	—
189	C₄×C₆₇	—	4	6	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
192	C₄×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
193	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
194	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
195	#60089	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	2	no	—
197	C₂×C₆₇	—	2	12	no	—
198	C₂×C₆₇	—	4	8	no	—
199	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₆₇	#3585	1	1	yes	characteristic
201	C₂×C₆₇	#3586	1	1	yes	characteristic
202	C₂×#585	C₂×C₆₇	1	1	yes	characteristic
203	C₂×#568	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#2898	C₂×C₆₇	1	8	yes	complemented
206	#2897	C₂×C₆₇	1	8	yes	complemented
207	#2896	C₂×C₆₇	1	6	yes	complemented
208	C₂×#417	C₂×C₆₇	1	6	yes	complemented
209	C₆₇	#60089	1	1	yes	characteristic, direct product factor, Sylow
210	#585	—	2	8	no	—
211	#585	C₄×C₆₇	1	16	yes	complemented
212	C₂×2₋¹⁺⁴	—	2	1	no	—
213	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
214	C₂×H(5,𝔽₂)	—	2	3	no	—
215	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	—
216	C₂²×#41	—	2	4	no	—
217	C₂²×#41	C₂²×C₆₇	1	6	yes	—
218	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
219	#568	—	2	8	no	—
220	C₂×#131	C₂²×C₆₇	1	8	yes	complemented
221	C₂²×M₁₆	C₂²×C₆₇	1	6	yes	complemented
222	#520	—	2	4	no	—
223	#520	C₂²×C₆₇	1	8	yes	complemented
224	C₂×C₄×Q₈	—	2	1	no	—
225	C₂×C₄×D₄	—	2	3	no	—
226	C₂×#117	C₂²×C₆₇	1	3	yes	complemented
227	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	complemented
228	#419	C₂²×C₆₇	1	12	yes	complemented
229	#418	C₂²×C₆₇	1	12	yes	complemented
230	#417	C₂²×C₆₇	1	12	yes	complemented
231	#417	C₂²×C₆₇	1	24	yes	complemented
232	#417	C₂²×C₆₇	1	4	yes	complemented
233	#416	C₂²×C₆₇	1	24	yes	complemented
234	C₂×#103	C₂²×C₆₇	1	12	yes	complemented
235	C₂×#102	C₂²×C₆₇	1	12	yes	complemented
236	2₋¹⁺⁴	—	2	16	no	—
237	2₋¹⁺⁴	—	2	16	no	—
238	2₋¹⁺⁴	C₂×C₄×C₆₇	1	32	yes	complemented
239	H(5,𝔽₂)	—	2	16	no	—
240	H(5,𝔽₂)	—	2	48	no	—
241	H(5,𝔽₂)	C₂×C₄×C₆₇	1	32	yes	complemented
242	C₂×#41	—	2	64	no	—
243	C₂×#41	—	2	4	no	—
244	C₂×#41	—	2	12	no	—
245	C₂×#41	—	2	6	no	—
246	C₂×#41	—	2	6	no	—
247	C₂×#41	—	2	24	no	—
248	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
249	C₂×#41	C₂×C₄×C₆₇	1	48	yes	complemented
250	C₂×#41	D₄×C₆₇	1	8	yes	—
251	C₂×#41	D₄×C₆₇	1	4	yes	—
252	C₂×#41	C₂³×C₆₇	1	4	yes	—
253	C₂×#41	C₂³×C₆₇	1	12	yes	—
254	C₂×#41	C₂³×C₆₇	1	4	yes	—
255	C₂²×Q₈	—	2	4	no	—
256	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
257	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
258	C₂²×D₄	—	2	6	no	—
259	C₂²×D₄	—	2	12	no	—
260	C₂²×D₄	D₄×C₆₇	1	3	yes	—
261	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
262	C₂³×C₄	—	2	6	no	—
263	C₂³×C₄	—	2	3	no	—
264	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
265	#131	—	2	32	no	—
266	C₂×M₁₆	—	2	24	no	—
267	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
268	C₂×M₁₆	C₂³×C₆₇	1	12	yes	—
269	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
270	C₄×Q₈	—	2	8	no	—
271	C₄×D₄	—	2	24	no	—
272	#117	—	2	6	no	—
273	#117	C₂³×C₆₇	1	12	yes	—
274	C₂×C₄⋊C₄	—	2	3	no	—
275	C₂×C₄⋊C₄	—	2	6	no	—
276	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₂×C₂²⋊C₄	—	2	3	no	—
280	C₂×C₄²	—	2	3	no	—
281	#103	C₂³×C₆₇	1	24	yes	—
282	#103	C₂³×C₆₇	1	8	yes	—
283	#102	C₂³×C₆₇	1	24	yes	—
284	#102	C₂³×C₆₇	1	8	yes	—
285	C₂⁴	—	2	6	no	—
286	C₂⁴	C₂×D₄×C₆₇	1	3	yes	—
287	#41	—	2	32	no	—
288	#41	—	2	96	no	—
289	#41	—	2	48	no	—
290	#41	—	2	48	no	—
291	#41	—	2	8	no	—
292	#41	—	2	24	no	—
293	#41	C₂²⋊C₄×C₆₇	1	32	yes	—
294	#41	C₂²×C₄×C₆₇	1	16	yes	—
295	#41	C₂²×C₄×C₆₇	1	16	yes	—
296	#41	C₂²×C₄×C₆₇	1	48	yes	complemented
297	#41	C₂×D₄×C₆₇	1	16	yes	—
298	C₂×Q₈	—	2	32	no	—
299	C₂×Q₈	—	2	6	no	—
300	C₂×Q₈	—	2	12	no	—
301	C₂×Q₈	C₂²⋊C₄×C₆₇	1	8	yes	—
302	C₂×Q₈	C₂²×C₄×C₆₇	1	24	yes	complemented
303	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
304	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
305	C₂×D₄	—	2	48	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	12	no	—
308	C₂×D₄	—	2	96	no	—
309	C₂×D₄	—	2	6	no	—
310	C₂×D₄	—	2	12	no	—
311	C₂×D₄	C₂²⋊C₄×C₆₇	1	24	yes	—
312	C₂×D₄	C₂²×C₄×C₆₇	1	24	yes	complemented
313	C₂×D₄	C₂×D₄×C₆₇	1	12	yes	—
314	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
315	C₂²×C₄	—	2	6	no	—
316	C₂²×C₄	—	2	48	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	12	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	3	no	—
321	C₂²×C₄	—	2	12	no	—
322	C₂²×C₄	—	8	6	no	—
323	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
326	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
329	M₁₆	—	2	8	no	—
330	M₁₆	—	2	24	no	—
331	C₂×C₈	—	2	24	no	—
332	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
333	C₄⋊C₄	—	2	3	no	—
334	C₄⋊C₄	—	2	12	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	C₂⁴×C₆₇	1	6	yes	—
337	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
338	C₂²⋊C₄	—	2	12	no	—
339	C₄²	—	2	12	no	—
340	C₂³	—	2	4	no	—
341	C₂³	—	2	24	no	—
342	C₂³	—	4	3	no	—
343	C₂³	—	4	3	no	—
344	C₂³	—	8	6	no	—
345	C₂³	—	8	6	no	—
346	C₂³	C₂×C₂²⋊C₄×C₆₇	1	12	yes	—
347	C₂³	C₂²×D₄×C₆₇	1	6	yes	—
348	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
349	Q₈	—	2	24	no	—
350	Q₈	—	2	4	no	—
351	Q₈	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
352	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
353	D₄	—	2	16	no	—
354	D₄	—	2	48	no	—
355	D₄	—	2	24	no	—
356	D₄	—	2	4	no	—
357	D₄	C₂×C₂²⋊C₄×C₆₇	1	48	yes	—
358	D₄	C₂³×C₄×C₆₇	1	8	yes	—
359	C₂×C₄	—	2	24	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	16	no	—
362	C₂×C₄	—	2	48	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	6	no	—
365	C₂×C₄	—	4	6	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	12	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	24	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	12	yes	—
370	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
371	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
372	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
373	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
375	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
376	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
377	C₈	—	2	8	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	6	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	6	no	—
382	C₂²	—	2	8	no	—
383	C₂²	—	4	12	no	—
384	C₂²	—	4	4	no	—
385	C₂²	—	4	4	no	—
386	C₂²	—	8	24	no	—
387	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	12	yes	—
388	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
389	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
390	C₄	—	2	8	no	—
391	C₄	—	4	6	no	—
392	C₄	—	4	2	no	—
393	C₄	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
394	C₄	C₂²×C₂²⋊C₄×C₆₇	1	12	yes	—
395	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
396	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
397	C₂	—	2	2	no	—
398	C₂	—	2	12	no	—
399	C₂	—	4	8	no	—
400	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
401	C₂	C₆₇×#3585	1	1	yes	characteristic
402	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#585×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#568×C₆₇	C₂	semidirect
3. C₂×#520×C₆₇	C₂	semidirect
4. C₆₇×#2898	C₂	semidirect
5. C₆₇×#2897	C₂	semidirect
6. C₆₇×#2896	C₂	semidirect
7. C₂×#417×C₆₇	C₂	semidirect
9. #585×C₆₇	C₄	semidirect
18. C₂×#131×C₆₇	C₂²	semidirect
19. C₂²×M₁₆×C₆₇	C₂²	semidirect
21. #520×C₆₇	C₂²	semidirect
24. C₂×#117×C₆₇	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
26. #419×C₆₇	C₂²	semidirect
27. #418×C₆₇	C₂²	semidirect
28. #417×C₆₇	C₂²	semidirect
29. #417×C₆₇	C₂²	semidirect
30. #417×C₆₇	C₂²	semidirect
31. #416×C₆₇	C₂²	semidirect
32. C₂×#103×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
36. 2₋¹⁺⁴×C₆₇	C₂×C₄	semidirect
39. H(5,𝔽₂)×C₆₇	C₂×C₄	semidirect
47. C₂×#41×C₆₇	C₂×C₄	semidirect
94. #41×C₆₇	C₂²×C₄	semidirect
100. C₂×Q₈×C₆₇	C₂²×C₄	semidirect
110. C₂×D₄×C₆₇	C₂²×C₄	semidirect
195. #60089	C₆₇	direct (N × H)
203. C₂×#568	C₂×C₆₇	semidirect
204. C₂×#520	C₂×C₆₇	semidirect
205. #2898	C₂×C₆₇	semidirect
206. #2897	C₂×C₆₇	semidirect
207. #2896	C₂×C₆₇	semidirect
208. C₂×#417	C₂×C₆₇	semidirect
209. C₆₇	#60089	direct (N × H)
211. #585	C₄×C₆₇	semidirect
220. C₂×#131	C₂²×C₆₇	semidirect
221. C₂²×M₁₆	C₂²×C₆₇	semidirect
223. #520	C₂²×C₆₇	semidirect
226. C₂×#117	C₂²×C₆₇	semidirect
227. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
228. #419	C₂²×C₆₇	semidirect
229. #418	C₂²×C₆₇	semidirect
230. #417	C₂²×C₆₇	semidirect
231. #417	C₂²×C₆₇	semidirect
232. #417	C₂²×C₆₇	semidirect
233. #416	C₂²×C₆₇	semidirect
234. C₂×#103	C₂²×C₆₇	semidirect
235. C₂×#102	C₂²×C₆₇	semidirect
238. 2₋¹⁺⁴	C₂×C₄×C₆₇	semidirect
241. H(5,𝔽₂)	C₂×C₄×C₆₇	semidirect
249. C₂×#41	C₂×C₄×C₆₇	semidirect
296. #41	C₂²×C₄×C₆₇	semidirect
302. C₂×Q₈	C₂²×C₄×C₆₇	semidirect
312. C₂×D₄	C₂²×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#585×C₆₇	2
C₂×#568×C₆₇	2
C₂×#520×C₆₇	2
C₆₇×#2898	2
C₆₇×#2897	2
C₆₇×#2896	2
C₂×#417×C₆₇	2
#60089	67

Subgroups — C67×#60089 #30739340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60089 #30739340