Subgroups — C₂×C₆₇×#2378 #30713592

Order 17152.

2566 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2378	C₂	1	16	yes	maximal, direct product factor
3	C₂×#391×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#389×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#388×C₆₇	C₂	1	2	yes	maximal
6	C₂×#388×C₆₇	C₂	1	2	yes	maximal
7	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	C₄	1	16	yes	complemented
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
26	#391×C₆₇	C₂²	1	16	yes	complemented
27	#389×C₆₇	C₂²	1	16	yes	complemented
28	#388×C₆₇	C₂²	1	16	yes	—
29	#388×C₆₇	C₂²	1	16	yes	—
30	#387×C₆₇	C₂²	1	16	yes	complemented
31	#387×C₆₇	C₂²	1	16	yes	complemented
32	#381×C₆₇	C₂²	1	8	yes	—
33	#380×C₆₇	C₂²	1	8	yes	—
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
35	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#95×C₆₇	C₂²	1	2	yes	—
39	C₂×#95×C₆₇	C₂²	1	2	yes	—
40	C₂×#95×C₆₇	C₂²	1	2	yes	—
41	C₂×#95×C₆₇	C₂²	1	2	yes	—
42	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	—	4	4	no	—
44	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₆₇	—	4	4	no	—
47	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
57	C₂²⋊Q₈×C₆₇	C₂×C₄	1	64	yes	complemented
58	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
60	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	16	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
76	C₂×C₄²×C₆₇	—	4	4	no	—
77	C₂×C₄²×C₆₇	—	4	4	no	—
78	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
80	#95×C₆₇	C₂³	1	8	yes	—
81	#95×C₆₇	C₂³	1	4	yes	—
82	#95×C₆₇	C₂³	1	8	yes	—
83	#95×C₆₇	C₂³	1	4	yes	—
84	#95×C₆₇	C₂³	1	8	yes	—
85	#95×C₆₇	C₂³	1	8	yes	—
86	#95×C₆₇	C₂³	1	8	yes	—
87	#95×C₆₇	C₂³	1	8	yes	—
88	C₂⁴×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
94	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	4	16	no	—
96	C₂×Q₈×C₆₇	—	4	8	no	—
97	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	4	16	no	—
117	C₂²×C₄×C₆₇	—	4	8	no	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
119	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
131	C₄⋊C₄×C₆₇	—	4	8	no	—
132	C₄⋊C₄×C₆₇	—	4	8	no	—
133	C₄⋊C₄×C₆₇	—	4	8	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
136	C₂²⋊C₄×C₆₇	—	4	4	no	—
137	C₂²⋊C₄×C₆₇	—	4	8	no	—
138	C₂²⋊C₄×C₆₇	—	4	4	no	—
139	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
140	C₄²×C₆₇	—	4	8	no	—
141	C₄²×C₆₇	—	4	8	no	—
142	C₂³×C₆₇	—	2	8	no	—
143	C₂³×C₆₇	—	2	8	no	—
144	C₂³×C₆₇	—	2	8	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	4	8	no	—
150	C₂³×C₆₇	—	4	8	no	—
151	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
152	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
153	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
154	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
155	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
156	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
157	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
158	Q₈×C₆₇	—	4	16	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	16	no	—
162	C₂×C₄×C₆₇	—	2	16	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	4	16	no	—
182	C₂×C₄×C₆₇	—	4	8	no	—
183	C₂×C₄×C₆₇	—	4	8	no	—
184	C₂×C₄×C₆₇	—	4	8	no	—
185	C₂×C₄×C₆₇	—	4	8	no	—
186	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
187	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
188	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
189	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
190	C₂²×C₆₇	—	2	4	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	4	16	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	#518	1	4	yes	—
197	C₂²×C₆₇	#519	1	4	yes	—
198	C₂²×C₆₇	#520	1	4	yes	—
199	C₂²×C₆₇	#562	1	8	yes	—
200	C₂²×C₆₇	#563	1	8	yes	—
201	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
202	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
203	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
205	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
206	C₄×C₆₇	—	2	4	no	—
207	C₄×C₆₇	—	2	4	no	—
208	C₄×C₆₇	—	2	8	no	—
209	C₄×C₆₇	—	2	8	no	—
210	C₄×C₆₇	—	4	4	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₂×#2378	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₆₇	—	4	4	no	—
216	C₂×C₆₇	#2378	1	8	yes	direct product factor
217	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
218	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#562	1	2	yes	—
221	C₂×C₆₇	C₂×#563	1	2	yes	—
222	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
223	#2378	C₂×C₆₇	1	16	yes	direct product factor
224	C₂×#391	C₂×C₆₇	1	2	yes	complemented
225	C₂×#389	C₂×C₆₇	1	2	yes	complemented
226	C₂×#388	C₂×C₆₇	1	2	yes	—
227	C₂×#388	C₂×C₆₇	1	2	yes	—
228	C₂×#387	C₂×C₆₇	1	2	yes	complemented
229	C₂×#387	C₂×C₆₇	1	2	yes	complemented
230	C₂×#381	C₂×C₆₇	1	1	yes	characteristic
231	C₂×#380	C₂×C₆₇	1	1	yes	characteristic
232	C₆₇	C₂×#2378	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₂²⋊Q₈	C₄×C₆₇	1	16	yes	complemented
236	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	—
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
238	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
239	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
240	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
241	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
246	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
247	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
248	#391	C₂²×C₆₇	1	16	yes	complemented
249	#389	C₂²×C₆₇	1	16	yes	complemented
250	#388	C₂²×C₆₇	1	16	yes	—
251	#388	C₂²×C₆₇	1	16	yes	—
252	#387	C₂²×C₆₇	1	16	yes	complemented
253	#387	C₂²×C₆₇	1	16	yes	complemented
254	#381	C₂²×C₆₇	1	8	yes	—
255	#380	C₂²×C₆₇	1	8	yes	—
256	C₂×#95	C₂²×C₆₇	1	2	yes	—
257	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#95	C₂²×C₆₇	1	2	yes	—
259	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#95	C₂²×C₆₇	1	2	yes	—
261	C₂×#95	C₂²×C₆₇	1	2	yes	—
262	C₂×#95	C₂²×C₆₇	1	2	yes	—
263	C₂×#95	C₂²×C₆₇	1	2	yes	—
264	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
265	C₂²×Q₈	—	4	4	no	—
266	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
267	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
268	C₂³×C₄	—	4	4	no	—
269	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
271	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
272	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
273	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
274	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
275	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
276	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
277	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
279	C₂²⋊Q₈	C₂×C₄×C₆₇	1	64	yes	complemented
280	C₂×C₄⋊C₄	—	4	4	no	—
281	C₂×C₄⋊C₄	—	4	4	no	—
282	C₂×C₄⋊C₄	—	4	4	no	—
283	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
284	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	16	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
290	C₂×C₂²⋊C₄	—	4	2	no	—
291	C₂×C₂²⋊C₄	—	4	4	no	—
292	C₂×C₂²⋊C₄	—	4	2	no	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
295	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
296	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
298	C₂×C₄²	—	4	4	no	—
299	C₂×C₄²	—	4	4	no	—
300	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
301	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
302	#95	C₂³×C₆₇	1	8	yes	—
303	#95	C₂³×C₆₇	1	4	yes	—
304	#95	C₂³×C₆₇	1	8	yes	—
305	#95	C₂³×C₆₇	1	4	yes	—
306	#95	C₂³×C₆₇	1	8	yes	—
307	#95	C₂³×C₆₇	1	8	yes	—
308	#95	C₂³×C₆₇	1	8	yes	—
309	#95	C₂³×C₆₇	1	8	yes	—
310	C₂⁴	—	2	2	no	—
311	C₂⁴	—	2	4	no	—
312	C₂⁴	—	2	4	no	—
313	C₂⁴	—	2	2	no	—
314	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
315	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
316	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
317	C₂×Q₈	—	4	16	no	—
318	C₂×Q₈	—	4	8	no	—
319	C₂×Q₈	C₂²×C₄×C₆₇	1	16	yes	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	4	16	no	—
339	C₂²×C₄	—	4	8	no	—
340	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
341	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
342	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
343	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
344	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
345	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
347	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
351	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
353	C₄⋊C₄	—	4	8	no	—
354	C₄⋊C₄	—	4	8	no	—
355	C₄⋊C₄	—	4	8	no	—
356	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
357	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
358	C₂²⋊C₄	—	4	4	no	—
359	C₂²⋊C₄	—	4	8	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
362	C₄²	—	4	8	no	—
363	C₄²	—	4	8	no	—
364	C₂³	—	2	8	no	—
365	C₂³	—	2	8	no	—
366	C₂³	—	2	8	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	4	8	no	—
372	C₂³	—	4	8	no	—
373	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
374	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
375	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
376	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
377	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
378	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
379	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
380	Q₈	—	4	16	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	16	no	—
384	C₂×C₄	—	2	16	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	4	16	no	—
404	C₂×C₄	—	4	8	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	8	no	—
408	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
409	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
410	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
411	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	16	no	—
418	C₂²	#518×C₆₇	1	4	yes	—
419	C₂²	#519×C₆₇	1	4	yes	—
420	C₂²	#520×C₆₇	1	4	yes	—
421	C₂²	#562×C₆₇	1	8	yes	—
422	C₂²	#563×C₆₇	1	8	yes	—
423	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
424	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
425	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
426	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
427	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
428	C₄	—	2	4	no	—
429	C₄	—	2	4	no	—
430	C₄	—	2	8	no	—
431	C₄	—	2	8	no	—
432	C₄	—	4	4	no	—
433	C₄	—	4	4	no	—
434	C₄	—	4	4	no	—
435	C₄	—	4	4	no	—
436	C₂	—	4	4	no	—
437	C₂	C₆₇×#2378	1	8	yes	direct product factor
438	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
439	C₂	C₂×#519×C₆₇	1	1	yes	characteristic
440	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
441	C₂	C₂×#562×C₆₇	1	2	yes	—
442	C₂	C₂×#563×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2378	C₂	direct (N × H)
3. C₂×#391×C₆₇	C₂	semidirect
4. C₂×#389×C₆₇	C₂	semidirect
7. C₂×#387×C₆₇	C₂	semidirect
8. C₂×#387×C₆₇	C₂	semidirect
13. C₂×C₂²⋊Q₈×C₆₇	C₄	semidirect
26. #391×C₆₇	C₂²	semidirect
27. #389×C₆₇	C₂²	semidirect
30. #387×C₆₇	C₂²	semidirect
31. #387×C₆₇	C₂²	semidirect
57. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
214. C₂×#2378	C₆₇	direct (N × H)
216. C₂×C₆₇	#2378	direct (N × H)
223. #2378	C₂×C₆₇	direct (N × H)
224. C₂×#391	C₂×C₆₇	semidirect
225. C₂×#389	C₂×C₆₇	semidirect
228. C₂×#387	C₂×C₆₇	semidirect
229. C₂×#387	C₂×C₆₇	semidirect
232. C₆₇	C₂×#2378	direct (N × H)
235. C₂×C₂²⋊Q₈	C₄×C₆₇	semidirect
248. #391	C₂²×C₆₇	semidirect
249. #389	C₂²×C₆₇	semidirect
252. #387	C₂²×C₆₇	semidirect
253. #387	C₂²×C₆₇	semidirect
279. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
437. C₂	C₆₇×#2378	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×C₂²⋊Q₈×C₆₇	2
C₆₇×#2378	2
C₂×#391×C₆₇	2
C₂×#389×C₆₇	2
C₂×#388×C₆₇	2
C₂×#387×C₆₇	2
C₂×#381×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2378	67

Subgroups — C2×C67×#2378 #30713592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2378 #30713592